Some Useful Analytic Models For Steamflood Processes: P.J. Closmann, Consultant, Houston, Texas, USA

 

  

                             

   

No. 1998.037

Some Useful Analytic Models for Steamflood Processes P.J. Closmann, Consultant, Houston, Texas, USA

Abstract

applied mathematical to predict performance. However, analytic methodsmethods have lately been somewhat overshadowed by the development of numerical simulators 4 and engineers' increasing dependence on them. These latter methods have the advantage of applying Buckley-Leverett theory, thus permitting the calculation of saturation distribution. They can also account for chemical and phase changes, as well as the introduction of other fluids besides steam or water. Nevertheless, analytic methods and models can still play a useful role in the engineer's bag of tricks. Following are some of the simpler models that have been found useful.

 In this paper various various analytic models ar aree consider considered ed and their  usefulness illustrated. Examples are presented of a number of   practical applications of the familiar Marx and Langenheim model. The fundamental uses and limitations of the frontal dis placement or piston-like approac approach h are enunciated. Most of  these methods depend on control of the process at the injection well. The Marx and Langenheim approach has been extended to models that correct for low quality steam and can be simplified by use of some published results. Some of the obvious advantages of numerical simulation are pointed out, including the ability to incorporate irregular  geometry, heterogeneity, individual well constraints, and 

Frontal Displacement and Marx and Langenheim Models

chemical and phase changes. But analytic models continue to  find usefulness in provi providing ding insight into into the steamflood mechanism and the relative importance of various reservoir and  operational factors.

The earliest model to be applied to steamflooding was that of  Marx and Langenheim5 (ML) for predicting the growth of the steam zone in the reservoir during steam injection into a single well. Steam zone growth is important because of the effective displacement of oil from the steam zone, as shown by laboratory studies.2,3 This model assumed that:

 Analytic models can describe the lo long-term ng-term continuation of  steam injection in the mature steam drive. This mode of operation usually continues for most of the project life. The particular model described in this paper considers the steam zone as already established. Careful control of the heat consumption during this phase benefits the project economics. Calculation of the oil production by an analytic method permits monitoring of project performance on an ongoing basis and assists the engineer in adjusting some of the operating variables, such as steam injection rate, steam quality, and steam vent  rate. Modern innovations, such as horizontal wells, can be included in the method, which provides a simple computational tool for the practicing engineer.

1.

Steam penetrate penetratess a singl singlee la layer yer of of uniform uniform thickness thickness..

2.

Temperature emperature and pr pressur essuree rem remain ain cconsta onstant nt th through roughout out tthe he injection interval.

3.

The displac displacemen ementt fro front nt is vert vertical ical and defines defines the m movin oving g boundary at which the temperature falls from its steam zone value to the initial reservoir temperature.

4.

Heat losses losses ar aree normal normal to tthe he bo boundar undaries ies of the the steam steam zone at the under- and overburd overburden. en.

Oil production was then assumed to be that displaced by the steam zone, unless pushed out of the pattern beyond the producers.

Introduction 2

Steam injection was proposed originally for heavy oil and bitumen reservoirs because of the significant lowering of oil viscosity as a function of temperature and the consequent improvement in flow.1 Laboratory experiments were performed to measure the effectiveness of the oil recovery by steam displacement.2,3 Based on laboratory results and theory analytic models were formulated for monitoring and analyzing the progress of steamfloods in the field. Originally their usefulness was due to the lack of readily available numerical methods for making the necessary calculations. As a result, the engineer was forced to rely upon his intuition and classical

V s = 87.6

r s i s H s ( r C   ) s h s

K ( r C   ) o b (T s - T r )

V( t D ) = As h s (1)

Based on the above assumptions the following formulas were derived: where tD

V(t D ) ML = e erfc

1

t d - 1 + 2 t D / p

(2)

 

                                                   

and

tD =

can be distributed according to the inferred flow patterns in the reservoir.

