Fundamentals of Fluid Flow in Porous Media v1!2!2014!11!11.Pdf0

FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA Apostolos Kantzas, PhD P. Eng. Jonathan Bryan, PhD, P. Eng. Saeed Taheri, PhD

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CHAPTER 1........................................................................................................................................... 11 INTRODUCTION ....................................................................................................................................... 11

2.

CHAPTER 2........................................................................................................................................... 15 THE POROUS MEDIUM............................................................................................................................ 15 HOMOGENEITY ................................................................................................................................... 16 ANISOTROPY ....................................................................................................................................... 18 POROSITY ............................................................................................................................................ 18 PORE SIZE DISTRIBUTION .................................................................................................................... 28 SPECIFIC SURFACE AREA ..................................................................................................................... 32 COMPRESSIBILITY OF POROUS ROCKS ................................................................................................ 33 PERMEABILITY ..................................................................................................................................... 37 SATURATION ....................................................................................................................................... 50 FORMATION RESISTIVITY FACTOR ...................................................................................................... 53 MULTI-PHASE SATURATED ROCK PROPERTIES ................................................................................... 60 RELATIVE PERMEABILITY ................................................................................................................... 100

3.

CHAPTER 3......................................................................................................................................... 171 MOLECULAR DIFFUSION ....................................................................................................................... 171 Introduction ...................................................................................................................................... 171 Fick’s Law of Binary Diffusion ........................................................................................................... 171 Diffusion Coefficient ......................................................................................................................... 174

4.

CHAPTER 4......................................................................................................................................... 204 Immiscible Displacement ...................................................................................................................... 204 Introduction ...................................................................................................................................... 204 Buckley-Leverett Theory, .................................................................................................................. 204 Water Injection Oil Recovery Calculations........................................................................................ 215 Vertical and Volumetric Sweep Efficiencies...................................................................................... 230

5.

CHAPTER 5......................................................................................................................................... 242 Miscible Displacement .......................................................................................................................... 242 Introduction ...................................................................................................................................... 242 FLUID PHASE BEHAVIOR.................................................................................................................... 245 First Contact Miscibility Process ....................................................................................................... 255

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Multiple Contact Miscibility Processes ............................................................................................. 257 Determination of Miscibility Condition............................................................................................. 267 Fluid properties in miscible displacement ........................................................................................ 277 The Equation of Continuity ............................................................................................................... 305 The Equation of Continuity in Porous Media .................................................................................... 306

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Figure ‎1-1: World supply of primary energy by fuel type. .......................................................................... 11 Figure ‎1-2: Simplified Illustrations of Vertical and Horizontal Wells .......................................................... 12 Figure ‎1-3: Elementary Trap in Sectional View ........................................................................................... 14 Figure ‎2-1: Dependence of Permeability on Sample Volume ..................................................................... 16 Figure ‎2-2 –A Probability Density Function can be used to find the homogeneity or heterogeneity type of a porous medium ........................................................................................................................................ 17 Figure ‎2-3 – Microscopic Cross Section Image of a Porous Medium .......................................................... 18 Figure ‎2-4 – Dead-end pore ........................................................................................................................ 20 Figure ‎2-5 - Storage and connecting pore model for shale or any other type of rock with interconnected pore systems ............................................................................................................................................... 20 Figure ‎2-6 - The Various Cross-Sections of Connecting Pores .................................................................... 21 Figure ‎2-7 – a) Three-dimension distribution of connecting; b) Complete storage-connecting pore system. ........................................................................................................................................................ 21 Figure ‎2-8 – Typical ordered porous medium structures ........................................................................... 23 Figure ‎2-9 – Effect of sorting and grain size distribution on porosity......................................................... 24 Figure ‎2-10 – Porosimeter Based on Boyle - Mariotte’s Law ..................................................................... 27 Figure ‎2-11 – Colored Thin Section Microscopic Image.............................................................................. 27 Figure ‎2-12 - Schematic Shape of pore and Pore Throat ............................................................................ 29 Figure ‎2-13 – Sieve Analysis Tools .............................................................................................................. 29 Figure ‎2-14 - Schematic representation of pores. ...................................................................................... 31 Figure ‎2-15 – Tomographic image. ............................................................................................................. 32 Figure ‎2-16 – Specific Surface Area............................................................................................................. 32 Figure ‎2-17 – Porosity Reduction as an Effect of Compaction Increment by Depth .................................. 34 Figure ‎2-18 – a) Experimental Equipment for Measuring Pore Volume Compaction and Compressibility 36 Figure ‎2-19 – a) Formation Compaction Component of Total Rock Compressibility ................................. 36 Figure ‎2-20 – Flow through a Pipe .............................................................................................................. 38 Figure ‎2-21 – Metallic cast of pore space in a consolidated sand .............................................................. 39 Figure ‎2-22 – Schematic Drawing of Darcy Experiment of Flow of Water through Sand ........................... 39 Figure ‎2-23 – Linear Flow through Layered Bed ......................................................................................... 42 Figure ‎2-24 – Linear Flow through Series Beds........................................................................................... 44 Figure ‎2-25 – Plot of Experimental Results for Calculation of Permeability ............................................... 46 Figure ‎2-26 – Permeability of Core Sample to Three Different Gases and Different Mean Pressure ........ 48 Figure ‎2-27 - Effect of Permeability on the Magnitude of the Klinkenberg Effect ..................................... 48 Figure ‎2-28 – ASTM Extraction Apparatus .................................................................................................. 53 Figure ‎2-29 - The Influence of Pore Structure on the Electrical Conductivity ............................................ 55 Figure ‎2-30 – Formation Resistivity Factor vs. Porosity .............................................................................. 56 Figure ‎2-31 - Apparent Formation Factor vs. Water Resistivity for Clayey and Clean Sands ..................... 57 Figure ‎2-32 - Water-Saturated Rock Conductivity as a Function of Water Conductivity ........................... 58 Figure ‎2-33- Illustration of surface tension (Surface molecules pulled toward liquid causes tension in surface). ...................................................................................................................................................... 60 Figure ‎2-34- Simplified Models for Interfacial Tension Determination. ..................................................... 62 Figure ‎2-35- Pressure relations in capillary tubes....................................................................................... 62 3

Figure ‎2-36- Illustration of Wettability ....................................................................................................... 63 Figure ‎2-37- Equilibrium of Forces at a Liquid-Gas-Solid Interface. ........................................................... 64 Figure ‎2-38- Rock-Fluid-Fluid Interactions Effect on the Contact Angle..................................................... 66 Figure ‎2-39- liquid drop spreading on a solid surface ................................................................................ 67 Figure ‎2-40- Amott Wettability Test ........................................................................................................... 68 Figure ‎2-41- Amott Index Ternary Diagram ................................................................................................ 69 Figure ‎2-41- Amott Index Calculation ......................................................................................................... 69 Figure ‎2-42- USBM Index Calculation ......................................................................................................... 70 Figure ‎2-43- Pressure Relation in capillary Tube ........................................................................................ 72 Figure ‎2-44- Dependency of Water Column to (a). Capillary Radius, (b). Wettability................................ 73 Figure ‎2-45- Principle radii for wetting fluid and spherical grain ............................................................... 74 Figure ‎2-46- Wetting and non-Wetting fluid distribution about inter grain contact of sphere. ................ 75 Figure ‎2-47- Non-Wetting fluid entering the capillary tube. ...................................................................... 76 Figure ‎2-48- Non-Wetting fluid entering the non-uniform capillary tube. ................................................. 76 Figure ‎2-49- Non-Wetting Fluid enter to a bubble and exit it. ................................................................... 77 Figure ‎2-50- Capillary Pressure versus wetting phase saturation .............................................................. 78 Figure ‎2-51- Variation of Capillary Pressure with Permeability .................................................................. 79 Figure ‎2-52- Flow into a Constriction (cone). ............................................................................................. 79 Figure ‎2-53- Flow out of a Constriction ...................................................................................................... 80 Figure ‎2-54- Flow in a Capillary Tube. ......................................................................................................... 81 Figure ‎2-55- porous diaphragm capillary pressure device.......................................................................... 83 Figure ‎2-56- Centrifugal apparatus ............................................................................................................. 84 Figure ‎2-57- Mercury Injection Method ..................................................................................................... 85 Figure ‎2-58- Pore size distribution from mercury injection test................................................................. 86 Figure ‎2-59- Dynamic Measurement of Capillary pressure. ....................................................................... 87 Figure ‎2-60- Capillary Pressure Curve. ........................................................................................................ 88 Figure ‎2-61- Contact Angle Hysteresis during the Displacement ............................................................... 89 Figure ‎2-62- Dynamic Contact Angle Behavior. .......................................................................................... 89 Figure ‎2.63: Static values of advancing and receding contact angles at rough surfaces versus values at smooth surfaces (where E refers to smooth surface measurements........................................................ 90 Figure ‎2-64- Non-Wetting fluid Enter to a capillary tube with square cross section. ................................ 91 Figure ‎2-65- Side view after snap-off .......................................................................................................... 91 Figure ‎2-66- Trapping in a porous media. ................................................................................................... 92 Figure ‎2-67- Typical non-wetting phase trapping characteristics of some reservoir rocks. ....................... 92 Figure ‎2-68- Pore Doublet Model. .............................................................................................................. 93 Figure ‎2-69- Imbibition and Drainage mechanisms in a pore doublet model ............................................ 94 Figure ‎2-70- Pore doublet model for illustration for displacement and trapping of oil. ............................ 95 Figure ‎2-71- Trapping of a droplet in a capillary tube. ............................................................................... 97 Figure ‎2-72- J-function correlation of capillary pressure data in Edwards Jourdanton field...................... 99 Figure ‎2-73- Typical relative permeability curve. ..................................................................................... 103 Figure ‎2-74- Typical Gas-Oil Relative Permeability Curve......................................................................... 105 Figure ‎2-75- Relative Permeability Curve, (a) Drainage, (b) Imbibition .................................................... 105 4

Figure ‎2-76- Hafford Relative Permeability Apparatus ............................................................................. 107 Figure ‎2-77- Fluid Saturation during Steady-State Test ............................................................................ 108 Figure ‎2-78- Unsteady state apparatus. ................................................................................................... 109 Figure ‎2-79- (a) Unsteady State Water Flood Procedure, (b) Typical Relative Permeability Curve ......... 110 Figure ‎2-80- (a) Average Water saturation vs. Water Injection, (b) Injectivity Ratio ............................... 111 Figure ‎2-81: Amott Ternary Wettability Diagram ..................................................................................... 119 Figure ‎2-82: Comparison between capillary pressure and relative permeability curves ......................... 122 Figure ‎2-83 Relative permeability curves of Berea Sandstone. Strongly Water-wet conditions (Sankar 1979) ......................................................................................................................................................... 125 Figure ‎2-84 Relative permeability of water wet and oil wet systems. ..................................................... 134 Figure ‎2-85 Comparison between experimental and predicted values (after 3). .................................... 140 Figure ‎2-86 Compare with results of Corey et al. (after 11) ..................................................................... 145 Figure ‎2-87 Compare with results of Dalton et al. (after 11).................................................................... 145 Figure ‎2-88 Stone’s Method 2 predictions and Corey et al.’s experimental data (after 12) ................... 146 Figure ‎2-89 Stone’s prediction of Sor and Holmgren-Morse’s data (after 12) ........................................ 146 Figure ‎2-90 Effect of capillary number on relative permeability (after 9) ................................................ 153 Figure ‎2-91 Low IFT systems (after 21) ..................................................................................................... 155 Figure ‎2-92 High IFT systems (after 21) .................................................................................................... 155 Figure ‎2-93 Effects of viscous force on Swir (after 22). .............................................................................. 156 Figure ‎2-94 Effects of flow rate on relative permeability (after 22) ......................................................... 158 Figure ‎2-95 Effect of viscosity on relative permeability (after 26) ........................................................... 160 Figure ‎3-1 - Simple diffusion experiment.................................................................................................. 172 Figure ‎3-2 – Diffusion across a thin film ................................................................................................... 173 Figure ‎3-3 – Prediction overall diffusion from intrinsic diffusion ............................................................. 177 Figure ‎3-4 – Diffusion process in a control volume with a concentration dependent diffusion coefficient .................................................................................................................................................................. 178 Figure ‎3-5 – In a porous medium fluid generally flowing at about 45o with respect to average direction of flow ....................................................................................................................................................... 180 Figure ‎3-6 – Pressure decay test cell. ....................................................................................................... 182 Figure ‎3-7 - Refraction of light at the interface between two media. ...................................................... 183 Figure ‎3-8 - Sample of light refraction results a) initial time b) after diffusion occurred ......................... 184 Figure ‎3-9 - (a). Hydrogen nuclei behave as a tiny bar magnets aligned with the spin axes of the nuclei. (b). Spinning protons with random nuclear magnetic axes in the absence of an external magnetic field. .................................................................................................................................................................. 185 Figure ‎3-10 – Line up nuclear spins in an external magnetic field............................................................ 186 Figure ‎3-11 – Polarization/Relaxation curve. ............................................................................................ 187 Figure ‎3-12 – the Tipping process. ............................................................................................................ 187 Figure ‎3-13 – Net magnetization return to equilibrium by turning off the B1, (the arrow represent net magnetization) .......................................................................................................................................... 188 Figure ‎3-14 – de-phasing (loss of phase coherence) during T2. ............................................................... 189 Figure ‎3-15 – Spin-echo sequence. ........................................................................................................... 190 Figure ‎3-16 – CPMG pulse sequence. ....................................................................................................... 191 5

Figure ‎3-17 - The amplitudes of the decaying spin echoes yield an exponentially decaying curve with time constant T2. ....................................................................................................................................... 191 Figure ‎3-18 - The echo train (echo amplitude as a function of time) is mapped to a T 2 distribution (porosity as a function of T2). .................................................................................................................... 192 Figure ‎3-19 – Typical NMR spectrum for pure bitumen, pure solvent, and a mixture of them. .............. 194 Figure ‎3-20 – two samples of NMR calibration for bitumen-solvent mixture,. ........................................ 195 Figure ‎3-21 – Diffusion coefficient as a function time, NMR experiment result. ..................................... 197 Figure ‎3-22 - Schematic view of CAT scanning using x-ray. ...................................................................... 199 Figure ‎3-23 – Calibration curves for the CAT scanner, (a) Liquid calibration curve, (b)Liquid-solid calibration curve. ...................................................................................................................................... 199 Figure ‎3-24 – Image sample of diffusion process ..................................................................................... 200 Figure ‎3-25 – Medium Domain ................................................................................................................. 201 Figure ‎3-26 - Sample of diffusion in sand saturated with oil. ................................................................... 202 Figure ‎3-27 - Average diffusion coefficients for pentane, hexane and octane in heavy oil. .................... 202 Figure ‎3-28 - Comparison of the diffusion coefficients of pentane in heavy oil in absence/presence sand. .................................................................................................................................................................. 203 Figure ‎4-1 – Semilog plot of relative permeability ratio versus saturation .............................................. 205 Figure ‎4-2 – Fractional flow curve ............................................................................................................ 206 Figure ‎4-3 – Horizontal bed containing oil and water. ............................................................................. 207 Figure ‎4-4 – Cubic reservoir under active water drive.............................................................................. 208 Figure ‎4-5 – Water fractional flow ant its derivative ................................................................................ 209 Figure ‎4-6 – Fluid Distribution at 60, 120, 240 days ................................................................................. 210 Figure ‎4-7 - Water saturation distribution as a function of distance, prior to breakthrough .................. 211 Figure ‎4-8 - Tangent to the fractional flow curve from Sw = Swc ............................................................... 212 Figure ‎4-9 – xD vs. tD for a linear waterflooding. ....................................................................................... 213 Figure ‎4-10 – Saturation Profile at tD = 0.28 ............................................................................................. 214 Figure ‎4-11 – Saturation History at xD = 1, producing face of the medium .............................................. 214 Figure ‎4-12 - Saturation distribution after 240 days................................................................................. 215 Figure ‎4-13 - Application of the Welge graphical technique to determine: (a) The front saturation, (b) Oil recovery after breakthrough .................................................................................................................... 219 Figure ‎4-14 - Water saturation distribution as a function of distance between injection and production wells for (a) ideal or piston-like displacement and (b) non-ideal displacement ...................................... 219 Figure ‎4-15 - (a) Microscopic displacement (b) Residual oil remaining after a water flood .................... 221 ‎Figure ‎4-16 – (a) Relative Permeability Curves, (b) Fractional Flow Curve............................................... 223 Figure ‎4-17 – Graphical determination of front saturation and water fractional flow. ........................... 223 Figure ‎4-18 – ............................................................................................................................................. 224 Figure ‎4-19 – dimensionless pore volume oil recovery vs. dimensionless pore volume water injection 225 Figure ‎4-20 - Fractional flow plots for different oil-water viscosity ratios ............................................... 227 Figure ‎4-21 - Water Saturation Distributions in Systems for Different Oil/Water Viscosity Ratios ......... 228 Figure ‎4-22 - Typical injection/production well configurations and associated flooding patterns .......... 229 Figure ‎4-23 - Schematic representation of the two components of the volumetric sweep: (a) areal sweep; (b) vertical sweep in stratified formation. .................................................................................... 231 6

Figure ‎4-24 – a) Bottom coning at oil-water or gas-oil contact, b) Edge coning at oil-water or gas-oil contact ...................................................................................................................................................... 237 Figure ‎4-25 - Stable Cone. ......................................................................................................................... 237 Figure ‎4-26 – Flow rate versus time.......................................................................................................... 239 Figure ‎5-1 – Miscible Displacement, a) secondary recovery, b)Tertiary recovery. .................................. 243 Figure ‎5-2 - A typical phase diagram for a pure component. ................................................................... 245 Figure ‎5-3 – Typical P-T diagram for a multicomponent system. ............................................................. 246 Figure ‎5-4 – (a). Phase diagram of ethane-normal heptane, (b) Critical loci for binary mixtures. ........... 246 Figure ‎5-5 – Typical P-X diagram for the Methane-normal Butane system. ............................................ 248 Figure ‎5-6 – Pressure-composition diagram for mixture of C1 with a liquid mixture of C1-nC4-C10. ..... 249 Figure ‎5-7 – ternary phase diagram for a system consisting of components A, B, and C which are miscible in all proportions. ...................................................................................................................................... 251 Figure ‎5-8 – All mixture of M1 and M2 would be along line . ...................................................... 252 Figure ‎5-9 – Example ‎5-2 a) Ternary diagram, b) Chemicals to be mixed. ............................................... 252 Figure ‎5-10 - Error! Reference source not found...................................................................................... 253 Figure ‎5-11 – Ternary Phase Diagram. ...................................................................................................... 254 Figure ‎5-12 – Pressure effect on the miscibility, P1 0.5, krw at S*or is less than 0.3 (S*or is the waterflood residual oil saturation), The amount of connate water saturation, Swc, is usually greater than 20% (Swc  20 to 25% PV) and has an effect on relative permeability behaviour, (kro)primary drainage > (kro)imbibition for the same Sw, The krw curve exhibits no hysteresis (e.g., (krw)pd = (krw)imb), As permeability increases, (krw/kro) becomes higher for a given saturation.

Oil-wet system:      

krw = kro at Sw < 0.5, krw (S*or) > 0.5, Swc has no effect on relative permeability behaviour, if Swc < 20%, krw(Sw)|primary drainage > krw(Sw)|imbibition, kro curve exhibits no hysteresis, As permeability decreases, (krw/kro) becomes higher for a given water saturation.

For both oil and water-wet systems, the sum of (krw + kro) is always less than 1, when the same base permeability is used for normalization (i.e., (krw + kro) < 1). The main reasons for this are: 1) dendritic to flow pore space; 2) the connectivity of the network of each of the phases in the occupied pores is much smaller than that of the porous medium under single phase flow; and 3) pressure of immobile (trapped) fluids.