4K ( rC   ) o b 2 s

Other applications are possible with the ML approach. A useful situation is that of a multilayered reservoir in which relatively uniformly sized layers are separated by approximately the same size shale or impermeable barriers.9 The requisite geometry for deducing the analogous Marx and Langenheim result is given in Ref. 9. The model assumptions are similar to those outlined above. This simple approach provides a rela-

(3)

2

h  ( r C   ) s

The oil displaced by steam is given by Vo = Vsf(Soi - Sor)

(4)

tively quick way to estimate steam with its responding oil displacement. On zone the growth, other hand, if corthe reservoir is highly nonuniform in layer thickness and layer separation, the numerical simulator is probably the best tool to make calculations including specific reservoir heterogeneities and irregular reservoir boundaries. In all these situations the quantity of oil displaced remains a direct function of the steam zone size developed during injection.

This equation proved to be surprisingly useful in analyzing behavior in field projects. The size of the developed steam zone correlates well with the oil produced. Good examples include the Inglewood field in California6 (Figure 1 ), the Schoonebeek field in The Netherlands7 (Figure 2 ), and the Tia Juana reservoir in Venezuela8 (Figure 3 ). As long as a single zone of uniform thickness is involved, the formula gives reasonable results. More complicated is the behavior of the injected steam when the effect of gravity is important, since the steam injection process is fundamentally gravity unstable. The effect of gravity results in the steam rising preferentially to the top of the reservoir with a displacement front that may be considerably slanted from the vertical. This situation requires the development of other formulas for estimating the effective effecti ve steam zone thickness, to be discussed below below.. Steeply dipping reservoirs pose a particularly difficult problem, since steam tends to rise and the condensate to drain downward. All All of these situations are automatically handled by numerical methods.

Another application is that of steam injection into a preheated reservoir. For example, a long term steam drive in one sand of the Kern River Canfield project resulted in conductive heating to an overlying sand, with beneficial effects on subsequent steamflood oil recovery.10 In cases of preheating the steam zone can be expected to propagate further than if no previous heating had taken place. If an approach similar to that of Marx and Langenheim is taken, it is necessary to make an assumption of the distribution of temperature due to heat that leaked off to the surroundings. 11 Depending on the assumption made for this distribution, the analytic solution may or may not be tractable and conveni convenient ent to use. In this case the performance based on a prior history of heating, with varying temperatures and heating rates, may be more readily handled with a numerical simulator, unless a straightforward and simple algorithm is available.

Nevertheless, there are some special uses of the ML formula. Consider the non-dipping reservoir in which there is a pronounced directional preference for flow, as occurs in some stream beds. In this case the formula may be applied to a zone of uniform thickness but of non-circular shape, such as an ellipse (Figure 4 ). Here the fundamental heat balance has been maintained, the correct area for heat loss is determined, and the most likely areal shape can be determined by the engineer's best judgement based on field observations.

One last example of the application of the ML approach is provided by the case of steam short-circulating from injector to producer.12 Such cases are unusual, but when they occur, analytic methods provide a simple means of estimating the extent of heterogeneities in the reservoir. The case of steam

In viscous oil reservoirs stimulation of the production wells is usually important for significant oil production. This occurs only after steam (heat) breakthrough to the producers. The steam breakthrough time is therefore important and is related to the areal sweep efficiency and the steam zone area for a small (or even zero) steam zone thickness. In this case (as well as the case of a fracture) the Marx and Langenheim formula reduces to the following:9

AS = 197.7

r s is H s

(T s - T r )

tD

injection into a cylinder can be used to estimate the size of  channel diverting steam to its point of detection. This is an example where an analytic solution provides the engineer a quick and reasonably reliable way to diagnose the field behavior, whereas a numerical simulation would require some ingenuity in selecting both block sizes and the proper input geometry. The above examples apply to single well models. In the case of multiple wells, the spreading of interfering and overlapping steam zones is a situation in which numerical simulators have a definite advantage over analytical models, provided enough is known of the important reservoir parameters.

(5)

K ( rC   ) oz

which gives the area of the high temperature “steam zone” generated by flowing steam in an infinitesimally thick interval interval.. It also provides a means of estimating the lower time limit for steam to blanket the top of the reservoir. As before, the area

2

 

                                                                         

Modifications to the Marx and Langenheim Model

V(t D ) H V(t D ) ML

It was recognized early that the ML model had certain fundamental limitations, i.e.,: 1.

It do does es no nott acc account ount for conduc conduction tion and conv convecti ection on ahead ahead of the condensation front.

2.

It do does es not not pr properl operly y acc account ount for the effe effect ct of steam quality.

 

]

1.05

1 - e erfcSQRTt D f h v ] H[2 t 1 - e SQRTt D

(8)

for tD > tcD where

f  = fraction of injected heat as latent heat. hv

t cD N  =  t cD

r w C w (1 - S o )

(9)

( r C   ) oz

The concept of the critical time was also analyzed by Wang Wang 17 and Brigham in the following manner. From a fluid dynamics viewpoint Mandl and Volek Volek defined the critical time as the time when the heat front velocity equals the temperature front velocity of an adiabatic hot water displacement at the same mass injection rate and temperature as the steam displacement. Wang and Brigham suggest the alternative of defining the critical time as the time when w hen the heat front velocity equals

tion of the CF is calculated by an approximate formula presented by Mandl and Volek.

the injectedatfluid flow velocity of an adiabatic hot water displacement the same mass injection rate and temperature as the steam. With this definition, they conclude that the critical time will be smaller than that defined by Mandl and Volek by a factor of 1.2–5, depending on the ratio of the temperature front velocity to the fluid velocity.17 Calculated in this manner, the critical time is given by the relation Wang and Brigham present some data to support their theory. 17 It is interesting that there has been relatively little experimental laboratory work to verify and/or improve the above theories, other than that of the original authors.

This model, with an improved estimate for the steam zone growth, is the basis of a method proposed by Myhill and Stegemeier14 for calculation of the steam zone and the accompanying oil displacement. Their results were presented in the form of a graph with thermal efficiency plotted against dimensionless time for various values of the ratio of latent heat to sensible heat. An alternate approach was proposed by Hearn, 15 in which the thermal equivalence of the steam zone and hot water zone ahead of the CF was used in the original Marx and Langenheim heat balance. Results were derived which show how the condensation front lagged the ML formula after a certain criti-

Effects of Gravity

cal time. Convenient formulas which use Hearn's method were presented by Burger et al.16 They can be rearranged and presented as follows:

The tendency of steam to rise due to gravity was early recognized by steamflooders. Methods of predicting the steam zone shape were presented by Neuman18 and by van Lookeren.19 A model to include steam overlay as well as steam vent rate and conditions at both injection and production wells was presented by Hsu.20 The method of van Lookeren, which gives a shape factor based on the steam injection rate to characterize the tilted steam/liquid interface, is especially convenient in making calculations. The equations were validated with scaled, physical models. For less viscous oils, the sloping steam zone interface may prove to be stable. In general, the average steam zone thickness may be estimated from the following:

(6)

The steam zone volume ratio V(tD)H /V(tD)ML = 1

tD

Derfc

These limitations do not exist for the numerical simulators. Nevertheless, Nevertheles s, they can be relatively easily addressed by means of two theories which were developed to improve the ML calculation. The theory of Mandl and Volek 13 assumes that until a critical injection time is reached the steam condensation front (CF), although always slowing due to heat losses, leads the convective convecti ve heat flow and tends to sharpen the conductive temperature profile ahead of the front. At the critical time all the latent heat of injected steam is used to supply heat losses from the steam zone to cap and base rock and to provide the latent heat content of the steam zone. After the critical time the convectivee heat flow due to condensate leads the steam condensavectiv tion front. Until this critical time is reached, the original ML theory is suitable (although conduction ahead of the CF is ignored by both theories). After After the critical time, the propaga-