TWO PHASE RELATIVE PERMEABILITY LITERATURE SURVEY. Experimental Research and Relative Permeability Predictions In the last forty years, there have been many experimental investigations on relative permeability measurements. Shankar (1979) gives a detailed discussion on the experimental techniques that apply for these measurements. He also gives a review of the experimental work done and of some attempts to predict relative permeability up until the early seventies. McCaffery (1973) also gives a detailed discussion on the proposed methods of prediction of relative permeability characteristics up until that time.

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There are certain factors that influence the experimental measurements of permeability and the relative permeability characteristics in general. These factors are:    



The Boundary or End Effects The Hysteresis Effect The Migration or Partial Water Saturation Effect The Flow Rate Effect The Effect of Gas Expansion

as they were discussed by Geffen et al (1951)and Shankar (1979). In later research works, the effect of many parameters on relative permeability and/or effective permeability characteristics was examined. These parameters are:        

Wettability Capillary number Pore geometry Heterogeneity of the system Anisotropy of the system Fluid viscosity Interfacial tension (IFT) Temperature.

One parameter of major importance in relative permeability characteristics is wettability. McCaffery (1973) examined its effect on the relative permeability characteristics of porous media made from Teflon using different fluids in various wetting conditions. Shankar (1979) examined the effect of wettability in sandstones and he reported experimental results that cover a range of wetting conditions as well. Some results are shown in Figure 2-83. The effect of interfacial tension on relative oil/water permeabilities for the case of consolidated porous media has been investigated by Amaefule and Handy (1982). The samples used were fired Berea cores. They also tested temperature effects both in steady and unsteady state cases. They have found that oil/water relative permeabilities are greatly affected at IFT values less than 0.1 mN/m. They provide some empirical equations that fit their data. Keelan (1976) examined the gas-water relative permeability characteristics during imbibition. He reported plots of relative permeability as a function of saturation and its variations due to pore geometry. He correlated his results using the end-point relative permeability and the trapped gas saturation. Early studies by Johnson et al (1959) provided the first calculations of relative permeability based on data from displacement experiments. Honarpour et al (1982) report empirical equations for the estimation of two-phase relative permeability in consolidated rocks. Torabzadeh and Handy (1984) report data for relative permeability for high and low tension systems at temperatures ranging from 220C to 1750C. They state that temperature affects relative permeability especially at low IFTs and describe the effects of temperature at low and high IFTs. They also state that the relative permeability of water and oil will increase with decreasing IFT for constant temperature, while the rate effect is not important after a critical minimum point. 125

Figure 2-84 Relative permeability curves of Berea Sandstone. Strongly Water-wet conditions (Sankar 1979)

Yadav et al (1984A) (1984B) studied the distribution of two immiscible fluids in the pore space of sandstones. This distribution can give us information about the percentage of each pore that is occupied by each phase and therefore the microscopic information that is necessary for relative permeability studies at the pore level. Naar and Henderson (1961) correlated relative permeability characteristics during imbibition to the ones from drainage or drainage capillary curves for two-phase flow. A similar work for three-phase flow is given by Naar and Wydal (1961). Downie and Crane (1961) and Odeh (1959) examined the effect of viscosity on relative permeability. Although their results contradict each other, it is shown that viscosity must be taken into account when the relative permeability values are calculated. However, this value is a property of the medium and it is not a function of the fluid viscosity. Morgan and Gordon (1970) investigated the effect of pore geometry on water-oil relative permeability. They grouped porous media in categories according to the volume and the interconnectivity of the pores and they provide characteristic curves for each different case. 126

Morrow and McCaffery (1977) present results of experimental studies of the effect of wettability on relative permeability, capillary pressure and spontaneous imbibition. They classify the uniformly wetted systems in Wetted (contact angle O0-620), Intermediately-Wetted (contact angle 620-1330) and NonWetted (contact angle 133 0-1800). Sigmund and McCaffery (1979) introduced a nonlinear least square procedure to analyse the recovery and pressure response to two-phase laboratory displacement tests. They state that two-parameter relative permeability curves determined by a least squares one-dimensional simulation history match could describe quantitatively the dynamic two-phase flow characteristics of various rock types with differing degrees of heterogeneity. They determined the relative permeability characteristics in the form of two saturation exponent parameters. They made simulator calculations and found out that capillary forces significantly affect the pressure and recovery response data obtained from dynamic displacement test. Capillary forces are most likely to have influence on low rate displacements in the drainage direction. Batycky et al (1981 B) presented a procedure to test the results of the above work. They performed tests in both high-rate displacement and low-rate displacement and they found out that for low-rate displacement there has to be consideration of effects due to capillary forces and the "end-effects". These effects can extend over significant portions of the core in systems with strong wetting characteristics. They also found that there is high possibility of a significant error in the simulation of relative permeability curves if the capillary forces are neglected. Thomeer (1983) proposes a relationship to determine air permeability from the mercury/air capillary pressure data. Similar relationships are reported by Swanson (1981) for brine permeability. Bardon and Longeron (1980) studied the influence of IFTs, flow rates and viscosity ratios on Fontainebleau sandstone. They found out that if the lFT is greater than 0.04 mN/m, the relative permeability can be predicted through a simulation but, for very low IFTs, there are great changes in relative permeability values and the prediction is not satisfactory. The macroscopic anisotropy and the heterogeneities of a porous medium are a common phenomenon in oil reservoirs. The terms of directional, vertical and horizontal permeabilities appear in many papers that describe the effect of heterogeneity in permeability and reservoir behaviour. Warren and Price (1961), Parsons (1972), Rose (1982), and Aguilera (1982) (for the case of naturally fractured reservoirs), give descriptions and methods on the aspects that govern those systems. Another area of great interest is the one that deals with low permeability cores. Such cores can be provided from tight gas reservoirs, but the measurements must take into consideration the Klinkenberg effect as well as the effect of confining pressure. There have been quite a few papers in this area although no modeling has been attempted, as far as we know. Reports have been given by Thomas and Ward (1972), Jones and Owens (1980), Walls (1982) (who suggests that a network approach might be the most appropriate modelling approach for this kind of media), Kassemi (1982), Keighin and Sampath (1982) and Freeman and Bush (1983). Finally, Jones and Roszelle (1978) suggest that some graphical techniques can be used for simple and accurate determination, of relative permeability. 127

Relative Permeability Simulations During the past forty years, many investigators tried to simulate the capillary and transport phenomena in porous media. Many different methods were used. These methods are classified as macroscopic approaches or pore level (microscopic) approaches. Greenkom (1981) gives a review of the theoretical advances in the study of flow through porous media and discusses the phenomenological approaches for phenomena that occur in a porous medium under steady state conditions. He also provides an extensive list of references. According to Dullien (1979) and Chatzis (1980), the early models that deal with the porous media are classified as:

  

Phenomenological models (i.e. Carman-Kozeny's 'Hydraulic Radius Theory'), Geometrical models (i.e., serial type, parallel type, and serial-parallel type capillaric models), and Statistical models, "cutting and random rejoining models", (Haring and Greenkom (1970)).

Macroscopic approaches published include Slattery (1968), Casulli and Greenspan (1982), Gilman and Kassemi (1983) for the case of naturally fractured reservoirs, and Ramakrishnan and Wasan (1984) in connection with the effect of capillary number. The pore level or network model approaches are based on the pioneering work ofFatt (1956), who stated that a network of tubes is a valid model of porous media. He also was the first to state that the relative permeability characteristics of porous media are a direct consequence of the network structure of these media. Later, the network principles were combined with the aspects of percolation theory and gave the stochastic network models. This approach is the basis of the work of Chatzis (1976) (1980), Chatzis and Dullien (1978) (1982) (1982B), Larson et al (1981), Mohanty et al (1980), Koplik (1982), Heiba et al (1982) (1984), Mohanty and Salter (1982), Diaz (1984), and Kantzas (1985). The various models have many similarities but some crucial differences. Details on the above models are given below.

Theoretical Developments Pertinent to Relative Permeability Studies the Network Approach for Modelling Porous Media A porous medium can be represented as a random network'of pores. The construction of such a network must follow some theoretical considerations (e.g., porosity, pore to pore coordination number, pore shapes and pore sizes, etc.). Many of these aspects are based on the percolation theory and its applications to flowing porous media. Some introductory information about percolation theory concepts is given in the sections that follow. For more details on percolation theory and its applications in porous media, see Shante and Kirkpatrick (1971), Kirkpatrick (1973), Chatzis (1976), (1980), Larson (1977), and Larson et al (1981).

Percolation Theory There are many physical phenomena in which a fluid is "spreading" randomly in a medium. The terms "spreading", "fluid" and "medium" are not necessarily used in their strict sense. Except for the spreading itself, external causes such as gravity forces, for example, may control the process and affect the 128

random mechanism. The random mechanism may be ascribed to either the medium or the fluid, depending on the nature of the particular problem. Broadbent and Hammersley (1957) introduced the term "Percolation Process" for the process that ascribes the random mechanism to the medium. This term came to distinguish the above mathematical analysis from the ones that are confined to the random mechanism of a process generally ascribed to the fluid which are labeled as "diffusion processes". Percolation theory has been a very important tool in the theoretical development of the conductivity of random mixtures of conducting and non-conducting materials (Kirkpatrick 1973, Shante and Kirkpatrick 1971). In percolation theory a new terminology is introduced. The "medium" is defined to be an infinite set of abstract objects called "atoms", or "nodes", or "sites". A fluid flows from the source site along paths connecting different sites. These paths are called "bonds", and can be oriented or unoriented. A bond is defined as oriented when it permits the flow only in a specified direction. The fluid that flows along a bond will wet its two end points. The latter is used in porous media simulations. The coordination number, z, of a network of sites connected by bonds is defined to be the weighted average number of bonds leaving a node in a network (Chatzis 1980). The random mechanism can be assigned to the medium in two distinct ways, leading to two different percolation problems. The first is the bond percolation problem in which each bond has a constant probability of transmitting fluid and the bonds "transmitting" fluid are assigned at random everywhere in the network. This probability is independent of the existence of other bonds at the level of a site. The second is the site percolation problem in which a site. A, has a certain probability of allowing, fluid reaching A to flow on, along bonds leaving A. In this case^every bond between two open sites is to flow and the random mechanism is assigned to the sites. When a site is missing, all bonds connecting it to its neighbours are missing too. There are quite a few works that are based on either problem (e.g., Chatzis (1976), (1980)). In the field of porous media the conventional methods treated drainage as a bond problem and imbibition as a site problem. Chatzis (1980) used the site problem for drainage by taking the bonds into account. This is known as the bond correlated site problem in which the sites are assigned at random and the event of a bond being open is correlated with the event of two adjacent sites being open. Network models of pore structure with pore body sizes randomly distributed over the sites and the pore throat sizes-assigned according to a correlation scheme have been found to be sound models of simulating capillary pressure curves in sandstones (Chatzis and Dullien (1982), Diaz (1984)). Network models of pore structure obeying the bond percolation problem have been found to be unrealistic in simulating pore structure and flow behaviour (Chatzis (1980)). The most important finding of percolation theory is the critical percolation threshold. This is defined as the minimum fraction of bonds (bond percolation threshold, Pbc)' or tne minimum fraction of sites (site percolation threshold, Pgc), that must be present in the network so that the "medium" is conducting to flow. A background to the approach taken in applying percolation theory to random network models of pore structure at the University of Waterloo is given next.

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Network Modeling at Waterloo (Waterloo Network Model) WATNEMO is based on the work done by Chatzis (1976), (1980), the extensions given by Diaz (1984) and Kantzas (1985). A random network involves three key parameters: the average coordination number z, the pore body size distribution, and the pore throat size distribution. A 3-D random model is generated using the cubic lattice as a complete topological representation of an irregular network. The case of 2-D networks is not considered in this work since, as was pointed out by Chatzis (1976), 2-D networks cannot be used as physically sound models of real porous media. The main reason for this is that, in a 2-D network, it is impossible to have nodes and bonds fully occupied by one phase to form a continuum, while the other phase fully occupies nodes and bonds that a continuum as well (i.e., bicontinua cannot exist). Using the fact that the real network of pores in a porous medium is in 3-D, one can create many different types of 3-D networks characterized by different values of the coordination number. It has been found that it is very convenient to work-with regular networks that have a cubic lattice arrangement and a coordination number of six. This simplification introduces the "physical' assumption that the pores of the medium are well connected, and facilitates straightforward computer algorithms. Taking the above fact into account, the other parameters to be defined are the size distributions of pore throats, assigned to the bonds, and of the pore bodies assigned to the nodes of the network. The network models generated using bond correlated site percolation have been found to bemore suitable models of pore structure in sandstones (Chatzis (1980), Diaz (1984)). The principle of generating such networks is simple. The nodes of the network are assigned with indices 1,2 ....n at random. The nodes assigned the index 1 represent the largest pore bodies and the nodes assigned index n are the smallest pore bodies. A pseudo-random number generation technique can be used to provide a random setting of those indices in the sites of the network. The next step is to define the bonds by assigning an index equal to the larger index of the two nodes it connects. The accessibility properties of random network models using the bond correlated site percolation scheme have been investigated at Waterloo with special attention given to cubic networks (Chatzis (1980), Diaz (1984), Kantzas (1985)). Generalized number based accessibility functions have been obtained and applied to model capillary pressure saturation relationships for mercury-air systems and water-oil systems with entrapment. Having specified the number based pore body size distribution, the pore throat size distribution and pore geometry, there are algorithms available to convert the number based accessibility data to volume based capillary pressure curve data (Chatzis (1980), Chatzis and Dullien (1982), Diaz (1984), Kantzas (1985)). In addition to the simulation of capillary pressure curves, the WATNEMO has been applied successfully to model the "dendritic" non-wetting phase saturation during primary drainage in mercury-air experiments (Chatzis and Dullien (1982)) and relative permeability characteristics in mercurypermeametry experiments. The work of Diaz (1984) expanded the capability of WATNEMO to model fluid distributions of the non-wetting phase as well as the wetting phase, as a function of saturation and saturation history, while Kantzas (1985) used the same principles for the simulation of two phase relative permeabilities as described below.

130

The conductivity properties of random network models of interest to us for the simulation of relative permeability behaviour in oil-water systems have not been explored in detail. Results presented by Kirkpatrick (1973) for the cubic networks are limited by the fact that all bonds in the network allowed to conduct flow had the same conductivity value. There are no analytical methods for the calculation of the conductivity properties of random networks with size distribution of conductivities or with variable coordination numbers. Effective medium theory approximations (Kirkpatrick 1973, Koplik 1982) have been applied by Larson et al (1981 A), Heiba et al (1982) to two phase flow problems of relevance to relative permeability behaviour. Mohanty and Salter (1982) have also looked at the application of the conductivity properties of random networks to simulate relative permeability behaviour, but their simulation results were not in good agreement with experimental results. Moreover, the information provided in their paper is not sufficient to warrant a detailed criticism. In all of the past studies involved with the simulation of relating permeability characteristics, several tacit assumptions are made. These include: 1) The nodes in the network simulation have no resistance to fluid flow; 2) Only the bonds in the network carry the information of the resistance to flow in pore networks; and 3) With the exception of Chatzis and Dullien (1982), a node in the network can be simultaneously part of the non-wetting phase network as well as part of the wetting phase network (e.g., Fatt (1956C), Heiha et al (1992), Winterfeld et al (1981)). In addition to the above assumptions, several inconsistencies arise in transforming the relative conductivities of random networks into the form of relative permeability curves. For example, for the conductivity of a pore with diameter D, g(D) may be taken to be proportional to D cubed or D to the power of four, while the volume of such a pore V(D) is proportional to D to the power of 0.84 instead of V(D) proportional to D cubed or D squared as it should be (e.g., Heiba et al (1982), Soo and Slattery( 1983)). Part of this work is devoted to clarify some of these inherent inconsistencies in past studies. To do so, however, the capability of WATNEMO to model relative permeability behavior had to be developed. This was the main objective of Kantzas (1985) - the development of software for the investigation of the conductivity properties of random networks of the bond correlated site percolation type. The following sections form the basis for understanding the development of this software for simulating immiscible two-phase flow problems in porous media with applications to predictions of relative permeability behavior.

Conductivity and Permeability, the Main Algorithm The term conductivity is one of the most commonly used terms that appear in all the texts and studies that deal with flow problems. It describes the difficulty of flow through a certain medium, where the flow refers to momentum, energy, mass, electricity, etc. The most well known conductivity is the electrical conductivity, which is described by Ohm's law, (2-138) Where, 131

I is the electric current (in Amperes), V is the voltage drop (in Volts), G is the electricalconductivity of a resistor (in I/Ohm). Eq. (2-138) can be rewritten in terms of current density J and charge density F as follows: (

)

(2-139)

Thermal conductivity is also another well-known conductivity and it is defined by Fourier's law, (

)

(2-140)

Where, The heat flux, The thermal conductivity, ( )

The temperature gradient along the x direction.

In porous media, the conductivity to flow (or permeability) is given by Darcy's law. Darcy's law and equations (2-139) and (2-140)are obviously similar. Many researchers who have seen this similarity tried to give solutions for the permeability simulation problems using algorithms originally designed for the electrical conductivity analogs. This strategy is very reasonable. The same strategy is used in Kantzas (1985).

First, let us consider the case of calculating the overall conductivity of a network of resistors. This can be done by applying Kirchhoffs' laws ∑

(2-141)

at any node i, and for any closed loop of resistors: ∑

∑

(2-142)

where m is the number of bonds that form a closed loop. The second law applies in such cases where we have many sources of electricity within a loop. If we consider a network that contains only resistors then application of the first law can give ∑

∑

(2-143)

This says that we have one equation for every node in the network. A group of equations can be formed that describes the whole network, such that: 132

[ ][ ]

[ ]

(2-144)

Where, [ ]

The conductivity matrix,

[ ]

The voltage drop vector,

[ ]

The current vector.

This way of approaching the problem is different from Dodds' approach (1968) that used the resistances instead of the conductivities resulting in the inverse set of equations: [ ][ ]

[ ]

(2-145)

where R is the resistance matrix. For the optimum numerical solution, eq. (2-145) is preferred because of less memory space and less execution time requirements (George and Liu (1981)). So, for a network that consists of n nodes, a set of n linear equations is created in which all the elements of the current vector have the value of zero. The linear set however, is not positively definite. Therefore, it is not that easy to get a solution for eq. (2-145), and the solution most probably will not be unique. Also, this solution will not provide any information for the overall conductivity of the network, which is the main aim of this work. So, it is necessary to have an additional independent equation that relates the overall conductivity of the network, Gt, the overall voltage drop, Vt, across the network and the overall current, It. This set up provides the extra information that relates It, Vt and Gt. ∑

∑

(2-146)

where e denotes the resistors that connect the network to the external node. If the value of the overall current (It) is defined, then a linear and positive definite set of equations is obtained which can be used for obtaining the solution for the voltage vector. If, in addition to the above, It = 1, then the overall conductivity of the network can be automatically calculated, since (2-147) Therefore, if the conductivity of each resistor is known, the overall conductivity of the network can be calculated. This calculation was performed by Kantzas (1985) and the result was a set of relative permeability curves that provided a good agreement with experimental results in Berea Sandstone.

Three Phase Relative Permeability Three-phase flow situations occur when gas is injected into a reservoir or when a reservoir is produced at a pressure below its bubble point. The flow of one phase relative to the other phases is represented by the relative permeability of that phase. Therefore, relative permeability is an essential variable in the prediction of reservoir performance in reservoir simulation studies.