Critical time: tCD = 1.877[(1 - f hv)- 0.645 - 1]2

f h v

=[

(7)

for tD £ tCD and

h

st

= 0.5A RD h f 

where the shape factor is

3

(10)

 

                             

ARD = 50.39[

m s ( r s i s ) f s 2

( r o - r s )h k s r s

]

Models to estimate the oil production under these conditions were presented by Miller and Leung24 and by Closmann.25,26 A number of analytic models are compared in Ref. 25. The method of Ref. 25 and Ref. 26 depends on a modification of the gravity drainage theory of Matthews and Lefkovits27 to include both temperature and oil viscosity variation in the gradually decreasing oil zone. An additional term to account for the steam flow-induced pressure drop is also included in Ref. 26. The basic equation which defines the model is as follows:

(11)

These theories to account for gravity override permit the engineer to make an easily obtained estimate of reservoir performance. Their accuracy can be checked and the results rendered more accurate by the use of numerical simulation. For steeply dipping reservoirs (dip greater than about 20– 30°) the above models may sometimes be applied. Knowledge of the injected steam will usually permit estimating the actual oil displaced. For many of these cases the numerical simulators have a definite advantage, since they can also account for the separation of the gaseous and liquid (condensate) phases.

re 2

2

I q o B o (1 - r  / r e ) rw

h

e

h

  2pk o 6Dr g I I hw 0

Steam Soak or Steam Stimulation

dZ mo

dr  = r h

dh + Ifromp w I 0

dZ

(12)

dp>

mo

To evaluate this equation, the temperature distribution assumed is the steady-state form

Although a number of steam soak models have been developed 21, the usefulness of analytic models for this process is somewhat limited. The success of an analytic approach depends on the ability to model the reservoir drive mechanism. This can vary widely from one reservoir to another (e.g.,

T - Tr Ts - Tr

gravity drainage, depletion, compaction). One example of the complexity that may occur is shown by the “foamy oil” zone and its interference with the viscous fingering of water and gases passing through from the cold reservoir at Cold Lake, Alberta, Canada.22 Such a mechanism presents a severe challenge to the formulator of analytic models. At the present time numerical simulators, with their inputs involving specified constraints, boundary and initial conditions, and adaptability to various drive mechanisms, have a definite advantage over the analytic methods developed thus far. The process involving repeated soaks or stimulations is especially well represented by numerical simulation, which can account for the distribution of heat in succeeding cycles as well as the withdrawal of fluids.

vZ / = exp ( - vZ  / a )

(13)

and the oil viscosity distribution has the form suggested by Butler:28

n os

=[

no

T - Tr Ts - Tr

]

m

(14)

F (a ,b )  =  6(1 + b )( 1 + 2ba M ) 2

2

2

 / [( 1 + 2ba ) a + 4(1 + b)a ] >

(15)

M

The integration of Eq. 11 and the evaluation of the constant “m”, as well as the calculation of the steam phase pressure drop are described in Ref. 26. The result may be put in the following useful form:

Mature Steamfloods Once steam has broken through to the producers and effectively blankets the reservoir, the effectiveness of direct displacement is reduced. Oil is then produced by gravity (usually the dominant mechanism) and also a “drag” effect (usually small) due to the flowing steam pressure gradient applied to the oil zone. The steam pressure gradient can be especially important during high rates of steam venting at the producers. This “mature” phase of the steamflood is generally reached relatively relativ ely quickly (1–3 years) after steam injection begins and generally lasts for the life of the flood, perhaps from 10–20 years. An analysis of this process was presented by Vogel, 23 who showed how to minimize the steam requirements by matching heat requirements to the slowly changing heat losses, thus optimizing the project performance.

a = 2527

S r o   m os R a AfD fDS 3

(16)

k o Dr mr o s h e Drm where

F = 1.289H10

-4

  mq o B o h e aA D S

and the shape factor is

4

(17)

 

                                                                                                                                                                   

aM=

 1   1 ] >   61 + [ 1 2 2ln( r e / r  / r w )  - 1

rates, therefore, are not as great as would be obtained if steam had been present at the wells the entire time.

(18)

The steam phase pressure drop Dps is included in the term ß: ß = 2.307 Dps /(heDp)

Summary and Conclusions

(19)

The above frontal displacement models can usually give a reliable estimate of steam zone development in the reservoir. Some which the author has found useful are listed in Table 1.

The pressure drop may be augmented by the inclusion of  additional pressures due to agents designed to block or limit

On the basis of the cases considered, and based on available literature sources, it is reasonable to conclude that:

flow in zones with bypassing steam. The oil production rate is calculated from the function F(a,ß), where values of “a” are determined for various assumed increments in the oil level he. qo = 7758 aAf DS F(a,ß)/(m he Bo)

1.

Analytic models models are useful in in many practical situations, especially when a measure of displaced oil (but not necessarily produced oil) is desired.

2.

The Marx and Lange Langenheim nheim fformula ormula can be used to assess the growth of steam zones in slightly dipping reservoirs and to give an indication of steam penetration when steam flow is areally non-uniform.

3.

Analytic models have certain limitations, and num numerical erical simulation is the tool of most usefulness in non-uniform, stratified, and steeply dipping reservoirs and in multiwell cases.

4.

Analytic models which depend depend on calculating oil produc-

(20)

In this method “history-matching” or fitting to the observations is accomplished by adjusting the absolute reservoir permeability (usually somewhat uncertain) within physically reasonable values. The procedure is readily adapted to spreadsheet computation. These results can be applied to horizontal as well as vertical wells.25,26 An equation similar to Eq. 15 is obtained: 2

F H (a H ,b )   =  6(1 +b   )( 1 + 2ba H ) 2

2

 / [( 1 + 2ba ) a + 4(1 +b   )a ] > H

H

tion by considering phenomena at the producers, such as gravity drainage and steam venting, can be useful for estimating oil production in mature steamfloods.

(21)

H

where

a H = 0.18225

F H   =  2.807

aX 2 r m DS e o osvarphi

Nomenclature (22)

mk o r os Dr h e

3

mq o B o h e

(23)

aLX e D S

and an analogous shape factor aH = 0.78868 is used. The oil rate is calculated from the function F H(aH,ß) as follows: qo = 0.3562 aLXefDS FH(aH,ß)/(mBohe)

(24)

The analytic method pertinent to the SAGD (Steamassisted Gravity Drainage) process, using mostly horizontal wells, has been described in detail by Butler.28 An up-to-date appraisal of SAGD is provided in the review article by Batycky.22 Some applications of the present method are illustrated in Figure 5 for vertical wells producing by gravity drainage with vented steam in the Kern River field,26,29 and in Figure 6 for horizontal wells producing by gravity drainage alone in the Midway Sunset field.26,30 In Figure 6 the calculated oil production rates lie above the observed values. During production from this reservoir of 35–50° dip, these wells were not always maintained at steam temperature. The observed production

A

Drainage area per well, L2, acres

As

Steam zone area, L2, ft2

a aH

Dimensionless reservoir parameter Dimensionless reservoir parameter, horizontal wells

Bo

Oil formation volume factor, RB/STB

Cw

Specific heat of water, L2 /(t2T), Btu/(lbm°F)

f s f hv

Steam quality, fraction Fraction of heat of steam as latent heat

F(a,ß) FH(aH,ß)

Dimensionless rate function, F(a) when ß=0 Dimensionless rate function, horizontal wells

G H He

Acceleration due to gravity, L/t2, ft/sec2 Height of oil level along oil/steam interface, L, ft = average height of oil level, L, ft Height of oil level at outer boundary of drainage area,

L, ft Hei

Initial height of oil level below steam zone, L, ft

Hf 

formation thickness, thickness, L, ft

Hl

Sensible heat of liquid water in injected steam, L 2 /t2,

H

Btu/lbm Height of liquid level in producer, L, ft

w

5

K

Absolute permeability, L2, d

Ko

Effective oil permeability, L2, d

 