133

Relative permeability is a direct measure of the ability of the porous medium to conduct flow of one fluid when two or more fluids are present. The flow of the fluids in the medium is controlled by the pore geometry, wettability, fluid distribution, and saturation history. Wettability is a controlling factor in determining three-phase relative permeability characteristics through its effect on the fluids distribution and flow of the three phases. The fluids distribution in water wet and oil wet systems is quite different. When a system is water wet, water fills all the small pores and exists as a film in the larger pores. When the medium is oil wet, the reverse is true; oil will occupy the small pores and exists as a film in the larger pores. It is very rare for a reservoir to be gas wet, so gas wet systems will not be discussed. It is generally believed that reservoirs are originally water wet. As oil migrates into the reservoir, crude oils come in contact with the rock surface and adsorption can occur which alters the wettability of the rock1. This can lead to many different forms of wettability. The wettability of a system can range from strongly water wet to strongly oil wet. When the rock has equal preference for both oil and water, the system has neutral or intermediate wettability. Fractional wettability is a condition when different areas of the core have different wetting preferences. There is also a special case of fractional wettability, called mixed wettability, where the small pores are water wet and the large pores are oil wet. There are two methods of evaluating the relative permeability of each phase: steady state and unsteady state. Each method has its own advantages and disadvantages, however most researchers say that the unsteady state method should not be used. The experimental procedure to evaluate relative permeability of two-phase flow is easy, so a lot of relative permeability data were collected. However, for three-phase flow, the procedure is quite complicated so not many experiments were carried out. Therefore, relative permeability characteristics of three-phase system are not fully understood.

Generally, the relative permeability can be obtained for imbibition (wetting phase displacing nonwetting phase) or drainage (nonwetting phase displacing wetting phase). At the same wetting phase saturation, the fluid distribution in the pores will be different, thus it is expected that relative permeability in drainage and imbibition will also be different. Many researchers published contradictory findings, so hysteresis between drainage and imbibition in three-phase flow is still being studied. Most researchers have seen that in strongly wetted mediums, the wetting phase and nonwetting phase show very little hysteresis. However, for the intermediate wetting phase significant hysteresis was seen. Due to the complex nature of three-phase relative permeability experiments and the lack of agreement between the limited data, many models were developed to generate three-phase relative permeability values. These models have often not compared with all the available experimental data. There were many experiments performed with two-phase flow to study the effects of interfacial tension, temperature, flow rate, and viscosity on the relative permeability characteristics. Some researchers found that these parameters affect relative permeability while some reported otherwise. Experiments on three-phase flow to study the effects of these parameters are very rare to non-existent. The effects of these parameters on two-phase relative permeability are included in the discussion, since it is likely that these parameters will affect three-phase flow in a similar manner. 134

The flow of fluid in carbonates is also investigated. Most carbonates are usually oil wet, this means that relative permeability should be different than that of sandstones, which are usually water wet.Relative permeability in heavy oil systems is also being studied.

Relative Permeability in Two-Phase Systems The wetting fluid in a uniformly wetted system is located in the smaller pores and as a thin film in the larger pores, while the non wetting phase is located in the centers of the larger pores. In general, at a given saturation, the relative permeability of a fluid is higher when it is the non wetting fluid1. This means that the water relative permeability is larger in an oil wet system than it would be in a water wet system. This occurs because the wetting fluid usually travels through the small pores, while the nonwetting fluid moves through the larger pores, which results in better flow of the nonwetting fluid. Also, at low nonwetting phase saturation the nonwetting phase will be trapped as discontinuous blobs in the larger pores which block the flow of the wetting fluid, thus decreasing its relative permeability. On the other hand, the relative permeability of the nonwetting phase at low wetting phase saturation is very high since it travels through the center of the larger pores. This nonwetting phase relative permeability could be as high as the absolute permeability, which indicates that the wetting phase does not significantly restrict the flow of the nonwetting phase. In a water-oil system, as the system becomes more oil wet, relative permeability of water (krw) increases, relative permeability of oil (kro) decreases, and the cross-over point occurs at a lower water saturation (see Figure 2-85)1. 1

Relative Permeability

Water Wet Oil Wet

0

Oil Water

Oil

Water

0

Water Saturation

1

Figure 2-85 Relative permeability of water wet and oil wet systems.

These special characteristics allowed Craig to generalize a few rules of thumb1, which indicate the differences in the relative permeability of strongly water wet and strongly wet oil cores. These rules are presented in Table 1 below.

135

Table 6. Craig’s Rules of Thumb for Determining the Wettability

Water wet

Oil wet

> 20-25 % of PV

< 10-15% of PV

Sw when krw = kro

> 50%

< 50%

krw at max Sw

< 30%

> 50% (can be 100%)

Swc

Exceptions have been found to these rules, therefore wettability experiments should be performed separately to evaluate the wettability of the reservoir instead of relying only on Craig’s rules of thumb. Also, Craig’s rules of thumb can only distinguish water wet conditions from oil wet; they cannot identify the degree of wettability, or neutrally wet, fractionally wet or mixed wet conditions. Pore geometry can also have a strong effect on the measured relative permeability curves, affecting the cross over point in two-phase flow and the irreducible water saturation1. If the pore medium consists of a significant number of small pores, the irreducible water saturation is relatively large. Anderson has also mentioned the significant differences in Swc found in rocks with large, well interconnected pores compared to rocks containing smaller and less well connected pores. In the water wet cores, the smaller pores are filled with water, thus a larger number of small pores increases the irreducible water saturation, but if these pores are less well connected, the water flow is not better. When comparing these two samples, the rock with many smaller pores has a larger irreducible water saturation and the cross-over point for the relative permeability occurs at higher water saturation.

Relative Permeability in Three-Phase Systems In three-phase systems, there is no obvious distinction in relative permeability of water wet and oil wet systems. Most researchers can identify the characteristics of the wetting phase and the nonwetting phase. However, the pattern for the intermediate wetting phase is not fully identified yet. For fractionally wet, intermediate and mixed wet system, special characteristics of relative permeability are not known at all. Most researchers agree that the relative permeability of the wetting phase depends on its saturation only. Anderson mentioned that many researchers found that in a water wet system, there was a good agreement between the wetting phase relative permeability of two-phase systems (water in the water oil tests and oil in gas-oil tests)1. However, there have been some experiments which show that the wetting phase relative permeability depends on saturations of the other phases. Some researchers even found that the trapped gas saturation could affect both the water and oil relative permeabilities. There are other experiments which show that the wetting phase relative permeability was affected by the nonwetting phase saturation in water wet systems. Most researchers still insist however that the relative permeability of the wetting phase should be a function of its saturation only. They suggest that some of these systems that showed significant difference in the relative permeability of the wetting phase were not strongly water wet.

136

Three-phase relative permeability data for water wet sandpacks have been reported by several people. However, there is very limited three-phase data for oil wet cores. The data for water wet cores and sandpacks show generally consistent behavior; gas and water primarily depend on the gas or water saturation respectively, and are weak functions of the saturations of other fluids present. These data indicate that a strongly wetting phase (water) and a nonwetting phase (gas) are not affected by the other phases. However, Schneider et al. reported that the three-phase relative permeability of gas (krg) values are smaller than the two-phase krg in drainage2. This is because of flow interference between the gas (nonwetting) phase and the oil (intermediate wetting) phase in a water wet system containing three phases. Corey et al., however, saw from their experiments that gas relative permeability was essentially the same in two-phase and three-phase systems3. The intermediate wetting phase (oil) appears to be influenced by interactions with the other phases, but the nature of the interaction is not very distinctive. In some cases the dependence of oil relative permeability on the saturations of the other phases is apparent and in some cases it is not. The experimental data often shows so much scattering that no conclusion can be reached. Even though the evidence is not as overwhelming as for the wetting phase, many researchers agree that the relative permeability of the nonwetting phase is a function of its own saturation only. In water wet systems with trapped gas present, Schneider et al. found that the maximum value of kro is larger than the value in the case without gas2. They explained that this is because gas is the nonwetting phase with respect to oil. Also, the trapped gas saturation resulted in lower krw. This is not consistent with the general belief that for water wet system; water relative permeability is a function of water saturation only.

DiCarlo et al. performed three-phase experiments and they have seen other characteristics4: 





The permeabilities of the most wetting fluid are always similar. In both cases of water wet and oil wet media, krw and kro can be described by a power law, kr S, where  = 5. However, a lower saturation is reached in the oil-wet medium than in the water-wet medium. This is consistent with Craig’s generalization that for oil wet systems Swi is lower than Swi in water wet system. For the strongly and uniformly water wet and oil wet media, it is expected that the configuration of the most wetting fluid should be similar in both systems at the same saturation, thus they should have the same relative permeability characteristics. At high saturations, the permeabilities of oil (nonwetting phase) in water wet media and water (nonwetting phase) in oil wet media are similar. Both can be described by a power law, kr S, where  = 4. DiCarlo et al. believed that at high saturations the arrangement of oil filled pores in a waterwet medium and the water filled pores in oil wet medium are quite similar. At low saturations, the permeabilities of oil in water wet media and water in oil wet media are very different. The oil relative permeability remains finite at low saturations while the water relative permeability drops off quickly and approaches zero. At low saturation phases may remain connected through wetting layers in crevices in the pore space. It is this connectivity which controls relative permeability at low saturation. It is possible that the pore scale configuration and connectivity of oil and water is very different for water wet and oil wet media. 137



The gas relative permeability for the oil-wet medium is approximately half its value in a water wet medium. In water wet media, the water phase will occupy the smallest pores and crevices while the gas phase occupies the large pore spaces since gas is more nonwetting than oil. In oil wet media, the water and gas phases compete for the largest pores. It is possible that at a specific gas saturations in the oil wet system, the gas is in smaller pathways leading to a lower permeability.



For the fractionally wet sand, the oil, water and gas relative permeabilities are between the oil, water and gas relative permeabilities in the water wet and oil wet sands.

Schneider et al. mentioned that for systems having no strong wetting preference, characteristics of three-phase flow can be very different than for two-phase flow2. This suggests that the relative permeability might be a function of other phases as well.

Drainage and Imbibition Relative Permeabilities Many people believe that in strongly wetted systems, the relative permeability of the wetting phase is usually a function of its own saturation, which means that it is not a function of saturation history. Anderson reported that the wetting phase relative permeability is very similar for both two and threephase in strongly wetted systems at a specific wetting phase saturation. This implies that the wetting phase distribution in two-phase is very much similar to the one in three-phase systems. Therefore the hysteresis between the drainage and imbibition of the wetting phase is very small. In three-phase systems, many people assumed that gas is almost always a nonwetting phase. There were other experiments, which show that there was hysteresis for the wetting phase in both water wet system and oil wet system. It is possible that for the cases where hysteresis was seen in the relative permeability of the wetting phase, the medium was not strongly wet. In general, the majority of the results show that there is little or no relative permeability hysteresis in the wetting phase. Most of the results that show significant hysteresis in the wetting phase relative permeability were obtained from unsteady state methods. Craig and others believe that there are problems with the unsteady state relative permeability measurement in strongly wetting systems1. As mentioned before, for the nonwetting phase in a strongly wet medium, many people agree that its relative permeability is a function of its saturation only, thus there is no hysteresis. However, there was significant hysteresis seen for the relative permeability of the intermediate wetting phase.

138

Methods of Relative Permeability Measurements Honarpour and Mahmood provide a detailed assessment of the steady state and unsteady methods of measuring relative permeability5. They believe that steady state provides the most reliable relative permeability data. In this method, two or three fluids are injected simultaneously at a constant rate or pressure drop. The system is then allowed to reach equilibrium. At this point, the saturations, flow rates, and pressure gradients are measured. Darcy’s law is then used to calculate the effective permeability of each phase. The injection rate or pressure drop is then changed and the cycle is repeated again until enough points are collected to plot the relative permeability curve. This method is very time consuming, because equilibrium at each flow rate may take several hours to days to achieve. Unsteady state is the quickest method of obtaining relative permeability. In this method, the in-situ fluids are displaced at a constant rate with the effluent volumes being monitored continuously. Equilibrium is not achieved, so the entire set of relative permeability curves can be obtained in a few hours. However, there are many difficulties associated with this method, such as capillary end effects, viscous fingering and channeling in heterogeneous medium, which cannot be properly accounted for. Capillary end effect is a phenomenon in which the saturation is high at the inlet or outlet of the core. To minimize the boundary effect, the fluids can be flowed through the core at high rates to make the capillary forces insignificant compared to the viscous forces due to the flow of fluids. The fluid dispersion in the porous media increases with the flow rate, thus minimizing the boundary effect at the input end6. However, the use of high flow rate can aggregate viscous fingering problems. Also, in this method viscous oils are usually used to prolong the period of two-phase production (at flow breakthrough, no more information on relative permeability can be obtained). Again, this will further enhance the viscous fingering problem. Craig and other researchers recommend that the unsteady state method not be used with strongly water wet cores. They believe that the combination of high velocities and high viscosities will cause a strongly water wet medium to behave as an oil wet system during waterflood. During a waterflood, the injected water will move rapidly through the larger pores, bypassing a large number of pores that are filled with viscous oil, causing early breakthrough. Usually, in waterflooding of an oil wet system, it was seen that breakthrough recovery is low and additional oil can still be recovered. Thus, the performance of a waterflood with viscous oil is very similar to a waterflood of an oil wet system. Also, in comparison with the steady state relative permeability of two-phase, the calculated unsteady state relative permeabilities measured on the same core appear to be more oil wet. During the process of core cleaning and handling, the wettability of the core can be altered, which can significantly affect the relative permeability. The cleaned core is usually more water wet than the actual native state wettability, thus the most accurate relative permeability measurements are obtained on native state core when the core’s wettability is reserved. When these cores are not available, restored state core could be used. The process of core wettability restoration must be handle with care since any error can be resulted in significant wettability alteration. There are other problems associated with obtaining data from core. Heaviside et al. said that the core sample might not truly represent the reservoir because the core samples are only a small fraction of the 139

reservoir, and may therefore be statistically unrepresentative7. Also, the sample may not be in the same state or have the same properties as when it was in the undisturbed reservoir. When the core plug is brought up to the surface, the light ends hydrocarbon can be liberated, causing the oil to be more heavy7. Some components can be deposited on the pore surface, making the rock more oil wet.

Three-Phase Relative Permeability Correlations Due to the complex nature of the experimental process to obtain three-phase relative permeability data, many models were developed to predict these values instead. These models can be divided into three catagories: those that are based on bundle of capillary tubes theory, those that are based on channel flow theory (Stone’s methods and their modifications), and other miscellaneous models. Bundle of capillary tubes theory There are many models which used this theory in the derivation of the relative permeability model. The models that will be discussed are Corey et al., Naar and Wygal, and Land. Corey, Rathjens, Henderson and Wyllie: The first model of three-phase relative permeability was published by Corey et al. in 19563. The model assumed that fluid flow in reservoirs can be represented by a bundle of capillary tubes. The flow paths through the medium can be approximated by the equivalent hydraulic radius of the capillary tubes. A tortuosity correction was included to account for the differences in path length of tubes of different sizes. Corey’s model also assumed that the wetting and nonwetting phase relative permeabilities are independent of the saturations of the other phases. In other words, relative permeability of the wetting phase is a function of wetting phase saturation and relative permeability of nonwetting phase is a function of nonwetting phase saturation. In this model, the intermediate wetting phase (oil) occupies the pores between those occupied by water and gas which are intermediate in size. Its flow can be interfered with by both the wetting and nonwetting fluids. Relative permeability of the intermediate wetting phase is proportional to the area of the pores occupied by this phase relative to the pores occupied by other phases. Corey also assumed that the relationship between saturation and drainage capillary pressure is represented by, (

)

(2-148)

(

)

(2-149)

Where, Pb = fitting parameter Pc = capillary pressure S = saturation Slr = residual liquid saturation for gas oil displacement (obtain from experiment) The model for the intermediate wetting phase (oil) is given as:

140

(2-150) Where, Sl = total liquid saturation Sw = water saturation The derivation for this model is based on the hypothesis that the residual oil saturation of a system containing water >= Slr will be zero. Corey compared the values calculated from his model to the experimental values which he collected from experiment on Berea sandstone. This is shown on Figure 2-86.

Figure 2-86 Comparison between experimental and predicted values (after 3).

Figure 2-86 shows that the match is quite good. One advantage of this method is that it requires only the value of residual liquid saturation to predict the oil relative permeability. However, this model does not allow for the adjustment of the end points of relative permeability in oil-gas and oil-water system. Therefore, its estimation of these end point relative permeabilities may be very different from the values obtained from the experiments.

Naar and Wygal: This model for imbibition was published in 19618. It is also based on the concept of flow in straight capillaries. It has random interconnections of pores and storage capacity, which makes trapping of the nonwetting phase by the invading wetting fluid possible. Again, relative permeabilities to the wetting and nonwetting phases are considered to depend only on their own wetting and nonwetting phase saturations, while the intermediate phase relative permeability depends on all threephase saturations. The relative permeability to water (wetting phase) can be obtained by assuming that oil and gas combine to form a single nonwetting phase; thus the system is two-phase. To compute the 141

relative permeability to gas, the system can be treated as a two-phase system with water and oil forming the single wetting phase. The relative permeability to oil is more complex. When the water invades the porous medium, part of the oil is blocked and the rest invades the pores that are full of gas. However, some of the gas phase is blocked while the remainder is pushed out. Because some of the oil is trapped, the relative permeability of oil decreases. In this model, the oil phase is more wetting than the gas phase, thus oil exists in smaller pores than gas does. So oil will invades gas in slightly larger pores first. Since some oil moves from small pores into larger pores, the permeability of oil increases. This model was derived by assuming that the capillary pressure curve is approximated by (2-151) Where C is a constant and S* is the reduced saturation. The relative permeability of the intermediate wetting phase can be predicted by:

k ro  S of*3 (S of*  3S fw*) S of* 

S *fw 

S ob 

(2-152)

S o  S ob 1  S wi

(2-153)

S w  S wi   S ob

(2-154)

1  S wi

S S 1 1  S wi  w wi 2  1  S wi

  

2

(2-155)

Where, So = oil saturation Swi = initial water saturation Sw = water saturation At the time of the model’s development, Naar and Wygal did not compare the model predictions with any experimental data. Land: Land has seen some evidence showing that the initial gas saturation controls the residual gas saturation, which is trapped after imbibition9. Residual gas saturation was seen to increase with increasing initial gas saturation. When the initial gas saturation is unity (ie. the reservoir contains only gas), the residual gas saturation is a maximum. Land published a model for imbibition in 1968, which takes into account the effect of initial saturations. This model assumes that the maximum residual hydrocarbon saturation is constant, whether the hydrocarbon is gas or oil, and that the residual hydrocarbon is related to the initial hydrocarbon saturation. Land assumed that in three-phase systems the gas relative permeability is the same as for two-phase systems. He reasoned that this is because the gas is a nonwetting phase, thus it occupies the same 142

pores regardless of the nature of the liquids present. Also, it was assumed that for a water wet medium, the imbibition relative permeability to water is the same function of water saturation for both two- and three-phase systems. Land’s model is very similar to that of Naar and Wygal. However, Land believed that Naar and Wygal wrongly placed trapped the gas in the largest pores rather than the smaller pores which are likely to be invaded first by the wetting phase. Land’s model also includes the dependence of relative permeability on saturation, saturation history and the capillary pressure.   *2  kro  Sof  1  S gf   



* *   Shr   (max)  1  S hr (max) S gf  ln  *  * *  2 2   Shr(max) 1  Shr(max) S gi   * * 2  S w  Sob  2   C  1 1   * * * * *  S*   hr(max)  1  Shr(max) S gf Shr(max)  1  Shr(max) S gi 

 



S gf* 

1 * *  S g  S gc  2

S w* 

S w  S wc 1  S wc

S g*  C



 S

* g

*  S gc



2









 4 * * S g  S gc  C 





* S gc  S gi* 

(2-157)

(2-158)

(2-159)

1  S wc

1 (S )

(2-156)



Sg

1

(2-160)

1 * S gi 2

(2-161)

* gc max

S gi* 





S gi

(2-162)

1  S wc

Where * ( S gc ) max =residual gas saturation after imbibition process which started with 100% gas

S 

* hr max

=normalized maximum trapped hydrocarbon saturation

*

( S ob is defined earlier in Naar and Wygal) If water saturation increases and oil saturation decreases while the gas saturation remains constant (or is increasing), the main equation simplifies to



k ro  S of*3 2(S w*  S or* )  S of*



(2-163)

When all gas in the porous medium is trapped,

143

k ro  S

*3 of

2S

* w

S

* of



*  *2 2  * 1 S gr    S  S gr  S gr  ln *  C  C S gl   *2 of

(2-164)

Land did not compare this model’s prediction with any experimental values. Parker, Lenhard, and Kuppusamy: This model was developed in 1987 and it is very similar to Corey’s10. However, in this model the flow path is assumed to be proportional to the square of the mean hydraulic radius of the pores occupied by that phase. In this model,



k ro  S o*1 / 2 1  S w*1 / m

  1  S  m



2 *1 / m m l

(2-165)

With

S l* = normalized liquid saturation The fitting parameter n can be obtained by a curve fit of capillary pressure or by fitting two-phase relative permeability. Therefore this model will tend to give good results for cases where there is a satisfactory fit of the two-phase data. Stone’s methods and their modifications: The methods proposed by Stone are the ones most commonly used for prediction of three-phase relative permeability. Many people have attempted to improve the estimations of Stone’s methods. Theory and assumptions used in Stone’s models: Stone’s method 1 and 2 are probability based models11,12. Both methods are based on the channel flow theory, which states that in any flow channel there is at most only one mobile fluid. As a consequence, the wetting phase is located in the small pores and the nonwetting phase in the large pores, and the intermediate phase occupies the pores in between. Thus at equal water saturations the fluid distributions will be identical in a water-oil system and in a water-oil-gas system, as long as the direction of change of water saturation is the same in both. This implies that water relative permeability and water-oil capillary pressure in the three-phase system are functions of water saturations alone. Also, they vary the same way in the three-phase system as in the two-phase water oil system. The gas phase relative permeability and gas oil capillary pressure are the same functions of gas saturation in the three-phase system as in the two-phase gas oil system.