                

k ro

Relative permeability to oil

L

length of horizontal well, L, ft

Volumetric heat capacity of oil zone, m/Lt 2T,

()oz

Btu/(ft3°F) 2 2

Lv

Latent heat of vaporization of steam, L  /t , Btu/lbm

M

Exponent in oil viscosity equation, Eq. 13

Volumetric heat ccapacity apacity of overburden, m/Lt 2T,

()ob

Btu/(ft3°F)

2

pe

Steam pressure at outer boundary, m/Lt , psi

pw

Steam pressure at well, m/Lt2, psi

Dps

Pressure drop in steam zone = p e-pw, m/Lt2, psi 3

qo

Oil production rate, L  /t, STB/D

r re

Radial distance from center of well, L, ft Radial distance to outer boundary, L, ft

rw

Well radius, L, t

R

ln re /rw - 0.5

So

Oil saturation in oil zone, fraction

Soi

Initial oil saturation, fraction

Sor

()s

Volumetric heat capacity of steam zone, m/Lt 2T,

f 

Btu/(ft3°F Porosity, fraction

SI Metric Conversion acre ´ 4.  4 .046856 bbl ´ 1  1..589873 Btu ´ 1  1..055056

Factors E-01 = ha E-01 = m3 E+00 = kJ

Residual oil saturation in steam zone, fraction

Btu/(ft3 /°F) ´ 6.706611 Btu/lbm ´ 2.  2.326 Btu/(lbm °F) ´ 4.  4.1868 cp ´ 1  1..0

E+01 E+00 E+00 E-03

=kJ/m3·K = kJ/kg = kJ/(kg·K) = Pa·s

DS

Cha Change in oil sa sattura rattion ion afte fter p paassa ssage of ste steam/ m/oi oill interface, = So - Sor, fraction

darcy ´ 9.  9.869233 ft ´ 3  3..048

E-01 E-01

= µm 2 =m

t tcD

Time, t, days Dimensionless critical time by Mandl & Volek 

tcD’

Dimensionless critical time by Wang & Brigham

ft2 ´ 9  9..290304 °F 5/9(°F-32) psi ´ 6  6..894757

= m2 = °C E+00 = kPa

tD

Dimensionless time

Tr Ts

Original reservoir temperature, T, deg.F Steam temperature, T, deg.F

DT

Ts - Tr, T, deg.F

v V(tD)H

Linear v veelocity of of ffaalling oi oil ssu urface, L L//t, fftt/D Dimensionless steam zone volume by Hearn method

V(tD)ML

Dimensionless steam zone volume by Marx &

E-02

References 1.

Pr Prat ats, s, Mi Mich chae ael, l, 19 1986 86.. Thermal Recovery, SPE Monograph Volume 7, pp. 6-15.

2.

Willman, Willman, B.T., B.T., Valleroy Valleroy,, V.V., .V., Runberg, G.W., G.W., Cornelius, A.J., and Powers, L.W., 1961. “Laboratory Studies of Oil Recovery by Steam Injection,” Injection ,” J. Pet. Tech. July, pp. 681–690.

3.

Closm Closmann, ann, P.J. an and d Seb Seba, a, R.D R.D.,1983. .,1983. “Labo “Laborator ratory y T Tests ests on Heavy Oil Recovery by Steam Injection,” SPEJ, June, pp. 417–426.

Langenheim Vs 

Steam zone volume, L3, ft3

Z

Distance coordinate, L, ft

a aM

Thermal diffusivity of reservoir, L2 /t, ft2 /D Dimensionless reservoir factor (Eq. 6)

ß

Dimensionless pr pressure dr drop parameter = 2.307 Dps /(heDp)

4.

µ µ os

Oil viscosity, m/Lt, cp Dynamic oil viscosity at steam temperature, m/Lt, cp

Coats, Coats, K.H., K.H., 1978. 1978. “A “A Highly Highly Im Impli plicit cit Stea Steamflo mflood od Model,”” SPEJ, October, pp. 369–383. Model,

5.

µ s

Dynamic steam viscosity, m/Lt, cp

vo

Kinematic oil viscosity, L2 /t, cm2 /sec

Marx Marx,, J.W J.W.. and and La Langenh ngenheim, eim, R.H., 1959. “Rese “Reservoi rvoirr Heating by Hot Fluid Injection,” Trans. AIME, Vol. 216, pp. 312–314.

6.

vos

Kinematic oil viscosity at steam temperature,

po

L2 /t, cm-2 /sec Oil density at reservoir temperature (Eq. 10),

Ble Blevins, vins, T.R., Aselt Aseltine, ine, R R.J., .J., an and d Kirk, Kirk, R.S., 1969. “Analysis of a Steam Drive Project, Inglewood Field, California,”” J. Pet. Tech. September, nia, September, pp. 1141–1150.

7.

Dijk, C., 1968. 1968. ““Steam Steam-Dri -Drive ve Projec Projectt in the Schoo Schoonebee nebeek  k  Field, The Netherlands,” J. Pet. Tech. March, pp. 295– 302.

8.

de Haan, Haan, H.J. H.J. and S Sche chenk, nk, L. L.,, 1969. 1969. “Pe “Perfo rforma rmance nce and and Analysis of a Major Steam Drive Project in the Tia Juana Field, Western Venezuela,” J. Pet. Tech. January, pp. 111– 119: Trans., AIME, Vol. 246.

9.

Closm Closmann, ann, P.J., 19 1967. 67. “S “Steam team Zone Gro Growth wth During During Multiple-Layer Steam Injection,” Injection,” SPEJ, March, pp. 1–10.

3

m/L ,

3

lbm/ft

o

Average oil density, m/L3, gm/cm3

pos

Oil density at steam temperature, m/L 3, lbm/ft3

ps

Steam density (Eq. 9), m/L3, lbm/ft3

psis

Steam injection rate, L3 /t, B/D(CWE) B/D(CWE)

pw

Density of water, m/L3, lbm/ft3

Dp

Den ensi sity ty dif iffe fere renc ncee bet etw ween oil an and d steam team phas ases es,, m/L3, lbm/ft3

6

 

10. Resti Restine, ne, J.L., Gra Graves, ves, W W.G., .G., and Elias Elias,, R., 1987. “Infill Drilling in a Steamflood Operation: Kern River Field,” SPE Reservoir Engineering, May, pp. 243–248.

21. Pra Prats, ts, Mich Michael ael,, 1986. 1986. Thermal Recovery, SPE Monograph Volume 7, pp. 113–124.

11. Closm Closmann, ann, P P.J., .J., 1968. “St “Steam eam Zone Gro Growth wth in a Preheated Reservoir,” Reservoir,” SPEJ, September September,, pp. 313–320.

22. Batycky Batycky,, J.P J.P., ., 1997. ““An An Assessment of In Situ Oil Sands Recovery Processes,” J. Can. Pet. Tech., Vol. 36, No. 9, October, pp. 15–19.

12. Closm Closmann, ann, P P.J., .J., 1984. “Steam “Steam Zone Growth Growth in Cylindr Cylindrical ical Channels,”” SPEJ, October, pp. 481–484. Channels,

23. Vogel, J.V J.V., ., 1984. “Simplified “Simplified Heat Cal Calculations culations for Steamfloods,”” J. Pet. Tech. July, pp. 1127–1136. Steamfloods,

13. Mandl, G. and Volek, C.W C.W., ., 1969. “Heat and Mass TransTransport in Steam-Drive Processes,” SPEJ March, pp. 59–79;

24. Miller, M.A. and Leung, W.K.,1985. “A “A Simple Gravity Gravity Override Model of Steamdrive,” Steamdrive,” SPE 14241, presented at

Trans. AIME, Vol. 246. 14. Myhil Myhill, l, N.A. and Stegeme Stegemeier ier,, G.L., 1978. “Steam-Driv “Steam-Drivee Correlation and Prediction,” J. Pet. Tech., Feb., pp. 173– 182.