These two models require two sets of two-phase data: water oil displacement and gas oil displacement. Hysteresis can be taken into consideration by using the appropriate two-phase data. Stone’s method 1: This method was published in 197011. It assumes that the flow of oil is interfered with by the presence of gas and water, and that the effects of gas and water are independent. Oil relative * permeability is a function of normalized oil saturation S o ,  w and  g , which depend on the water and

gas saturation, respectively. 144

 k  k rog    S o*  w  g k ro  S o*  row*  *    1  S w   1  S g   S g* 

Sg 1  S wc  S om

(2-166)

(2-167)

S o* 

S o  S om 1  S wc  S om

(2-168)

S w* 

S w  S wc 1  S wc  S om

(2-169)

Where, w = probability that an oil filled pore is not blocked by water g = probability that an oil filled pore is not blocked by gas krow = relative permeability of oil in oil water system krog = relative permeability of oil in oil gas system Som = minimum residual oil saturation Sg = gas saturation So = oil saturation Sw = water saturation Swc = connate water saturation = Swir = Swi Stone mentioned that Som can be expressed as a function of fluid saturation but since sufficient data is not available to correlate this relationship, a constant value is set for Som instead. Stone compared the simulated values from the model with the data of Corey et al. and Dalton et al. (see Figure 2-87 and Figure 2-88).

145

Figure 2-87 Compare with results of Corey et al. (after 11)

Figure 2-88 Compare with results of Dalton et al. (after 11)

In Figure 2-87 and Figure 2-88, the dashed curves are obtained with Som = 0.1, while the solid curves correspond to Som = 0. From the figures it can be seen that the value of Som only has an affect on the small values of relative permeability. Also, data obtained by Corey et al. has significant scattering, so it is hard to say whether the match is good or not. Overall, Stone’s method 1 fits the data of Dalton et al. better than Corey et al.’s data. Stone’s method: In the three-phase system, the sum of all three relative permeabilities is always less than or equal to unity. Stone assumes that it achieves its maximum value in the cases where So = 1 - Swc, Sw = Swc and Sg = 0 and So = 1 - Sgc, Sw = 0, Sg = Sgc. For these cases, kro = 1 and krg = krw = 012.

k ro  k row  k rwo k rog  k rgo   k rwo  k rgo

(2-170)

Where, krow = relative permeability of oil in oil-water system krwo = relative permeability of water in oil water system krog = relative permeability of oil in oil gas system krgo = relative permeability of gas in oil gas system Stone claimed that this method predicts better values than method 1, especially at low oil saturations. It is also not required to estimate Som in this method; the model can predict the value of Sor. Comparison of Stone’s method 2 predictions with Corey et al.’s data is shown in Figure 2-89.

146

Figure 2-89 Stone’s Method 2 predictions and Corey et al.’s experimental data (after 12)

At low water saturation, the agreement is good. However, at higher water saturation, the predicted values are too low. Stone also compared the predicted Sor with Holmgren-Morse residual oil data of sandstone, see Figure 2-90.

Figure 2-90 Stone’s prediction of Sor and Holmgren-Morse’s data (after 12)

These data shows that as the initial free gas saturation increases, residual oil saturation decreases and residual gas saturation increases. Stone’s method 2 consistently predicts higher Sor. Dietrich and Bondor: It has been observed in experiments on water-wet systems that the water relative permeability is approximately a function of water saturation only, and that the gas relative permeability is a function of the gas saturation. However, the oil relative permeability seems to be a function of both the water and gas saturation. This model (proposed in 1976) takes into account the reduction in oil relative permeability caused by the presence of water and gas13. 147

Dietrich and Bondor realized that the oil permeability at connate water saturation predicted by Stone’s method 2 would be applicable only if krow and krog can be one. However, this is not always possible. Thus, they rewrite the Stone’s model as,

k ro 

k row  k rwo k rog  k rgo  k rocw

 k rwo  k rgo

(2-171)

Where krocw is the relative permeability of oil at connate water in oil water system. Hirasaki: Hirasaki modified Stone’s method 1 in 197313. Hirasaki’s model assumes that both water and gas may be flowing simultaneously in the pore space with the oil. He derives a three-phase oil relative permeability expression by considering the total reduction in oil relative permeability caused by the water and gas.

k ro  k rowk rog   S g 1  k row 1  k rog 

(2-172)

Aziz and Settari: This model was published in 197910. Aziz and Settari suggested the use of the absolute permeability as the basis for calculating relative permeability and use krocw as a partial normalizing factor. For method 1:

 k / k   k rog / k rocw  k ro  k rocwS o*  row rocw   * *  1  S w    1  S g  

(2-173)

k ro  k rocwS o*  w  g

(2-174)

For method 2,



k ro  k row  k rwo k rog  k rgo   k rwo  k rgo



(2-175)

Aziz and Settari said that Stone’s method 2 usually predicts too low oil permeabilities and method 1 usually predicts too high oil permeabilities. They believed that the use of  as a free parameter is a convenient way to improve the predicted permeability. Fayers and Mathews: This model was published in 198414. Fayers and Mathews believed that the value of Som should be better predicted to improve the estimations of Stone’s method 1 (Som is a constant in Stone’s method 1). They suggested the use of a saturation dependent value of Som for Stone’s method 1,

S om  S orw  1   S org   1

Sg 1  S wc  S org

(2-176) (2-177)

In the presence of trapped gas,

S om  S orw  0.5  S g

(2-178)

148

Where, Sorg = residual oil saturation in gas oil system Sorw = residual oil saturation in water oil system Baker said that this provides an improved fit to the residual oil data. Aleman: In 1986, Aleman suggested another approximation of Som for Stone’s method 110. 

S om

  Sg  S w  S wc   S orw   S   org  1  S wc  S orw  1  S wc  S org 



(2-179)

The free parameters  and  are used to fit the curvature of the zero oil permeability isoperm Aleman et al. later developed a statistical structural model for prediction of two-phase relative permeability based on a local volume average approach to a bundle of capillaries model15. In this model, pores are randomly distributed but there are no pore interconnections. It assumed that the saturation change is in the direction of decreasing intermediate wetting phase saturation. All other assumptions are the same as the ones made by Stone in the development of his methods. Aleman expanded this model for three-phase flow with the intermediate wetting phase relative permeability:

k ro  k ro(I )  

(2-180)

Where kro(I) is the relative permeability to oil predicted by Stone’s method 1 and  is a correction term.









* * * * * * S *o k row  1 k rog  1 k row k rog  k rwo k rgo

k

* rgo



1 k

* Where k row 

* row

k

* rwo

  k

* row



1 k

* rgo

k



* rog



(2-181)

k row * * * , and other parameters such as k rog , k rwo , k rgo are defined in a similar way. * 1  Sw

This model is sensitive to the value of Som, and may predict incorrect oil permeabilities for values of Som which are too small. The problem may be due to assumptions made in the model about the distributions of fluids. Larger values of Som give more reasonable predictions. Aleman recommended that this model should be used only if the predicted isoperms shows reasonable match with experimental data. Other methods Other methods included in this report are the models that were proposed by Baker, Pope, Blunt, and Moulu et al. Saturation-weighted interpolation: This method was proposed by Baker in 198810. This is a simple model of oil relative permeability that is based on saturation-weighted interpolation between water-oil and gas-oil data.

k ro 

S w  S wc k rwo  S g  S gr k rog S w  S wc   S g  S gr 

(2-182)

149

For the wetting and nonwetting phase,

k rw 

S o  S or k rwo  S g  S gr k rwg S o  S or   S g  S gr 

k rg 

S o  S or k rgo  S w  S wc k rgw S o  S or   S w  S wc 

(2-183)

(2-184)

These equations assume that the end points of the three-phase relative permeability isoperms coincide with the two-phase relative permeability data. The saturation-weighted interpolation method may give erroneous results if the two-phase relative permeability curves being interpolated between are very different. This is especially a problem if the end point saturation (Sorw and Sorg) of the oil-water and oil-gas curves differ significantly. Pope: Pope published his model in 198816. This model does not require two-phase data:







k ro  k rocw a S o 1  S w







 1  a S o 1  S g 

 

(2-185)

Where, So  Sw 

Sg 

1  S w  S g  S or 1  S wc  S gr  S or S w  S wr 1  S wc  S gr  S or

S g  S gr 1  S wc  S gr  S or

(2-186) (2-187) (2-188)

Where Sgr is the residual gas saturation. The parameters , , ,  and a can be obtained by matching the two-phase data, such that  =  +  = eow,  =  +  = eog, and a=1/2. eow and eog are obtained by curve fitting the oil-water and oilgas relative permeability data sets into exponent functions. Empirical model: This model was developed by Blunt and published in 200017. It is based on the saturation-weighted averages of the two-phase relative permeabilities, which includes gas and oil trapping. The model also allows for drainage of oil at low saturations. In this model, six relative permeability sets are required: krwo and krow from oil-water displacement, krog and krgo from gas-oil displacement, and krgw and krwg from gas-water displacement. If the medium is water wet or mixed wet and oil is spreading then layer drainage is also considered. In this case, the bulk oil relative permeability, kob, is obtained by:

 S g S o*  S ol  Min  * , S o   S g 

(2-189)

150

k ol 

* k rg k rog * rgo

S ol2

*2 o

k S

(2-190)

k ob S ob   k rog S o   k ol S ol 

(2-191) *

Sol represents the oil saturation in layer. S o is the saturation of oil when the layer drainage regime * starts, and k rog is the corresponding oil relative permeability. If experimental data does not indicate

* * * layer drainage, it is assumed that S o  S orw . S g is the gas saturation when So = S o (all oil resides in * * layers). Note that S o and S g in this model are defined differently than before.

If the medium is not at the condition described above, then kob = krog(So). When trapping is considered: S gr 

S gmax

1

Cg 

(2-192)

1  C g S gmax S grw



1

(2-193)

S gmax

S gf 

1 S g  S gr   2 

S hr 

S hmax 1  C h S hmax

S

 S gr   2

g

 4 S g  S gr  Cg 

(2-194) (2-195)

S hmax is the largest hydrocarbon saturation during displacement S hf 

1 S h  S hr   2

S h  S hr 2 



 4 S h  S hr  Ch 

(2-196)

 S  S k S   b S  S   b S  S k S  S  S   S  S  S   S  S a k S   b k S  S  S   S  S 

S ofb  min S o  S ol , max S hf  S gf  S ol ,0

k rw 

a S o

k ro  

S

w

o

 S oi   a g



g

 S wi  ao k row

gr

ofb

w

rwo

wf

o

o

oi

g

g

gr

wi

g



o

ob

o

oi

g

gr

rwg

wf

(2-198)

gr

ofb

o

rgo

ofb

gr



So  k ol S ol    ao k row S hf   bo k rgw S hf  So  S g k rg  

g

(2-197)

(2-199)

S w  S wi   S o  S oi  S w  S wi a g k row S gf   bg k rgw S gf   S o  S oi a g k ob S gf   bg k rgo S ogf 



Sg So  S g

a k S   b k S  g

row

hf

g

rgw

(2-200)

hf

151

   ai  max min    

i

 

g0 0

   bi  max min  o 0  0  



i

   ,1,0   

(2-201)

  ,1,0  

(2-202)



crit  N cap   ,1,0  N cap  









  max min 

  max min 1 

crit   N cap ,1,0 N cap   

(2-203)

(2-204)

Where, o0

= reference oil density

g0

= reference gas density



0

= reference density different between oil and gas

crit N cap = critical capillary number

Ncap = capillary number Ch is the Land trapping constant for hydrocarbon. In strongly water wet medium, it can be assumed that Ch = Cg = Co. In other cases, it can be assumed that Ch = min ( Co , Cg). Mathematical model: Moulu et al. developed a mathematical model to simulate the three-phase flow in porous media in 199918. The model assumes that the reservoir consists of fractal pores with gas in the center, and in the case of water wet pores, the water is residing near the wall. A fractal is defined as a shape made of parts similar to the whole in some way. This model assumes that krw is a function of Sw only and krg is a function of Sg and Sw. Essentially in this model:   

The fluids flow together in the same fractal pore with the wetting phase along the rock walls, the gas phase in the center and the third phase in between The flow of each fluid is given by their relative permeabilities, which are analytical functions of a linear fractal dimension DL Capillary pressure is taken into account when calculating the relative permeabilities by using DL

DL is the slope obtained from log-log plot of the capillary pressure curve. This plot will give two different slopes. The sharp slope is used for the wetting phase flow (krw) at the wall and for the intermediate phase (kro). The other slope is use for gas flow (krg). For a water wet medium:

k rw  S

4  DL 2  DL w

S

4  DL 2  DL wi

(2-205)

152

 42 DDL 4  DL  k ro  k ro (2 Ph )  S L L  S w  S or  2 DL   



k rg  k rg max 1  S L 



4

1 2  DL

(2-206)

(2-207) (2-208)

For cases when the medium is not water wet,

m

WI  1 with WI = wettability index 2

k rw  mS w  S wi 

(2-209)

k ro  1  mS o  S oi   m  k row S L  S w  S or 

(2-210)

Where Soi = initial oil saturation The expression for gas relative permeability can still be used in this case since gas always shows its strongly nonwetting phase behavior. It is important to note that the values of DL used in krw (and kro) and krg are different. Comparison There are many correlations available for the prediction of three-phase relative permeability. These include the models of Stone, Hirasaki, Corey et al, Naar and Wygal, Land, Aleman, and Parker et al. The comparison by Baker shows that the models are often not very good predictors of the experimental data10. This means that there is a need for better relative permeability models of three-phase flow. Baker has shown that in most cases, saturation weighted interpolation between the permeabilities at the two-phase data set provided a better fit of the experimental data than other models. Baker has seen that most of the prediction methods fit the data sets equally well. This shows that each of the methods is capable of representing three-phase relative permeabilities in the high oil permeability region. Stone’s method 1, using saturation-weighted interpolation fits better than Stone’s method 2 for most cases. Hirasaki’s model is one of the worse. Parker’s model is generally equal to or better than Corey’s, Land’s, Naar and Wygal’s, and Aleman. Stone’s method 1 usually predicts too high oil permeability at low oil saturation, while Stone’s method 2 predicts too low oil permeability in the same region. With better values Som or  the predictions of Stone’s method can be improved. However, the value of Som or  are not always easily determined. Thus, saturation-weighted method seems to be a good model due to its simplicity and its ability to yield comparable results with Stone’s methods. Delshad and Pope claim that the model proposed by Pope agrees with several sets of data, which makes it more superior than other methods (Corey, Naar-Wygal, Land, Stone’s, Baker and Parker).

153

A comparison of all the methods presented is not seen in the literature, thus it is not known which method is most superior16. Effects of Low IFT or Nca on Relative Permeability In the literature, many researchers agree that as the interfacial tension (IFT) is reduced, relative permeabilities increases. There are people who believe that the relative permeability relationship with saturation is a linear function and there are people who report otherwise. Harbert noticed that the flow of low IFT fluids differs from that of conventional gas-oil-water systems in both sandstones and carbonates. He found that both water and oil relative permeability curves were found to shift upward, indicating that the two phases interfere less with each other as IFT is reduced19. This is shown in Figure 2-91.

Figure 2-91 Effect of capillary number on relative permeability (after 9)

In this figure, kro appears to be a straight line while krw still maintains its curvature (N in the figure is the capillary number, which is the same as Nca). Also, the change in krw as IFT changes is not significant. However, Sor did change; in this case it becomes less than 10%. For the lower capillary number, the water relative permeability curve is a function of saturation (no hysteresis). Harbert also mentioned that at the same Nca and IFT, a less permeable core has a higher residual saturation. This is expected because a core that was less permeability has pores that are not well connected, and the invading fluid cannot easily displace oil from these pores. Foulser et al. proposed a model in 1992 for relative permeability at high capillary number for the flow of two or more phases20. This model allows the residual oil saturation to be reduced as the capillary number is increased. As mentioned before, many people conventionally assumed that relative permeabilities are linear functions of saturations for low IFT fluids. However, this model assumes that relative permeability is non-linear, as shown in some reports. It also assumes that the flow of two or three phases can be

154

represented by the flow of droplets of fluid through the pore network. The geometry of the pore network was assumed to consist of capillary tubes. The relative permeability can then be derived as a function of saturation according to this equation,

k r  k ro  S* /    S *

2-211

S*  f ( N  1   /    f  ) / N

2-212



And 

Where, kr=relative permeability of phase 

k ro = end point relative permeability of phase 

 = viscosity of phase  f  = fractional flow of phase  N = number of droplets N defines the droplet size relative to the capillary tube length. When N = 1 flow is segregated and the relative permeabilities are linear functions of saturation. As N approaches infinity, a “mixed flow” regime is created, in which the droplet size is very small compared to the tube length, giving the relative permeabilities as a function of saturation. In this model the relative permeabilities are a function of phase viscosities as well as saturations.

Effects of Temperature on Relative Permeability Even though there are contradictory reports about the effects of temperature on relative permeability, many researchers agreed that as temperature changes, relative permeability also changes. However, they have not agree on the effect of temperature on the adsorption mechanisms. Handy et al Handy et al.’s experimental results on two-phase flow in Berea sandstone indicate that relative permeability curves are affected by temperature, especially for low IFT cases21. As temperature increases, relative permeability to oil increases and relative permeability to water decreases at a given saturation while residual oil saturation decreases and irreducible water saturation increases. This is shown in Figure 2-92 and Figure 2-93 below.

155

Figure 2-92 Low IFT systems (after 21)

Figure 2-93 High IFT systems (after 21)

Figure 2-92 and Figure 2-93 show the effects of temperature on high and low-tension systems respectively. In both figures, the cross-over point shifts to higher Sw as temperature increases. In general, as temperature increases oil is able to flow much easier than water can. These results suggest an increase in water wettability of sandstone with temperature. The figures also show that temperature effects are more significant for low tension than for high tension systems. The residual oil saturation decreases significantly at higher temperature, but only small changes in irreducible water saturation (Swir) are seen in the high-tension systems. At any temperature, Swir for the low-tension system is smaller than that observed for the high-tension system. Relative permeability to oil and water both increase with increasing temperature up to 100°C for the low-tension system. For the low tension system, the changes in Swir with temperature are not significant, but Swir values for low tensions were lower than those at high tensions. This implies that the increase in temperature and decrease in IFT have opposite effects on Swir; increasing temperature increases Swir but as IFT is decreases, Swir decreased. Therefore it is possible that Swir in the low-tension system at high temperatures is controlled by both the IFT and wettability changes due to temperature.