60th annual SPE meeting, Las Vegas, Nevada, September 22–25. 25. Closmann, P P.J., .J., 1995. “Simplified Gravity-Drainage OilProduction Model for Mature Steamfloods,” SPE Reservoir Engineering, May, pp. 143–148.

15. Hearn Hearn,, C.L., 1969. “Ef “Effect fect of La Latent tent Hea Heatt Conte Content nt of  Injected Steam in a Steam Drive,” J. Pet. Tech. April, pp. 374–375.

26. Closmann, P.J., 1997. “Simplified Oil-Production Model With Viscous Flow, Gravity Drainage, Skin, and Instantaneous Oil/Steam Ratio for Mature Steamfloods,” SPEJ, Vol. 2, No. 4, pp. 466–473.

16. Burg Burger er,, J., Sourieau Sourieau,, P P., ., and Combarnous, Combarnous, M., 1985 1985.. Thermal Methods of Oil Recovery, Gulf Publishing Co., pp. 169–171.

27. Matth Matthews, ews, C.S. an and d Lefkovits, Lefkovits, H.C H.C., ., 1956. “Gravity “Gravity Drainage Performance of Depletion-Type Reservoirs in the Stripper Stage,” Trans. AIME, Vol. 207, pp. 265–274.

17. Wang, Fred P P.. and Brigham, W.E., 1986. ““A A Study of  Heat Transfer During Steam Injection and Effect of Surfactants on Steam Mobility Reduction,” Report TR 55, performed for Dept. of Energy Under Contract No. DE– ACO3–81SF–11564, Stanford, California, August, pp. 65–84.

28. But Butler ler,, R.M., R.M., 199 1991. 1. Thermal Recovery of Oil and Bitumen, Prentice-Hall Inc., Englewood Cliffs, NJ, Chap. 7, pp. 285–359. 29. Kimbe Kimber, r, K.D., Deeme Deemer, r, A.R., A.R., Luce, T.H., T.H., and Sharpe Sharpe,, H.N., 1995. “A New Analytical Model for Assessing the Role of Steam Production in Mature Steam Floods,” paper SPE 296s57, presented at SPE Western Regional Meeting, Bakersfield, California, March 8–10.

18. Neuma Neuman, n, C.H.,1975 C.H.,1975.. “A Mat Mathemat hematical ical Mod Model el of the Steam Drive Process — Applications,” SPE 4757, presented at the California Regional Meeting, Ventura, Ventura, April 2–4. 19. van Loo Lookere keren, n, J., 1983. “Calculat “Calculation ion Meth Methods ods for Line Linear ar and Radial Steam Flow in Oil Reservoirs,” SPEJ, June, pp. 427–438.

30. Kuhac Kuhach, h, J.D. and Myhi Myhill, ll, N.A., 1995. 1995. “Optimizat “Optimization ion of a Mature Steam Flood Utilizing Horizontal W Wells, ells, Midway Sunset Field,” J. Can. Pet. Tech., September, Vol. 34, No. 7, pp. 36–41.

20. Hsu, C.F C.F., ., 1992. ““A A Pattern Steamdrive Steamdrive Model ffor or Personal Computers,” SPE 24076, presented at the Western Regional Meeting, Bakersfield, California, March 30– April 1.

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Table Tab le 1: Some Useful Analytic Methods.

Figure 1: Area heated vs time.6

8

 

Figure 2: Pore volume steam zone according to material balance and Marx and Lang enheim.7

Figure 3: Comparison of calculated and actual production.8

9

 

Figure 4: Non-radial steam zone.

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Figure 5: Model calculation of oil production rate in mature steamflood, illustrating gravity drainage rate and effect of steam venting.

Figure 6: Model calculation of oil production rate by gravity drainage for three horizontal wells — Midway Sunset field.

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