156

For both low IFT and high IFT systems, krw/kro decreases with increasing temperature at a given saturation (as temperature increases, krw decreases and kro increases, thus krw/kro decreases). Thus krw/kro shifts toward higher water saturations with increasing temperatures. Sufi et al. Sufi et al. found that in the temperature range of 20°C to 85°C, the relative permeability of oil and water obtained from unconsolidated clean Ottawa sand does not change22. In a separate experiment, they saw that as temperature increases, Sor decreases. This observation is consistent with Handy et al. Sufi et al. also said that as temperature increases, the viscosity ratio of the oil water system decreases. This reduction in the viscosity ratio is responsible for the change of the fractional flow curve as seen in the second experiment. Thus a change in Sor is not necessarily due to a change in wettability. They also found that the irreducible water saturation increases as the temperature of the system increases. This observation is similar to that of Handy et al. The changes in irreducible water saturation can be illustrated to depend only on the change in the viscous force, irrespective of whether it is caused by a temperature or a rate change. Figure 10 demonstrates this:

Figure 2-94 Effects of viscous force on Swir (after 22).

From this figure, it can be seen that the change in Swi due to temperature or flow rate follows the same pattern. This indicates that changes in irreducible water saturation with temperature can be caused by changes in viscous forces not necessarily changes in wettability.

Other theories Nakornthap et al. believed that the adsorbed layer of polar components in crude oil on rock surfaces is thermodynamically unstable, thus it can be desorbed at high temperature and the rock may become more water wet23. From the analysis of three-phase relative permeability data, Maini and Okazawa also found that the studies of cleaned sand shows that relative permeability does not change as temperature changes. They 157

mentioned that Polikar et al. reported that the temperature effects could be controlled by the characteristics of the porous medium24.

Effects of Flow Rate on Relative Permeability Theoretically, relative permeability is not a function of flow rate. However, there has been experimental data which shows that the relative permeability changes as the flow rate changes. In general, the relative permeabilities for both phases were found to increase with increasing flow rate. Leverett et al. were the first to report about the influence of flow rate on relative permeability, however they later attributed this to capillary end effects. Crowell et al. and Greffen et al. found that injection rate has no affect on the viscous flow of water and gas. Also, Labastie et al. reported that relative permeabilities were independent of flow rate except near residual oil saturation25. However, there are many others who reported that relative permeability does change as flow rate changes. Heaviside et al. believed that for an intermediate wettability medium, at low flow rates the oil segments would exist in the center of pores, and blocks the flow of water. However, at high pressure gradients or high flow rates water would be forced through the throats with the oil coating the pore surface. Thus relative permeability can change with flow rate7. Heaviside et al. also found that the viscous force does not affect residual oil saturation for strongly water wet chalk. In the experiment, the pressure gradient is increased but no change in production rate is seen7. Handy et al. reported that rate affects relative permeability below 120 cc/hr. Above this rate, no significant rate effect was observed21. Also, they have seen that the rate effect does not significantly affect Sor or end point relative permeabilities Sandberg et al., however, reported that the relative permeability of the water phase increased very slightly with flow rate. In contrast, the oil phase relative permeability increases significantly. He reasoned that the rate effect on the oil phase may be a result of some tendency for the oil phase to flow in slugs6. Sufi et al. performed a more in depth study regarding the effects of flow rate on relative permeability. They noticed a trend in the relative permeability curves from the experimental results, which is shown in Figure 2-95.

158

Figure 2-95 Effects of flow rate on relative permeability (after 22)

This figure shows that the relative permeabilities for oil remain unchanged, while the water relative permeability curves are low at low flow rate, and become independent of rate when the flow is high (greater than 240 cc/hr). Therefore, it was assumed by Sufi et al. that the rate of 240 cc/hr represents the minimum rate required for front stability. As mentioned before, Handy said that in their experiment when the flow rate is larger than 120 cc/hr, no effect is seen. Thus Sufi and Handy disagree about the value of the critical stability flow rate. It was found by Akin et al. that gas relative permeability increases with increasing total flow rate. They have also seen that the effects on brine and hexane relative permeability are much more compared to gas relative permeability. The oil and water capillary pressure values for higher flow rates were also greater at high brine saturation data. Thus it is expected that relative permeability should change with flow rate25. Heaviside et al. reported that for strongly wetted systems, at high IFT, there is a negligible rate dependency, even at very high flow rate. However, in cases of low IFT, rate dependency can be seen7.

Effects of Viscosity on Relative Permeability Researchers still haven’t reached a conclusion regarding the effect of viscosity on relative permeability. A number of researchers report that viscosity does not affect relative permeability at all, while others say that it does. Sandberg and Gournay studied the effects of oil viscosity on relative permeability of sandstone outcrop. They reported that the relative permeability of both phases is not a function of the nonwetting phase viscosity6. Odeh mentioned that Leverett had shown earlier that the wide range of viscosity had essentially no effect on relative permeability26.

159

Odeh developed a two-phase flow model which assumed that the porous medium consists of straight circular capillaries of different radii. He also assumed that there are no interconnections among the capillaries and no mass transfer across the oil water interface. During waterflooding, a layer of water is left between the oil and the walls of the capillary26. The model is as follow:

 n 4 o  1 n n rn3 1   r  k ro  S o f  k , x    w m  4  ro   mrm

(2-213)

1

n

 r S o f  k , x  ro

 nrn

4

 1   m   mrm4

(2-214)

1

Where, kro = relative permeability of oil o = viscosity of oil w = viscosity of water m = total number of capillaries in porous sample n = number of capillaries through which one phase flow k = a constant n = thickness of water film = rc - rn rc = radius of any capillary rn = radius of oil phase in the capillary

r  r  In Odeh’s model, f  k , x  is a series in x where rx is the radius of the smallest capillary which is filled ro  ro  with oil at an oil saturation So. This equation indicates that relative permeability to oil is a function of saturation as well as the viscosity ratio, the thickness of the molecular layer and pore sizes. Odeh performed experiments to investigate the effects of viscosity ratio on the relative permeability of the nonwetting phase (kro). The results are shown in Figure 12.

160

Figure 2-96 Effect of viscosity on relative permeability (after 26)

The figure shows that the maximum differences in relative permeability values due to viscosity ratio variation occur at the point of minimum brine saturation. This increase tends to a limit as the viscosity ratio becomes larger. As mentioned before, Sufi et al. found that as temperature increases, viscosity ratio (oil to water) decreases, which changes the fractional flow of water and oil22. This in turn changes the relative permeability curves. Thus it is possible that viscosity ratio does affect the relative permeability curves.

Carbonates Sandstones are usually being investigated in laboratory, thus the majority of the relative permeability data shows the effects that are specific to sandstones only. Very little data were obtained on carbonates in literature. From the limited literature, Schneider et al. showed that for oil wet carbonate samples, they found no effect on the oil (wetting phase) relative permeability when comparing two- and three-phase measurements2. This means that relative permeability of oil is a function of its saturation only. This observation is consistent with what other people have seen. Also, the water relative permeability was lowered by the trapped gas, showing the interaction between the two nonwetting fluids. Schneider et al. also reported that the flow behavior of the uniform porosity carbonate samples tested was similar to that of consolidated sandstones2. Thus rock type does not seem to influence the flow relationships other than through its wetting preference. However, due to the surface minerals most of the time carbonate rocks are oil wet, while sandstones are usually water wet. Also, carbonates usually have vugs. When large vugs exist in the reservoir, the core sample used in the experiment might not be representative of the reservoir. When laboratory data of carbonates is being used to predict threephase flow, a greater uncertainly has been added. Also, most of the relative permeability models were derived from the assumption that the medium if water wet. Attention must be paid to the assumptions of the model to select the appropriate one for carbonates. 161

It is expected that IFT will affect relative permeability and recovery in the same way regardless of the rock properties since IFT is a fluid property. The effects of temperature on relative permeability in carbonates have not been reported in literature. It can be expected that a change in temperature will change the characteristics of relative permeability of fluids in carbonates. However, no data is available to identify what those changes are. Again, nothing can be said about the effects of flow rate and viscosity on the flow of fluids in carbonates since there is no data available.

Relative Permeability of Heavy Oil Systems Relative permeability includes contributions from a number of different variables, each causing some resistance to flow. The resistance to flow of a given phase in a multiphase situation depends primarily on how this phase distributes itself within the porous medium in the presence of other fluids. The variables that affect fluid distribution in a two-phase system include: pore structure and pore size distribution, wettability, saturation history, interfacial tension, interfacial viscosity, viscosity ratio, density ratio, and flow rate24. For heavy oil systems, it may no longer be safe to assume that the local fluid distribution at a given saturation depends only on the first three factors listed above, and is independent of the viscosity ratio and fluid velocity involved as many people have assumed. Furthermore, while relative permeability of a fluid depends on its own distribution within the pore space of the medium, this may not be the case in heavy oil systems. Assuming that the residual oil is distributed within the porous medium in the form of small globules, if the viscosity of the oil is very high, these globules can behave like solid particles and may plug pore throats more efficiently than globules of a low viscosity of oil24. Also, the use of relative permeability model must also be carefully chosen since most of this model does not take viscosity into account. More experiments with heavy oil are required to shed more light regarding the effect of each parameter on relative permeability. It cannot be assumed that these parameters will affect relative permeability of oil in the same way described earlier.

Conclusions Relative permeability for water wet and oil wet systems is distinctively different for two-phase flow. In water oil system, as the system becomes more oil wet, relative permeability of water increases, relative permeability of oil decreases and the cross over point occurs at smaller water saturation. In three-phase systems where the medium is strongly wetted, the relative permeability of the wetting phase is a function of its saturation only. There is also some evidence that the relative permeability of the wetting phase and the non-wetting phase is a function of its own saturation only in strongly wetted medium. This indicates that the relative permeability of the wetting and non-wetting phase is a function of its own saturation only. This means that there is little or no hysteresis between drainage and imbibition relative permeability of the wetting phase and the non-wetting phase. However, for the intermediate wetting phase, its relative permeability has been seen to be a function of other saturations, as well as saturation history. Thus significant hysteresis in relative permeability of this phase is seen. The steady state and unsteady state method can be used to evaluate relative permeability of each phase. However, the steady state method yields more reliable data. Most people recommend that this method should always be used. 162

The experiment to find relative permeability of three-phase systems is very complex, thus many models were developed. Corey et al. published the first three-phase relative permeability model in 1956. This model was based on the assumptions that the pore space in the medium can be represented by a bundle of capillary tubes. Other methods that made the same assumption include: Naar and Wygal, Land, Parker et al. Stone’s models are based on the channel flow theory. He published method 1 in 1970 and method 2 in 1973. Since then many people have modified these models. These people include: Dietrich and Bondor, Hirasaki, Aziz and Settari, Fayers and Mathews, Aleman, and Parker et al. There were also other models published by Pope, Baker, Blunt and Moulu et al. Baker compared the models of Stone, Hirasaki, Corey et al., Naar and Wygal, Land Aleman and Parker et al. He found that Stone’s method 1 could fit data better if the estimation of Som is good. However, he found that his own method, which is a simple interpolation between the various set of data, yields the best fits. Delshad and Pope evaluated the predictions of Corey, Naar and Wygal, Land, Stone, Baker, Parker and Pope. They found that Pope’s model fits better than the rest. Up to this point, an extensive comparison of all the models with all the experimental data is not seen in literature, so it cannot be concluded which model is best. However, Stone’s methods are the most commonly used. There are some parameters that researchers have seen to affect relative permeability characteristics (of two-phase systems). Many have seen that as interfacial tension decreases, the relative permeability of water and oil increases. The reduction in IFT reduces the interference between two phases, making them able to flow better. The relationship between relative permeability and saturation of low IFT system is still a point of debate; many say that this relationship is linear, while others say that it is not. When temperature changes, the relative permeability also changes. Handy et al. reported that the effect of temperature is more significant in low IFT systems. In both high and low IFT, Sor decreases and Swir increases with increasing temperature. The cross over point also shifts to higher Sw values. Thus Handy et al. believed that as temperature increases, the system becomes more water wet. Nakornthap et al. explained that this increase in water wetness is due to the breakdown of the organic layer on the surface of the rock. Sufi et al. disagree with this. They believe that the change in relative permeability is due to the change in viscous forces, not a change in wettability. Flow rate was seen to affect relative permeability. Handy et al., Sandberg et al. and Sufi et al. reported that as the flow rate increases, relative permeability of water increases, while relative permeability of oil decreases. However, Handy and Sufi disagree with the critical stable flow rate. The effect of viscosity on relative permeability is also investigated. Sandberg said that oil viscosity has no effect on relative permeability. However, Odeh reported that the effect of viscosity is most significant at connate water saturation. Also, Sufi et al believed that changes in viscosity lead to changes in relative permeability. The studies of relative permeability and the effects of other parameters were mainly done on sandstones (or sandpacks) with conventional oil. For carbonate systems, if the rock properties are uniform, the same flow characteristics seen before can be expected. However, carbonates usually are oil wet while the majority of the models assumed that the medium is water wet. Thus care must be taken when choosing a model to predict three-phase flow. Also, carbonates have vugs, so the core sample evaluated might not be representative. With heavy oil systems, the flow characteristics might be different. The effects of these parameters on relative permeability may be different as well. More 163

experiments must be conducted to investigate the effects of each parameter on relative permeability. The selection of an appropriate model to predict relative permeability of viscous oil should be made with care.

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Larson, R.G., Scriven, L.E. and Davis, H.T., Percolation Theory of Two Phase Flow in Porous Media. Chem Engng Sci, 36, 57, (1981) Larson, R.B. and Morrow, N.R., Effects of Sample Size on Capillary Pressures in Porous Media. Powd Tech,30,lZ3,(1981). Larson, R.G., Scriven, L.E. and Davis, H.T., Displacement of Residual Non-Wetting Fluid From Porous Media. Chem Engng Sci, 36, 75, (1981). Lenormand, R., Zarcone, C. and Sarr, A., Mechanisms of the Displacement of One Fluid by Another in a Network of Capillary Ducts. J Fluid Mec, 135, 337, (1983) Leverett, M.G., Flow of Oil-Water Mixtures Through Unconsolidated Sands. Trans AIME, 132. 149, (1939) Levine, S. and Lowndes, J., Application of Percolation Theory to Capillary Pressure-Saturation Effects in a Porous Medium. Unpublished paper, (1983) Lin, C. and Slattery, J.C., Three-Dimensional Randomized. Network Model for Two-Phase Flow Through Porous Media. AICHEJ,28, 2,311, (1982) Loren, S.D., Permeability Estimates from NML Measurements. J P T, 24,923, (1972) Lucia, F.S., Petrophvsical Parameters Estimated from Visual Descriptions of Carbonate Rocks: A Field Classification of Carbonate Pore Space. JPT, 35, 629, (1983) Marle, C.M.. Multiphase Flow in Porous Media. Institute Francais Du Petrole, Gulf Publishing Company, (1981) McCaffery, F.G., The Effect of Wettability on Relative Permeability and Imbibition in Porous Media. Ph. D. Thesis, University of Calgary, (1973). Mann, R., Androutsopoulos, G.P. and Golshan, A., Application of a Stochastic Network Pore Model to Oil Bearing Rock with Observations Relevant to Oil Recovery. Chem Engng Sci, 36, 337, (1981) Mohanty, K.K.and Salter, S.J-, Multiphase Flow in Porous Media: II. Pore Level Modeling. SPE paper No. 11018, presented at the 57th Annual Fall Technical Conference and Exhibition of SPE of AIME held in New Orleans in LA (Sept. 26-29 1982) Mohanty, K.K., Scriven, T.H. and Davis, E.L., Physics of Oil Entrapment in Water Wet Rock. SPE paper No. 940, presented at the 55th Annual Fall Technical Conference of the Society of Petroleum Engineers of AIME, Dallas, Texas, (Sept. 21-24 1980) Moran, J.H. and Papaconstantinou, C.M., A Novel Dynamic Measurement of Permeability. Soc Pet E J, 2L 670, (1981) Morgan, J.T. and Gordon, D.T., Influence of Pore Geometry on Water-Oil Relative Permeability. J P T, 22, 1199,(1970) Morrow, N.R., Chatzis, L, Siegel, D., Taber, J.J., Corrections to In-Situ Measurements of Residual Oil for Flushing at the Wellbore. Pet. Soc. of CIM paper No 83-34-26 presented at the 34th. Annual Technical Meeting of the Petroleum Society of CIM, Banff, (May 10-13 1983) Morrow, N.R., and Chatiudompunth, Application of Hydraulic Radii Concept to Multiphase Flow. P.R.R.C. Report, 82-85, (1982)

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3 .

C H A P T E R

3

MOLECULAR DIFFUSION Introduction If a few crystals of a colored material like copper sulfate are placed at the bottom of a tall bottle filled with water, the color will slowly spread through the bottle. At first the color will be concentrated in the bottom of the bottle. After a day it will penetrate upward a few centimeters. After several years the solution will appear homogeneous. The process responsible for the movement of the colored material is molecular diffusion that often called simply diffusion, which is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of the particles. In gases, diffusion progresses at a rate of about 5 cm/ min; in liquids, its rate is about 0.05 cm/min; in solids, its rate may be only about 0.00001 cm/min. This slow rate of diffusion is responsible for its importance. In many cases, diffusion occurs sequentially with other phenomena. When it is the slowest step in the sequence, it limits the overall rate of the process. For example, diffusion often limits the efficiency of commercial distillations and the rate of industrial reactions using porous catalysts. It limits the speed with which acid and base react and the speed with which the human intestine absorbs nutrients. The result of diffusion is a gradual mixing of material. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration, but it is important to note that diffusion also occurs when there is no concentration gradient. In gases and liquids, the rates of these diffusion processes can often be accelerated by agitation. For example, the copper sulfate in the tall bottle can be completely mixed in a few minutes if the solution is stirred. This accelerated mixing is not due to diffusion alone, but to the combination of diffusion and stirring. Diffusion still depends on random molecular motions that take place over smaller distances. The agitation or stirring is not a molecular process, but a macroscopic process that moves portions of the fluid over much larger distances. After this macroscopic motion, diffusion mixes newly adjacent portions of the fluid. In other cases, such as the dispersal of pollutants, the agitation of wind or water produces effects qualitatively similar to diffusion; these effects, are called dispersion.

Fick’s Law of Binary Diffusion25 The description of diffusion involves a mathematical model based on a fundamental hypothesis or ‘‘law’’. Imagine two large bulbs connected by a long thin capillary (Figure 3-1). Both of bulbs are at the

25

“Diffusion, Mass Transfer in Fluid Systems”, E. L. Custler.

172

same pressure and temperature but are filled with two different gases (N2 upper bulb and CO2 lower one). The concentration of the carbon dioxide in the initially N2 filled bulb is measured with time to find how fast these two gases will mix. The concentration of CO2 varies linearly with time (Figure 3-1). So the amount of CO2 transferred could be determined from this graph at each time step. Carbon dioxide flux is defined as follows:

Figure 3-1 - Simple diffusion experiment

We can assume that the flux is proportional to the gas concentration difference and we can recognize that increasing the capillary tube length will decrease the flux, so:

(

)

3-1)

The new proportionality constant D is the diffusion coefficient. Its introduction implies a model for diffusion, the model often called Fick’s law. There is a similarity between Fick’s law and Ohm’s law for flux of electrons: (

)(

)

The diffusion coefficient D is proportional to the reciprocal of the resistivity. So the general form of Fick’s law is (3-2) Where, j is the diffusion flux, and the minus appears because of the opposite directions of diffusion flux and concentration gradient. ⁄ . Since Fick's law is derived for molecules moving in From eq. (3-2), we see that D has units Brownian motion, D is a molecular diffusion coefficient, which is called Do to be specific. The intensity (energy and freedom of motion) of these Brownian motions controls the value of D. Thus, D depends on the phase (solid, liquid or gas), temperature, and molecule size. It should be considered that the Fick’s Law could not be applied when the capillary is very thin or two gases react. Parallel to Fourier’s law for heat conduction Fick’s second law is developed as; 173

(

)

(3-3)

for one dimensional unsteady state diffusion, and for the constant area, A, it becomes the more known Fick’s second law equation: (

)

(3-4)

This equation can be applied only for isotropic media and when the potential for diffusion is only given by concentration gradients. The diffusion coefficient is also independent of concentration.

Example 3-1 Find the diffusion flux and concentration profile in a steady diffusion across a thin film.

Solution The objective is to determine how much solvent moves across the film and how the solvent concentration changes within the film. On each side of the film is a well-mixed solution of one solvent. The solvent diffuses from the fixed higher concentration, located at xl on the right-hand side. As the first step mass balance on a thin layer located at some arbitrary position x within the film is written: (

)

(

)

C0

(

)

x

Cl x

Figure 3-2 – Diffusion across a thin film

Because the process is in steady state, the accumulation is zero. The diffusion rate is the diffusion flux times the film’s area A. Thus (3-5) Divide this equation by the layer volume: →

(3-6)

Combining eq. (3-6) with Fick’s law equation (3-2) yields the following equation: (3-7)

174

Where C = solvent concentration in the layer. There are two boundary conditions for this differential equation: (3-8) (3-9) Analytical solution of eq. (3-7), according to these two boundary conditions, will be the concentration profile: (3-10) The resulted solution for the concentration profile shows that the profile is independent of the diffusion coefficient. Base on the Fick’s law the diffusion flux can be found by differentiation of concentration profile: (3-11)

Diffusion Coefficient The diffusion coefficient was introduced as a proportionality constant, the unknown parameter appearing in the Fick’s law. Often Do is used as the molecular diffusion nomenclature. Mass fluxes and concentration profiles in many situations can be found using Fick’s law equation and, most of the time, the results contained the diffusion coefficient as an adjustable parameter. There exist four important definitions of the diffusion coefficient depending on the nature of the diffusion process26: Overall Diffusion Coefficient (Mutual Diffusivity): It is denoted by DAB, and it refers to the diffusion of one constituent in a binary system (A and B). For liquids, it is common to refer to the limit of infinite dilution of A in B using the symbol, D°AB. Assume a bitumen-solvent vertical system where solvent is on the top of the bitumen. The diffusion rate of solvent through a unit area is described by eq. (3-2) as follows: (3-12) Where, = Diffusion coefficient of solvent, Cs = Solvent concentration, y = Vertical direction. As solvent diffuses into the bitumen, a diffusion flux occurs for oil molecules toward the solvent: (3-13)

26

Perry et al., 1999.

175

Where, = Diffusion coefficient of solvent, Cb = Solvent concentration. According to the volume conservation in the system, the diffusion flux of bitumen should be equal to the solvent flux, but in opposite direction: (3-14) Also: (3-15) From eqs ((3-13) to (3-15)): (3-16) The overall diffusion coefficient Dbs is a material property that describes the mobility of either component in the mixture. Self Diffusion Coefficient: It is denoted by DA’A and is the measure of mobility of a species in itself. Also when we have a binary system where A and B are the less mobile and more mobile components respectively, their self-diffusion coefficients can be used as rough lower and upper bounds of the mutual diffusion coefficient in some systems. That is, DA’A < DAB < DB’B. Intrinsic Diffusion Coefficient: When there is a considerable difference between the molecule size of solvent and oil, for example in the previous system of solvent-bitumen, during diffusion of solvent into bitumen (in the previous system), the small solvent molecules tend to diffuse faster than the large molecules of bitumen. The intrusion of solvent molecules into bitumen causes a buildup of pressure in bitumen part. This pressure gradient causes a bulk flow of bitumen solution toward the solvent-rich part. In this situation the net flow of solvent is the sum of the diffusional flux of solvent (

) and

the flux of solvent displaced by bulk flow ( ), which ‘u’ is measured in the direction of ‘y’. According the volume balance solvent, net flow should be equal to the bitumen net flow with diffusion and bulk flow: (3-17) Where, Bitumen Intrinsic Diffusion Coefficient, Solvent Intrinsic Diffusion Coefficient, Bulk Flow Velocity Intrinsic diffusion coefficient is dependent on the concentration as the overall diffusion coefficient. Bulk flow velocity could be found from eq. (3-17):

176

(3-18) When there is not much difference between the molecule sizes of solvent and oil the net flow velocity is almost zero, because of the equality between intrinsic diffusion coefficient of oil and solvent. Although for solvent-bitumen system which intrinsic diffusion coefficient of solvent is higher than bitumen, because of smaller molecular size, bulk flow will be most significant. Eq. (3-19) describe the volume conservation of solvent in a solvent-bitumen system: (

)

(3-19)

But the net flux of solvent was found with eq. (3-12). Equal eq. (3-12) with eq. (3-19), with attention to eq. (3-16) to find the following equation: (3-20) Substituting eq. (3-18), for ‘u’, into the eq. (3-20) will be ended with the following relation between the overall and intrinsic diffusion coefficient: (3-21) So the overall diffusion coefficient value is equal to the intrinsic diffusion coefficient of solvent in pure bitumen (Cs = 0) and intrinsic diffusion coefficient of bitumen in pure solvent (CB = 0). Ds and Db are concentration dependent as Dbs, but there is no way to find the equation of this dependency other than with experiment. Most of the time a linear relationship between the intrinsic and concentration is assumed. The minimum value of the solvent intrinsic diffusion coefficient is the overall diffusion coefficient when solvent concentration is zero and the maximum is solvent self-diffusion coefficient, and the same for bitumen. By knowing the minimum and maximum of the intrinsic diffusion coefficient of bitumen and solvent and assumption of a linear relationship between these coefficients and concentration a relation between overall diffusion coefficient and concentration could be found using eq. (3-21). Figure 3-3 shows a result for such calculation.

177

Diffusivity (cm2/s * 1e6)

6

DS DBS

4

2

0

D

0.25

0.50

CB

0.7 5

1.00

Figure 3-3 – Prediction overall diffusion from intrinsic diffusion

Regarding eq. (3-21) and Figure 3-3, the overall diffusivity reaches a maximum at an intermediate concentration. During the experimental measurement of diffusion coefficient, what we find is the overall diffusion coefficient. Tracer Diffusivity: Denoted by DA’B is related to both mutual and self-diffusivity. It is evaluated in the presence of a second component B, using a tagged isotope of the first component. In the dilute range, tagging A merely provides a convenient method for indirect composition analysis. As concentration varies, tracer diffusivities approach mutual diffusivities at the dilute limit, and they approach selfdiffusivities at the pure component limit. In some cases the diffusion coefficient can be reasonably taken as constant (e.g. dilute solutions) while, in some others it depends very markedly on concentration.

The diffusion coefficient as a function of concentration27 The diffusion coefficient can be considered constant in many occasions especially when we refer to the diffusion of gases. However when we move to complex and denser fluids the assumption of constant diffusion coefficient may become unreal. The assumption of constant diffusion coefficient for the solvent/heavy oil or bitumen systems should be based on three important conditions that need to be fulfilled in order to support the hypothesis. If one of the three conditions is not fulfilled it is highly expected that the diffusion coefficient will be function of concentration. The three conditions are: Dimensions and shape: The molecular diameter and molecular shape should be similar for the diffusing components. That means in this case that the molecular diameter and shape of the solvent should be similar to those of the heavy oil and bitumen. We know that large hydrocarbon chains are present in heavy oil and bitumen. Thus the assumption of similar size and shape is not valid especially when we

27

“The Diffusion Coefficient of Liquid and Gaseous Solvents in Heavy Oil and Bitumen”, U. E. Guerrero-Aconcha

178

take into consideration that the solvents used in the recovery process are light gases and small hydrocarbons molecules. Molecular interactions: The molecular interactions between the diffusing components should be negligible. That means the attraction and repulsion forces should not interfere in the diffusion process. However, it has been shown that the repulsive forces play the most important role in the diffusion process. Non reacting environment: There should be a non-reacting environment in the system. That means no transformations of any kind due to the components on the system and/or the system conditions (pressure and temperature). However the interaction of solvent with heavy oil or bitumen may in some cases cause organic deposition, mainly asphaltenes. Based on the concentration dependency of the diffusion coefficient Fick’s second law is modified as follows: (

)

(3-22)

Example 3-2 Proof eq. (3-22)

Solution Assume a control volume like as Figure 3-4. There is no source or sink in the control volume. Solvent diffuses in from left hand side and go out by diffusion from right hand side. Write material balance for solvent in over the control volume:

(

)

(

)

(3-23)

Where: V = Volume, A = Area available for diffusion, C = Solvent concentration, D = Diffusion coefficient, = incremental time.

( 𝐷

𝑉

𝜕𝐶 𝐴 ) 𝜕𝑥 𝑥

𝐴

𝑥 ( 𝐷

x

𝐴

𝜕𝐶 ) 𝜕𝑥 𝑥

𝑥

x+ 𝑥

Figure 3-4 – Diffusion process in a control volume with a concentration dependent diffusion coefficient

179

According to the dependency of diffusion coefficient on the concentration , because we have a diffusion process and the potential for this process is concentration difference between x and x+ . With a constant area: . Now divide both side of eq.(3-23) by : (

→

)

(

(

)

)

Effective Diffusion Coefficient Experiments and field data show that the diffusion process in porous media is slower than that of two liquids adjacent to each other in a vial. This results from the fact that the diffusion coefficient in porous media is smaller than the bulk diffusion coefficient; therefore, an effective diffusion coefficient is proposed, which is based on the average cross-sectional area open to diffusion and the distance traveled by molecules in porous media.

For a bundle of straight capillary tubes, the effective diffusion coefficient and the bulk diffusion coefficient are the same. However, the straight capillary model is not a very good representation of a porous rock. A lot of work has been done to predict the apparent diffusion coefficient in porous media. Carman (1939) has shown that, in porous rock, fluids must move on the average at about 45o to the direction of flow (Figure 3-5). Hence, when fluid has traveled a net distance, L, it has traveled an actual average distance of about √

. On the other hand he assumed a tortuosity ( )

equal to √ for the porous medium. Carman proposed the following ratio between the bulk and effective diffusion coefficients:

√

(3-24)

Where, Deff = Effective Diffusion Coefficient, D = Molecular Bulk Diffusion Coefficient. Here 0.707 is a correction for the actual diffusion process length. On the other hand in the porous medium the cross section area available for diffusion is not the total cross section of the medium. By assumption of the equality between areal and volume porosity it can be concluded that only a portion (equal to the porosity) of the total cross section of the medium is available for diffusion process. According to this assumption Penman (1940), after some experiments on a packing of spherical particles, proposed the next equation to relate the bulk and effective diffusion coefficient to each other:

180

(3-25) Where, is the porosity. In this formula, 0.66 is the correction for real length of diffusion process and is the correction for actual area available for diffusion.

Net Path Real Path Figure 3-5 – In a porous medium fluid generally flowing at about 45o with respect to average direction of flow

On the other hand grain shapes and sizes of a porous medium are not homogeneous all the time so the assumption of constant values such as 0.707 or 0.66 is not reasonable to use as the inverse of the tortuosity, so it could be more sophisticated to use the tortuosity in the formula instead of a constant value: (3-26) A more sophisticated and comprehensive approach was suggested by Brigham, Reed et al. (1961) and van der Poel (1962). These investigators recognized that there is an analogy between diffusion and the electrical resistivity factor with the following formula: (3-27) Where, F is the formation resistivity factor. Diffusion coefficient in the absence of the porous media, is sometimes called bulk diffusion coefficient in contrast to the effective diffusion coefficient at the presence of the porous medium.

Importance in petroleum engineering In 2004, Alberta’s Oil Sands were recognized by international media, for the first time, as part of global oil reserves. This established Canada as second only to Saudi Arabia in the hierarchy of potential oil producing nations. While oil sands extraction is more expensive than conventional sources, continuing technological advances are reducing the importance of those cost differences. Moreover, conventional oil production in Canada is declining, underscoring the importance of the oil sands as a vital source of North American supplies (Timilsina et al., 2005). To recover these resources steam injection is widely used for heavy oil and bitumen reservoirs. The advantage of the process is its high recovery factor and its high oil production rate. However, the high 181

production rate is associated with excessive energy consumption approximately 1 million BTU/barrel, CO2 generation, and expensive postproduction water treatment. Additionally steam injection has operational restrictions that do not allow its application in all types of reservoirs. In order to overcome the problems associated to steam injection additional techniques have been developed to recover the heavy oil and bitumen. Among those techniques the vapor injection process (VAPEX), the cyclic solvent injection and the co-injection of steam and solvent (SAS, SAP, ES-SAGD and LASER) are the ones with the most promising future, thanks to the viscosity reduction of the oil phase, the change in absolute and relative permeability and the upgrading of the oil phase. The above processes involve the injection of solvent into the oil reservoir. The objective of the solvent is mainly to reduce the viscosity of the heavy oil or bitumen by mixing with it. This mixing process is a mass transfer process and its velocity is controlled by the diffusion coefficient. Therefore the diffusion coefficient is one of the most important parameters for the proper characterization of the solvent based recovery processes. Accurate diffusion data for these processes are necessary to determine: -

The amount and flow rate of solvent required to inject into a reservoir, The portion of reserves that have been affected by the solvent undergo viscosity reduction, The time required by the reserves to become less viscous and more mobile as desired, The rate of live oil production from the reservoir.

To find the diffusion coefficient value we must depend largely on experimental measurements of these coefficients, because no universal theory permits their accurate a-priori calculation. Unfortunately, the experimental measurements are unusually difficult to make, and the quality of the results is variable.

Measurement techniques It is noteworthy that there is no well-established and universally applicable technique for measuring the molecular diffusion coefficient. Unlike the measurements of viscosity or thermal conductivity, for which standardized techniques and equipment are readily available, the measurements of mass transfer characteristics are often more difficult due to difficulties in measuring point values of concentration and other issues which complicate this transport process. Considerable efforts were made to determine diffusion coefficient for diffusion of solvent in the oil experimentally. Different experimental methods can be classified into direct methods and indirect methods. Also there are some empirical correlations based on the experimental results that could be used under some condition to find the dispersion coefficient.

Direct Methods: Direct methods evaluate the diffusion coefficient by measuring concentration of the diffusing species (solvent) as a function of depth of penetration. Such methods are often more reliable and include the wide variety of physicochemical methods like mass spectrometry, radio-active tracer technique, spectrophotometry etc. The diffusivity is estimated by using compositional analysis techniques (Schmidt 1989). The drawbacks of direct methods are expense, time consumption and many of them are systemintrusive.

182

Indirect Methods: Indirect methods measure the changes of one of the system parameters that depend on the diffusion rate. These parameters could be the rate of change of solution volume or movement of the gas-liquid interface, rate of pressure drop in a confined cell which is known as pressure decay method, rate of gas injection from the top to a cell in which the pressure and solution volume are kept constant, magnetic field characteristics, computed tomography (CT) analysis and dynamic pendant drop analysis. The advantage of these methods is that they do not need to determine the change in composition. In this section, the more common and modern methods will be introduced: Pressure Decay Method: Among indirect methods, the pressure decay method has attracted more attention due to its simplicity in terms of experimental measurements. In this method, gas (as a solvent) and oil are injected into a cell (Figure 3-6). The cell content is initially at a non-equilibrium state. As the experiment progresses, the gas dissolve into the oil and the pressure inside the cell decreases as a result. By recording the pressure and the level of the liquid in the cell, the amount of gas transferred into the oil can be determined. From this, the diffusion coefficient is calculated. In cases involving complex hydrocarbon mixtures with possible multiphase behaviors, the pressure decay method fails. This method, however, had been discredited by Luo and Gu28. They showed that, minor changes to assumptions related to boundary conditions led to orders of magnitude differences in reported values. This method was first applied by Riazi29 and for dissolution of methane in n-pentane. He took the nonequilibrium gas into contact with n-pentane in a sealed container at a constant temperature. He determined the final state by thermodynamic equilibrium. However, the time which was required to reach the final state was determined from the diffusion process in each phase. He assumed that at the gas-liquid interface, thermodynamic equilibrium exists between the two phases at all times. However the position of the interface as well as the pressure may change with time. The rate of change of pressure and the interface position as a function of time depends on the rate of diffusion in each phase and therefore on the diffusion coefficients. P

T = const. Lg

Gas LT

Lo

Oil

Figure 3-6 – Pressure decay test cell

28 29

Luo, P. and Y. Gu, Fluid Phase Equilibria, 2009 Riazi, M. R., J. Pet. Sci. Eng. 1996

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Other researchers have proposed and developed different mathematical solutions by modeling the interface boundary condition differently. Modeling the physics of the interface (when the pressure is declining) often requires complex mathematical solutions, and it is known that more simplified analysis based on assumption of constant equilibrium concentration at the interface introduces significant error in the estimation of the diffusion coefficient. To overcome some of these shortcomings different researcher used different models. Upreti and Mehrotra30 improved the pressure decay method to find concentration dependent diffusion coefficient. Their experimental apparatus primarily consists of a closed cylindrical pressure vessel used to hold gas over a layer of bitumen. A pump supplied the gas as a solvent at desired pressure into the vessel. The vessel was submerged in a constant temperature water bath, the pressure vs. time data were recorded by a pressure sensor. Using the logged pressure versus time data, the experimental mass of a given gas diffused into the bitumen was determined. The gas diffusivity was then calculated by fitting the calculated mass of the gas diffused into bitumen (given by a mass transfer model) to the experimental mass. The tuning parameter was the diffusion coefficient. Additionally a correlation was provided for the average diffusivity as a function of temperature. Refractive Index Method: Refractive index is the ratio of the velocity of wave propagation in a reference phase to that in the phase of interest. Normally, the refractive index used in diffusion measurement is taken as the ratio of the velocity of light in vacuum to the velocity of light in the relevant phase: (3-28) Where, c = velocity of light in vacuum,

 = velocity of light in the relevant phase, and n = refractive index. With the velocity of light in vacuum chosen as the reference, the refractive index is always greater than 1. For example, the refractive index of water is 1.33, meaning that light travels 1.33 times as fast in vacuum as it does in water. As light moves from a medium, such as air, water, or glass, into another it may change its propagation direction in proportion to the change in refractive index.

Figure 3-7 - Refraction of light at the interface between two media 30

Upreti, S., Lohi, A., Kapadia, R. and El-Haj, R., 2007

184

Eq. (3-29), Snell's law (also known as the Snell–Descartes law, and the law of refraction), is a formula used to describe the relationship between the angles of refraction and refraction index, when referring to light or other waves passing through a boundary between two different isotropic media, such as two fluid with different concentration of a solvent. (3-29) Where,

is the angle of refraction.

It is noted that, for a solution, different concentrations of a sample substance will lead to different refractive indices. As a result, from the angle of refraction, the concentration of a solution phase can be determined. Normally, in experimental measurements of refractive indices, a laser light is emitted through the diffusion cell and, according to the concentration of the solution at each elevation, the corresponding refractive angle of the laser beam is determined. As a result, the point at which the laser beam is captured by a CCD camera will represent the concentration inside the diffusion cell at that specific elevation. Figure 3-8 shows a sample picture of CCD during the diffusion process.

(a)

(b)

Figure 3-8 - Sample of light refraction results a) initial time b) after diffusion occurred

It is noted that the above method is suitable only for transparent fluids. As heavy oil, even when diluted, is opaque, such a method cannot be employed. NMR Method: Nuclear Magnetic Resonance (NMR) was mainly developed for chemical-physical-medical use. The principle of this method is to calculate the density of hydrogen protons. NMR is a phenomenon, which occurs when the nuclei of certain atoms are immersed in static magnetic field and exposed to a second oscillating magnetic field. Some nuclei experience this phenomenon, and others do not, depending on atomic composition. Based on the nature of magnetic resonance, NMR measurement can be made on any nucleus that has an odd number of protons or neutrons or both, such as the nucleus of hydrogen (H), carbon (C), and sodium (Na). For most of the nuclei found in earth formations, the nuclear magnetic signal induced by external magnetic fields is too small to be detected with NMR device such as borehole NMR logging tool. However hydrogen as the main atom of water and other hydrocarbon molecules produces a strong signal. So the strength of the signal could be used as a scale of the existing hydrogen.

185

Principles of NMR and Processing31,32 Polarization Process: The nucleus of the hydrogen atom is a proton, which is small, positively charged particle with an associated angular momentum or ‘spin’. The spin of this proton causes the proton to behave like a tiny magnet with a north and south poles (Figure 3-9, a). In the absence of an external magnetic field, the hydrogen nuclear spin axes are randomly aligned (Figure 3-9, b). This results in a net magnetization of zero. In the presence of an external magnetic field, the nuclear spins attempt to line up with the field either parallel or anti parallel to the net magnetic field (B0). According to quantum mechanics, the proton in a net magnetic field is forced into one of two energy states, high-energy or low-energy state. The protons that their processional axes are parallel to the net magnetic field are in the low energy state, which is the preferred state. On the other hand the protons are in the high-energy state when their processional axes are anti-parallel to the magnetic field.

+

N

S (a)

(b)

Figure 3-9 - (a) Hydrogen nuclei behave as a tiny bar magnets aligned with the spin axes of the nuclei. (b) Spinning protons with random nuclear magnetic axes in the absence of an external magnetic field.

The difference between the numbers of protons with high and low energy level produces the bulk magnetization ‘M’, which provides signal measured by NMR devices (Figure 3-10). The bulk (Macroscopic) magnetization ‘M’ is defined as the net magnetic moment per unit volume.

31 32

Behnaz Afsahi, M.Sc. Thesis, 2007 NMR logging, Principle and Application, G.R. Coates, L. Xiao, and M. G. Prammer, Halliburton Energy Services

186

+ + +

+

+ + + + + + + +

+ M=0

+ +

+ + +

+ + +

+

B0 M

+ +

Figure 3-10 – Line up nuclear spins in an external magnetic field

M is measurable and is proportional to the number of protons, the magnitude B0 of the applied magnetic field, and the inverse of the absolute temperature. After the protons are exposed to the static external magnetic field (B0), they are said to be polarized. Polarization does not occur immediately but rather grows with a time constant, which is the longitudinal relaxation time, T1: (

)

(3-30)

Where, t = the time that the protons are exposed to the B0 field, The magnitude of magnetization at time t, when the direction of B0 is taken along the z axis The final and maximum magnetization in a given magnetic field T1 is the time at which the magnetization reaches 63% of its final value, and three times T1 is the time at which 95% polarization is achieved. Figure 3-11 is a T1 relaxation or polarization curve. Different fluids, such as water, oil, and gas, have very different T1 relaxation/polarization times. Relaxation definition will be illustrated later. Tipping Process: After exposing the protons to the net static magnetic field (B0), apply an oscillating magnetic field (B1) perpendicular to B0, therefore, the magnetization M will precess farther and farther away from the zaxis. This process is called ‘Tipping’. According to the quantum mechanics point of view, if a proton is at the low-energy state (its processional axes are parallel to the net magnetic field), it may absorb energy provided by B1 and jump to the high-energy state. The application of B1 also causes the protons to precess in phase with one another. This change in energy state and in-phase precession caused by B1 is called nuclear magnetic resonance.

187

M(t)/M0

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

PolarizationTime / T1 Figure 3-11 – Polarization/Relaxation curve

The magnetization vector M can be thought of as having a component MZ along the z-axis (direction of B0), which called longitudinal magnetization, and a component Mxy perpendicular to the field named transverse magnetization. At equilibrium Mxy = 0 and MZ = M. when the protons are exposed to the oscillating magnetic field B1, MZ decreases and Mxy increases. The angle of deflection or rotation of sample’s net magnetization depends primarily on the product of amplitude (energy) of the B1 and the length of time that B1 is applied: (3-31) Where, Tip angle, B1 = Amplitude (energy) of the oscillating field, Time over which the oscillating field (B1) is applied The gyromagnetic ratio (measure of the strength of the nuclear magnetism) Each nucleus has a fixed

value. For hydrogen: ⁄

MHz/tesla.

Angular pulse terms such as π pulse (180o pulse) or a π/2 pulse (90o pulse), refer to the angle through which magnetization is tipped by B1 (Figure 3-12).

B1

M

𝜃

M

θ

𝑜

𝑜

𝜃

M

M M

τ

o

90 Pulse

M

o

180 Pulse

Figure 3-12 – the Tipping process

188

Relaxation Process: When the B1 field is turned off, the net magnetization decreases and system gradually returns to equilibrium. During this process, the protons gradually lose their extra energy and return to equilibrium by emitting radio waves and by transferring energy to surrounding molecules. The processes by which nuclei transfer energy to their surroundings to return to equilibrium state are called relaxation. The relaxation processes are exponential in time and are described by two time constants, T1 as the longitudinal magnetic relaxation time constant and T2 as the transverse magnetic relaxation time constant. These two constant values are seldom equal. Transverse relaxation is always faster than longitudinal relaxation; consequently, T2 is always less than or equal to T1. In general, for protons in solids, T2 is much smaller than T1. T1-Longitudinal relaxation time As the protons absorbed energy from B1, lift up to the high-energy state during T1 relaxation, any given spin can return to the ground state only by dissipating excess energy to the surrounding (lattice) (Figure 3-13). Therefore this process also called spin-lattice relaxation. During the T1 time the z-component of M returns to 63% of its original value.

(1)

(2)

(3)

(4)

Figure 3-13 – Net magnetization return to equilibrium by turning off the B1, (the arrow represent net magnetization)

T2-Transverse relaxation time During T2 relaxation, no energy is exchanged from the nuclei to lattice. Exchange of energy happens among nuclei. Therefore T2 also called spin-spin relaxation. Transfer relaxation corresponds to the loss of phase coherence or randomization of spins in transverse direction (x-y) direction, which causes the loss of transfer magnetization. T2 refers to the time required for the transvers component Mxy to decay to 37% of initial value. A 90o pulse B1 gives energy to the protons and M rotates entirely into the x-y plane (Figure 3-14). Coherence now exists in this transverse plane at the end of the pulse. The protons are all synchronized and precess at the same frequency. A transfer of energy can occur between these protons. Spin-spin relaxation refers to this energy transfer from an exited proton to a nearby unexcited proton. This energy exchange produces a gradual loss of phase coherence to the spins. As the coherence gradually disappears, the value of Mxy decreases toward zero (Figure 3-14). This loss of coherence is a consequence of T2. T2 relaxation is more efficient in large molecules since they reorientate more slowly than small molecules.

189

Mxy

(1)

(2)

(3)

(4)

Figure 3-14 – de-phasing (loss of phase coherence) during T2

When a wetting fluid fills a porous medium, such as a rock, both T2 and T1 are dramatically decreased, and the relaxation mechanisms are different from those of the protons in either the solid or the fluid. There are many different properties of the fluid and porous media that could be measured or explained by using the relaxation process and (T1, T2) values. Spin-Echo and CPMG pulse sequence Once the 90o B1 pulse is turned off, the proton begin to de-phase or lose phase coherency in B0 (Figure 3-14). As the net magnetization in the transverse plane decreases, a receiver coil that measures the magnetization in the transverse direction could detect a decay signal in this situation. If the magnetic field was really homogeneous (the amplitude is not a function of x, y or z), the signal would decay with a time constant T2. However, since the B0 has some inhomogeneity, the signal actually decays faster with the time constant T2*, which called Free Induction Decay (FID). The FID is very short, which is lasting a few milliseconds. Consequently in the small time interval between the two pulses, very little T1, some T2 de-phasing and substantial T2* occurs. The de-phasing resulting from T2* occurs at a constant rate since it arises from the spatial inhomogeneity of the magnetic field. T2 de-phasing on the other hand fluctuates randomly since it results from the interaction among the nuclei themselves. This type of de-phasing provides valuable sample information. In order to measure T2, the signals must be recombined. It can be done by applying an 180o pulse after the 90o pulse (after τ ms) to re-phase the proton magnetization vectors in the transverse plan (Figure 3-15). In effect, the phase order of the transverse magnetization vectors is reversed, so that the slower vectors are ahead of the faster vectors. The faster vectors overtake the slower vectors, rephrasing occurs, and a signal is generated that is detectable in the receiver coil. This signal is called spin echo. The echo time (TE) defined as the time between the 90o pulse and the re-phasing completion, which is 2τ.

Mxy o

Applying 90 B1 pulse at time 0.

190

De-phasing after turning off the B1

At time τ ms, a 180° B1 pulse is applied to reverse the phase angles and thus initiates re-phasing.

Re-phasing is complete, and a measurable signal (a spin echo) is generated at time 2τ ms.

Figure 3-15 – Spin-echo sequence

Only a single echo decay very quickly. One way for determining T2 from spin echo amplitudes is by repeating the spin echo method several times with very time τ. In CPMG method a series of 180o pulse are applied at intervals τ, 3τ, 5τ, 7τ, etc., following the 90

o

pulse. Echoes are then observed to form at times 2τ, 4τ, 6τ, 8τ, etc. because the de-phasing resulting from molecular interactions the protons can no longer be completely refocused, and the CPMG spin-echo train will decay. On multiple repetitions of the 180o pulse, the height of the multiple echoes decreases successively as a consequence of T2 de-phasing (Figure 3-16).

191

Received Signal Amplitude

TE = 2τ

o

90

o

180

2τ

o

4τ

180

6τ

o

8τ

o

180

180

180

o

10τ

Time (ms)

Figure 3-16 – CPMG pulse sequence

Amplitude

M0

0

50

100

150

200

250

Time (ms) Figure 3-17 - The amplitudes of the decaying spin echoes yield an exponentially decaying curve with time constant T2

Typical NMR Experiment: The above theory for an entire NMR experiment involves these procedures: the first step is to polarize the hydrogen protons in an external magnetic field, B0. The second step is to tip the magnetization from the longitudinal direction to a transverse plane by applying an oscillating magnetic B1 perpendicular to B0. The last is to measure their true relaxation back to equilibrium polarized direction. There are four important parameters to control these procedures: the length of the echo time (TE), the number of 1800 pulse (NE), the waiting time (TW) to re-polarize protons along B0 and the number of trains. As TE decreases, spin echoes will be generated and detected earlier and more rapidly, and effective signal-to-noise ratio is increased because of the greater density of data points. As TE increase, spin echoes will be generated and detected longer, but more B0 power is required. The first two parameters determine the length of a single NMR experiment. Waiting time is the time between the ends of one CMPG sequence to the start of the next CMPG sequence. The set up the waiting time depend on the sample. In general, if we simply try to capture the full signal from all the protons present in the sample, the wait time must be set long enough that all protons returns back to their equilibrium position. Then number of trains simply as how many times the entire process is repeated. All four parameters can be 192

controlled manually. Generally the more times experiment is repeated, the better the results should be. Usually for pure fluid, such as water, which only contains a single relaxing species, it is not necessary to use too many trains. For the fluid, such as crude oil, which contains several different relaxation hydrocarbon species, it is good to run more trains to get better results. One of the most important steps in NMR data processing is to determine the T2 distribution that produces the observed magnetization. This step is called echo-fit or mapping, is mathematically inversion process, the total measured NMR signal is inverted from a decay curve by Echo-fit software to give T2 spectrum (Figure 3-18).

Porosity

Echo Amplitude

In general, the strength of a received signal is directly proportional to the number of hydrogen protons present, which can be correlated to the amount of fluid present. The relaxation time T2 is a function of the viscosity of the fluid, or the confinement of the space where molecule is relaxing. For example heavy oil and bitumen are made of complex chains containing branches and rings, so the amount of hydrogen present in a given mass of oil will generally be less than the hydrogen present in the same mass of water. Generally the total amplitude (At) of a bitumen sample will be less than that of water of the same mass. On the other hand T2 value for heavy oil is much less than its value for water because of high viscosity of heavy oil.

Echo Time (ms)

T2 Distribution

Figure 3-18 - The echo train (echo amplitude as a function of time) is mapped to a T2 distribution (porosity as a function of T2)

In a NMR experiment, several parameters of interest are measured. The first is the total signal amplitude, which is the amplitude of both oil and solvent components. Amplitude refers to the measurable hydrogen protons in the sample and it is proportional to the sample mass. The second is amplitude index, which is simply the measurable NMR amplitude per gram of bulk fluid. Knowing the amplitude index allows for the amplitude of any given sample to be correlated to the sample’s mass: (3-32) Where, AI = amplitude Index, AP = amplitude of the fluid signal, m = mass of the fluid.

193

Not all NMR machines are operated at the same signal gain, so the value of oil amplitude index could vary from machine to machine. In order to normalize the measured amplitude index, the term relative hydrogen index is defined33: (3-33) Where, RHI = relative hydrogen index. Since oil and bitumen consist of different components, each with different relaxation rate, their signal will be detected with a broader relaxation range compared to pure substances with a single relaxation time like water. In other words the protons in crude oil do not all relax with a single value of T2, yielding a variety of relaxation times. The characteristic time for the oil relaxation is the geometric mean T2gm, of the oil spectrum34: (

∑ ∑

)

(3-34)

Where, The geometric mean T2 value (ms), The T2 value of a component in the mixture (ms), The amplitude of a component at a time constant T2i. For a fluid like oil, which consist of multiple components, all the components.

represents the mean relaxation rate for

Application of Low Field NMR in Diffusion Measurements35 The basis for using NMR is the fact that the relaxation spectra of a heavy oil or bitumen are distinctly different than those of a solvent. Figure 3-19 shows the spectra of pure viscous oil, of pure solvent, and of a mixture of solvent with bitumen that was prepared by mild heating and stirring. As mentioned before the amplitude indices of the bitumen signals (AIb) are lower than the amplitude indices of the signals from the solvent (AIS). It is because of the large molecular size of bitumen that lead to lower amount of hydrogen in unit volume compare to the solvent with smaller molecular size. On the other hand relaxation time for bitumen is much faster than the relaxation time for solvent because of high viscosity of bitumen. It can be seen that the spectrum of the mixture is distinctly different than the spectra of the individual components. This distinction forms the basis of the methodology used to study the diffusion of solvent in heavy oil using NMR.

33 34 35

Y. Wen, M. Sc thesis, 2004. B. Afsahi, A. Kantzas, 2007. Y. Wen, J. Bryan, A. Kantzas, 2005.

194

T2c

Figure 3-19 – Typical NMR spectrum for pure bitumen, pure solvent, and a mixture of them

From Figure 3-19, it is evident that during the diffusion of a solvent in bitumen the solvent spectrum shifts to faster relaxation times by increasing its viscosity during the mixing, while the bitumen spectrum shifts to longer relaxation times by decreasing of its viscosity. Two separate peaks of pure heavy oil/bitumen and solvent become one continuous multi-peak spectrum. It is assumed that the amount of solvent in bitumen is proportional to the expansion of the spectrum of the bitumen component (see Figure 3-19). If this assumption is correct, then there will be a critical relaxation time that will split the mixture spectrum into two components. Spectrum peaks with relaxation times faster than the critical relaxation time will correspond to bitumen and diffused solvent. Spectrum peaks with relaxation times slower than the critical relaxation time will correspond to pure solvent that has not yet diffused in bitumen. It then becomes important to define this critical relaxation time, T2c. T2c is defined as the maximum relaxation time observed in a thoroughly mixed solvent/bitumen mixture, which is shown in Figure 3-19. Usually, the spectra of heavy oil and bitumen are less than 10 ms, the spectra of solvents are beyond 1,000 ms, and the spectra of the mixtures of solvent with heavy oil or bitumen extend to some intermediate value. For bitumen-solvent system of Figure 3-19, the signal under the 100 ms was considered to be solvent diffusing in the bitumen, in other words the signal amplitude change up to 100 ms was attributed to solvent mass transfer. The signal beyond the 100 ms was considered to be pure solvent. Prior to starting the experiment, mixtures of known percentages of bitumen and solvent are prepared and the correlation between the NMR parameters and the concentration of bitumen in the mixture is determined (NMR parameters calibration with bitumen content). These correlations were used to calculate the concentration of the solvent that diffused in the heavy oil or bitumen during the test (Figure 3-20). (3-35) Where, Mass of diffused solvent, Bulk solvent signal amplitude, Pure solvent amplitude Index. 195

1

Ln[1/(RHI * T2gm)]

1/(AI * T2gm)

100 10 1 0.1 0.01 0.001

-1

y = 4.5954x2 + 2.5996x - 7.3702

-3 -5 -7 -9

0

0.2

0.4

0.6

0.8

1

Bitumen Fraction

0

0.2

0.4

0.6

0.8

1

Bitumen Fraction

Figure 3-20 – two samples of NMR calibration for bitumen-solvent mixture36,37

The mass of diffused solvent could be calculated as: (3-36) Where, Mass of initial pure solvent, Mass of diffused solvent. The total amplitude of diffused solvent is calculated using the following relation: (3-37) Where, Total pure solvent amplitude, Total bulk solvent amplitude. The combined amplitude index for the diffused solvent and bitumen/oil is calculated, using amplitude index definition: (3-38) Where, Signal amplitude of diluted bitumen, Mass of pure bitumen, Mass of diffused solvent, Amplitude index of diluted bitumen.

36 37

Y. Wen, J. Bryan, and A. Kantzas, 2003. D. Salama and A. Kantzas, 2005.

196

As the next step, geometric mean relaxation time for the diluted bitumen, T2gm, is calculated according to eq. (3-34). The solvent content (concentration) could be calculated form the generated correlation between NMR properties and bitumen content (such as Figure 3-20) at different times. The concentration of solvent determined through NMR spectra change is the overall concentration, ̅ , in the mixture area, which is a function of time and diffusion coefficient. The correlation between concentration ̅ , time t, and D is described in the following equations: (3-39) ∫

∫

(3-40)

At given t: ∫ ̅

∫

(3-41)

∫

As a result of eq. (3-41), one can construct models of ̅ with D and t, and pursue estimates of the diffusion coefficient. In the NMR test, it is assumed that the D value is constant during each small time interval. So the following equation could be used for the determination of the diffusion coefficient: ̅

(

√

)

(3-42)

Where, Starting concentration, t = the time of measurement, x = the equilibrium distance for the early times (for example first two days) of measurements, D = the diffusion coefficient that gives the best fit. Since there is no ability to obtain the correlation between concentrations with distance from NMR spectra, an overall constant diffusion coefficient is considered from the analysis of NMR data. x is determined independently using X-Ray CAT scanning. The reason for using early time data is because the interface between solvent and heavy oil/bitumen can only be maintained stable for a short time. That was the observation made from CAT scanning experiments38. Figure 3-21 shows a sample result of diffusion coefficient calculated using NMR experiment. Afsahi and Kantzas (2007) used the same method to find the diffusion coefficient in sand-saturated sample. They mixed the bitumen with sand thoroughly in a way that the final mixture of sand and bitumen had almost 35 to 40% porosity (40% bitumen, 60% sand). Then the bitumen-sand mixture was placed in the vial and compacted completely. Solvent was placed on top of oil-saturated sand in the vial. In order to keep the sand in place after dilution of heavy oil by the solvent they used a nylon mesh to separate the oil-saturated sand from the solvent. NMR measurements were taken frequently and 38

Y. Wen, A. Kantzas, and G.J. Wang, 2004.

197

Diffusion Coefficient, 106 cm2/s

changes in the spectra were related to the change of oil and solvent properties as solvent diffused into the oil-sand matrix. Another step of the procedure was the same as for the bulk diffusion that explained earlier. 100 80 60 40 20 0 0

6

12

18

24

30

36

42

Time, hr Figure 3-21 – Diffusion coefficient as a function time, NMR experiment result

Another interesting usage of NMR method, such as pore size distribution or wettability determination, will be illustrated later. Computer-Assisted Tomography: Computer-assisted Tomography (CAT) scanning using X-ray has been extensively used in research laboratories around the world for reservoir rock characterization and fluid flow visualization. Principles of CAT Scanning and Processing39 X-rays lose their energy as they pass through a medium, and this reduction depends on the density of the substance and the path length through that substance. CAT is based on emitting x-rays from a source which revolves around the object in consideration while one-dimensional projections of attenuated x-rays are collected by a detector on the other side of the source. These projections are collected as the sample travels through the scanner longitudinally and are used to reconstruct a twodimensional image of the object. Intensity values of attenuated x-rays are collected from small volumetric elements, called pixels. These elements are typically 0.40×0.40 mm in area and 3 cm in depth (along the direction of the x-ray beam) for a second generation CAT scanner. Once these elements are all assigned an intensity values after a complete radial and longitudinal scan, these data are processed by a computer. This processing constitutes the major part of the CAT. The inlet intensity and the outlet intensity are related through the following relationship: (3-43) Where: I = The intensity remaining after the X-ray passes through a thickness (kV),

39

L. Song, A. Kantzas, J. Bryan, 2010.

198

Io = The incident X-ray intensity (kV), µ = Linear attenuation40 coefficient, L = Path Length. This relationship applies only for a narrow mono-energetic beam of x-ray photons which travels across a homogeneous medium. If the medium in consideration is heterogeneous, the above equation holds true while replaced by the line integral of the linear attenuation coefficients. The modified form is: ( )

∫

(3-44)

The following equation relates the linear attenuation coefficients to the number stored in computer (known as the CT numbers or CTn), (3-45) Where: CTn = CT number, x-ray linear attenuation coefficient of the object scanned, x-ray linear attenuation coefficient of water. Linear attenuation coefficient (µ) is a function of the bulk density and the effective atomic number of the sample, given by: (3-46) Where: Bulk oil density (kg/m3) a = Energy-independent coefficient called Klein- Nishina coefficient b = Constant Z = Effective atomic number of the sample E = Mean photon energy (kV) When exposing a medium to x-rays, gathering the exiting x-rays from the medium (Figure 3-22), and averaging the intensity at each cross section, a transmitted intensity vs. elevation curve can be constructed (Figure 3-23). The resulted curve could be converted to a density curve. According to the relation between the x-ray intensity and density.

40

In physics, attenuation (in some contexts also called extinction) is the gradual loss in intensity of any kind of flux (X-Ray) through a medium.

199

Figure 3-22 - Schematic view of CAT scanning using x-ray

A series of calibration tests for liquid and solid samples of known densities are performed in order to correlate the CT numbers (as the intensity of the detected x-ray), generated by the scanner, to densities. In Figure 3-23 the calibration curves for liquid samples and liquid/solid samples are shown respectively. Using these calibration curves, the densities of the scanned samples can be back calculated.

(a)

(b)

Figure 3-23 – Calibration curves for the CAT scanner, (a) Liquid calibration curve, (b)Liquid-solid calibration curve41

In contrast to the refractive index method, this method has the ability to operate with opaque solutions such as bitumen + pentane. Diffusivity investigation using CAT scanning Wen and Kantzas42 used this method to monitor the concentration profiles at a bitumen-solvent interface. Solution of the solvent in the oil dilutes the oil and change the linear attenuation coefficient. This is the basic idea of using the CAT scanning for diffusion process study. For a diffusion study, as the first step heavy oil is adjacent to the solvent in a fixed volume cell (Figure 3-24). Because of higher oil gravity, solvent is on the top of the heavy oil in the cell. A fixed vertical-sectional position of the diffusion cell is scanned at fixed frequency during the diffusion process. Figure 3-24 illustrates the typical CT scan image of solvent diffusing into the heavy oil.

41 42

D. Salama and A. Kantzas, 2005. Y. Wen, A. Kantzas, 2005

200

Figure 3-24 – Image sample of diffusion process

The two-dimensional CT image shows the diffusion process along the vertical direction (x axis) of length of the diffusion cell. A central area (region of Interest, ROI) is cut as shown by a dotted line in Figure 3-24. The ROI is used for diffusion coefficient calculation and analysis. This study considers the diffusion process as a one-dimensional vertical diffusion process. Therefore, only an average CT number in the horizontal direction is calculated to represent the CT number in the center, and then the profile of CT numbers change with vertical distance for each “x” value is obtained. The changes in the CT numbers are related to the changes of oil densities as a solvent diffuses into the heavy oil, and CT number has a linear relationship with density. Thus, the CT number profiles can be converted to the density profiles. In addition, the mixture (solvent and heavy oil) density has a linear relation with the solvent content in the mixture. Therefore, using the following equation (eq. ((3-47)), normalized concentration profiles could be obtained from the densities. (3-47) Where: Normalized Concentration (volume fraction), Bulk oil density (kg/m3), Density of the solvent close to the interface (kg/m3) Initial Density of the oil (kg/m3) By assumption of linear relationship between concentration and diffusion coefficient43 the Fick’s second law equation (eq. (3-22)) could be converted to the following equation: (

)

(3-48)

Using CAT scanning we have density at each point for any measurement time. This was the base idea for Guerrero-Aconcha and Kantzas44 to find diffusion coefficient as a function of concentration. They assume that the diffusion process is in the x direction and at each cross section there is a uniform 43 44

Upreti, S.R., Mehrotra, A.K., 2000. Guerrero-Aconcha, U., Kantzas, A., 2009.

201

concentration. They divided the diffusion length to several control volumes and discretized eq. (3-48) explicitly and apply discretized equation on the control volumes, as follows: (

⁄

(

))

(

⁄

⁄

(

))

(

(3-49)

)

In this equation only the diffusion coefficients are the unknown values. To solve this equation for the domain, two boundary conditions are needed. Guerrero-Aconcha and Kantzas assumed a constant concentration at the interface between oil and solvent (point A) as the first boundary condition and a no flow boundary surface at the end of medium (point B). Solvent Diffusion Direction

A

i+1

i

i-1 A

i

i-1 x/2

B

i+1 x/2

x Figure 3-25 – Medium Domain

According to these two boundary conditions a discretized equation for the first and last control volumes are as follows: Boundary A: (

⁄

(

))

(

(

⁄

))

(

)

(3-50)

Boundary B: (

⁄

(

⁄

))

(

)

(3-51)

By arranging the discretization equations within the medium domain and at the boundary surfaces, the Equations ((3-49), (3-50) and (3-51)) can be written in matrix form as Ax=b. The components of vector x are the unknown diffusion coefficients. The diffusion coefficients can be obtained by solving the system of linear equations. Effective Diffusion Coefficient Using CAT scanning give us the opportunity to find effective diffusion coefficient for a porous media. The procedure is the same as the procedure of finding bulk diffusion coefficient except that the vessel is half filled with oil saturated sand and topped with solvent. In contrast to the bulk diffusion coefficient that the region of interest (ROI) was in both solvent and oil region (Figure 3-24), here ROI includes only the liquid volume in the solvent region because of the complexity of obtaining a smooth concentration profile that could be used in the calculation of diffusion coefficients when including the sand volume 202

(Figure 3-26). The diffusion calculations were made based on the assumption that the amount of oil that diffused in the solvent is equal to the amount of solvent that diffused in the oil/sand mixture but in opposite direction. The calculation procedure is the same as for bulk diffusion coefficient.

Figure 3-26 - Sample of diffusion in sand saturated with oil

Diffusion Coeficient, 106 cm2/s

Figure 3-27 shows the result of CAT scanning experiment for bulk diffusion coefficient of three different solvents in heavy oil. Each point is the average diffusion coefficient over the height of the region of interest (ROI).

12

Pentain

10

Hexane

8

Octane

6 4 2 0 0

0.5

1

1.5

2

2.5

3

3.5

Time, Day Figure 3-27 - Average diffusion coefficients for pentane, hexane and octane in heavy oil10

Figure 3-28 compares the bulk diffusion coefficient and effective diffusion coefficient of Pentane in heavy oil. Figure 3-28 provide evidence that the diffusion coefficients of hydrocarbon solvents in bulk oil are higher than in presence of sand.

203

Diffusion Coefficient, 106 cm2/s

15

D in bulk

12

D in presence of sand 9 6 3 0 0

1

2

3

4

5

6

7

Time, hr Figure 3-28 - Comparison of the diffusion coefficients of pentane in heavy oil in absence/presence sand10

204

4 .

C H A P T E R

4

Immiscible Displacement Introduction Fluid displacement processes require contact between the displacing and the displaced fluid. The movement of the interface between displacing and displaced fluids and the breakthrough time associated with the production of injected fluids at producing wells are indicators of sweep efficiency. This chapter shows how to calculate such indicators using the Buckley-Leverett theory.

Buckley-Leverett Theory45,46 One of the simplest and most widely used methods of estimating the advance of a fluid displacement front in an immiscible displacement process is the Buckley-Leverett method. The Buckley-Leverett theory [1942] estimates the rate at which an injected water bank moves through a porous medium. The approach uses fractional flow theory and is based on the following assumptions: 

Flow is linear and horizontal



Water is injected into an oil reservoir



Oil and water are both incompressible



Oil and water are immiscible



Gravity and capillary pressure effects are negligible

In many rocks there is a transition zone between the water and the Oil zones. In the true water zone, the water saturation is essentially 100. In the oil zone, there is usually present connate water, which is essentially immobile. Only water will be produced from a well completed in the true water zone, and only oil will be produced from the true oil zone. In the transition zone both oil and water will be produced, and at each point the fraction of the flowrate that is water will depend on the oil and water saturations at that point. Frontal advance theory is an application of the law of conservation of mass. Flow through a small volume element () with length ∆x and cross-sectional area “A” can be expressed in terms of total flow rate qt as:

(4-1)

45 46

“Principle of applied reservoir simulation”, John R. Fanchi “Applied Petroleum Reservoir /engineering”, B.C. Craft, M. Hawkins, 1991

205

Where q denotes volumetric flow rate at reservoir conditions and the sub-scripts {o,w,t} refer to oil, water, and total rate, respectively and fw and fo are fractional flow to water and oil (or water cut and oil cut) respectively: (4-2) (4-3)

(4-4)

4-5)

is a function of saturation. So for constant viscosity fw is just a function of saturation. Figure 4-1 is a plot of the relative permeability ratio, range of

, versus water saturation. Because of the wide

values, the relative permeability ratio is usually plotted on the log scale of semi-log paper.

Like many relative permeability ratio curves, the central or main portion of the curve is quite linear. As a straight line on semi-log paper, the relative permeability ratio may be expresses as a function of the water saturation by: (4-6) The constants “a” and “b” may be determined from the graph, such as Figure 4-1, or determined from simultaneous equations from known data of saturation and relative permeability. Relative Permeability Ratio

1000 100 10 1 0.1 0.01 0.001 0

0.2

0.4 0.6 Water Saturation

0.8

1

Figure 4-1 – Semilog plot of relative permeability ratio versus saturation

Substituting eq. (4-6) into eq. (4-5) will end with:

206

(4-7)

Fractional Flow of Water, fw

If the water fractional flow is plotted versus water saturation, an S-shaped curve will result that is named fractional flow curve. 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Water Saturation Figure 4-2 – Fractional flow curve

Assume that the total flow rate is the same at all the medium cross section. Neglect capillary and gravitational forces that may be acting. Let the oil be displaced by water from left to right. The rate the water enters to the medium element from left hand side (LHS) is: (4-8) The rate of water leaving element from the right hand side (RHS) is: (4-9) The change in water flow rate across the element is found by performing a mass balance. The movement of mass for an immiscible, incompressible system gives: Change in Water Flowrate = water entering – water leaving = =

(4-10)

This is equal to the change in element water content per unit time. Let Sw is the water saturation of the element at time t. Then if oil is being displaced from the element, at time (t + t) the water saturation will be . So water accumulation in the element per unit time is: (4-11) Where,

is porosity. Equating equations (4-10) and (4-11) results:

207

(4-12) In the limit as ∆t → 0 and ∆x → 0 (for the water phase): (

)

(

)

(4-13)

The subscript x on the derivative indicates that this derivative is different for each element. It is not possible to solve for the general distribution of water saturation Sw(x,t) in most realistic cases because of the nonlinearity of the problem. For example, water fractional flow is usually a nonlinear function of water saturation. It is therefore necessary to consider a simplified approach to solving Eq. ((4-13)). x

A Figure 4-3 – Horizontal bed containing oil and water.

For a given rock, the fraction of flow for water fw is a function only of the water saturation Sw, as indicated by Eq. (4-13), assuming constant oil and water viscosities. The water saturation however is a function of both time and position, which may be express as fw = F(Sw) and Sw = G(t,x). Then: (

)

(

)

( (

) )

(4-14) 4-15)

Now, there is interest in determining the rate of advance of a constant saturation plane, or front ( ) , where Sw is constant and dSw = 0. So from eq. (4-14): (

)

(

)

(4-16)

Substituting eqs (4-13) and 4-15) into eq. (4-16) gives the Buckley-Leverett frontal advance equation: (

)

(

)

(4-17)

208

The derivative (

) is the slope of the fractional flow curve and derivative ( ) is the velocity of the

moving plane with water saturation Sw. Because the porosity, area, and flowrate are constant and because for any value of Sw, the derivative (

)

is a constant, then the rate dx/dt is constant.

This means that the distance a plane of constant saturation, Sw, advances is proportional to time and )) at that saturation, or:

to the value of the derivative (( (

)

(

)

(4-18)

Where, is the distance traveled by a particular Sw contour is the cumulative water injection at reservoir conditions. In field units: (

)

(

)

(4-19)

Example 4-1 Assume a cubical reservoir under active water drive with oil production of 900bbl/day. The flow could be approximated as a linear flow. The cross sectional area is the product of the width, 1320 ft, and the true formation thickness, 20 ft, so that for a porosity of 0.25, eq. (4-19) becomes: (

)

Consider that because we assume the fluids are completely incompressible, so the oil production rate is equal to the total flowrate in the different cross sections of the reservoir. 0

x

Flow Direction

1320 ft

20 ft Water

Transition zone

Oil + Connate Water

Figure 4-4 – Cubic reservoir under active water drive

If we let x=0 at the first point of the transition zone, then the distances the various constant water saturation planes will travel in, say, 60, 120, and 240 days are given by:

209

(

)

(

) (

(4-20) )

The value of the derivative (

) may be obtained for any value of water saturation, Sw, by plotting fw

from eq. (4-7) versus Sw and graphically taking the slopes at various values of Sw. Assume you find a=1222 and b=12 from Figure 4-1 (intercept = 1222 = ‘a’ and slope of the straight line = 13 = ‘b’) for eq. (4-7). For example at Sw = 0.4, fw = 0.129. The slope taken graphically at Sw = 0.4 and fw = 0.267 is 1.66. The derivative ( ( (

) may also be obtained mathematically using eq.(4-7):

) (

)

(4-21)

)

Figure 4-5 shows the water fractional flow curve and also the derivative (

) plotted against water

saturation from eq. (4-21). Since Eq. (4-7) does not hold for the very high and for the quite low water saturation ranges (see Figure 4-1), some error is introduced below 30% and above 80% water saturation. Since these are in the regions of the lower values of the derivatives, the overall effect on the calculation is small. Fractional Flow of Water, fw

0.8

3

0.6

2 0.4

1

0.2

Derivative, dfw/dSw

4

1

0

0 0

0.2

0.4

0.6

0.8

1

Water Saturation Figure 4-5 – Water fractional flow ant its derivative

A plot of Sw versus distance using Eq. (4-20) and typical fractional flow curves leads to the physically impossible situation of multiple values of Sw at a given location. For example Figure 4-6 shows water saturation distribution according to eqs (4-20) and (4-21). For example, at 50% water saturation, the value of the derivative is 2.87; so by eq. (4-20), at 60 days the 50% water saturation plane will advance a distance of:

210

(

)

This distance is plotted as shown in Figure 4-6 along with the other distances that have been calculated using eqs (4-20) and (4-21) for other time values and other water saturations. These curves are characteristically double-valued or triple valued. For example, Figure 4-6 indicates that the water saturation after 240 days at 400 ft is 20, 39, and 69%. The saturation can be only one value at any place and time. What actually occurs is that the intermediate values of the water saturation have the maximum velocity (Figure 4-5 and eq. (4-17)), will initially tend to overtake the lower saturations resulting in the formation of a saturation discontinuity or shock front. Because of this discontinuity the mathematical approach of Buckley-Leverett, which assumes that Sw is continuous and differentiable, will be inappropriate to describe the situation at the front itself. The difficulty is resolved by dropping perpendiculars at point Xf (as flood front position) so that the areas to the right (A) equal the areas to the left (B), as shown in Figure 4-6. In other words a discontinuity in Sw at a flood front location Xf is needed to make the water saturation distribution single valued and to provide a material balance for displacing fluid.

Xf

Xf

A B Initial Water Saturation Initial Water Saturation

(a)

(b) Figure 4-6 – (a) Fluid Distribution at 60, 120, 240 days (b)Triple-valued saturation distribution (after Buckley and Leverett, 1942)

211

A more elegant method of achieving the same result was presented by Welge in 1952. This consists of integrating the saturation distribution over the distance from the injection point to the front, thus obtaining the average water saturation behind the front Sw, as shown in Figure 4-747.

Figure 4-7 - Water saturation distribution as a function of distance, prior to breakthrough

The situation depicted is at a fixed time, before water breakthrough, corresponding to an amount of water injection. At this time the maximum water saturation, Sw = 1 - Sor, has moved a distance X1, its velocity being proportional to the slope of the fractional flow curve evaluated for the maximum saturation which, as shown in Figure 4-5, is small but finite. The flood front saturation Swf is located at position x2 measured from the injection point. Applying the simple material balance: (̿̿̿̿

)

(4-22)

So: ̿̿̿̿

(4-23)

Where,

is cumulative water injection.

Using eq. (4-18): (

)

(4-18)

)

(4-24)

At breakthrough time: ( Where, Breakthrough time, Total injection rate, Medium length From eq. (4-24): 47

“Fundamentals of Reservoir Engineering”, L.P. Lake, 1978.

212

(

)

(4-25)

Where PVI is the pore volume injected. So: (

)

(4-26)

The average water saturation in the reservoir at the time of breakthrough is given by material balance as: ̿̿̿̿̿̿

(4-27)

From eqs (4-26) and (4-27): ̿̿̿̿̿̿ (

(4-28)

)

Therefore: (

)

4-29)

̿̿̿̿̿̿

i.e. the slope of the fractional flow curve at conditions of the front is given by eq. 4-29). To satisfy eq. (4-29) the tangent to the fractional flow curve, from the point Sw = Swc, where fw = 0, must have a point of tangency with co-ordinates Sw = Swf; fw = fwf, and extrapolated tangent must intercept the line fw = 1 at the point (Sw = ̿̿̿̿̿̿; fw = 1). See Figure 4-8.

̿̿̿̿̿ 𝑆𝑤𝑏𝑡 Swf, fwf

Figure 4-8 - Tangent to the fractional flow curve from Sw = Swc

The use of either of these equations ignores the effect of the capillary pressure gradient, ∂Pc/∂x.

213

This simple graphical technique of Welge has much wider application in the field of oil recovery calculations. As eq. (4-19) shows the velocity of every saturation front is constant, the graph of saturation location vs. time is set of straight lines starting from the origin. This graph is often plotted in dimensionless form. The equation can be made dimensionless by defining:

(4-30)

Where Normalized distance Pore volumes injected Eq. (4-19) becomes: (4-31) Figure 4-9 is a graph of dimensionless distance vs. dimensionless time for the movement of water saturation predicted by the frontal advance equation. Saturation Siw