Howard-Petroleum Engineers Handbook

Chapter 40

Estimation of Oil and Gas Reserves Forrest A. Garb, SPE, Gerry L. Smith ,** H.J.

H.J. Grt~y & Assocs. Gruy

6i Asaoca.

Inc.*

Inc.

Estimating Reserves General Discussion Management’s decisions are dictated by the anticipated results from an investment. In the case of oil and gas, the petroleum engineer compares the estimated costs in terms of dollars for some investment opportunity vs. the cash flow resulting from production of barrels of oil or cubic feet of gas. This analysis may be used in formulating policies for (1) exploring and developing oil and gas properties; (2) designing and constructing plants, gathering systems, and other surface facilities; (3) determining the division of ownership in unitized projects; (4) determining the fair market value of a property to be bought or sold: (5) determining the collateral value of producing properties for loans; (6) establishing sales contracts, rates, and prices; and (7) obtaining Security and Exchange Commission (SEC) or other regulatory body approvals. Reserve estimates are just what they are calledestimates. As with any estimate, they can be no better than the available data on which they are based and are subject to the experience of the estimator. Unfortunately, reliable reserve figures are most needed during the early stages of a project, when only a minimum amount of information is available. Because the information base is cumulative during the life of a property, the reservoir engineer has an increasing amount of data to work with as a project matures, and this increase in data not only changes the procedures for estimating reserves but, correspondingly, improves the confidence in the estimates. Reserves are frequently estimated (1) before drilling or any subsurface development, (2) during the development drilling of the field, (3) after some performance data are available, and (4) after performance trends are well established. Fig. 40.1 demonstrates (I) the various periods in the life of an imaginary oil property, (2) the sequence

of appropriate recovery estimating methods, (3) the impact on the range of recovery estimates that usually results as a property ages and more data become available, (4) a hypothetical production profile, and (5) the relative risk in using the recovery estimates. Time is shown on the horizontal axis. No particular units are used in this chart, and it is not drawn to any specific scale. Note that while the ultimate recovery estimates may become accurate at some point in the late life of a reservoir, the reserve estimate at that time may still have significant risk. During the last week of production. if one projects a reserve of 1 bbl and 2 bbl are produced, the reserve estimate was 100% in error. Reserve estimating methods are usually categorized into three families: analogy, volumetric, and performance techniques. The performance-technique methods usually are subdivided into simulation studies, material-balance calculations, and decline-trend analyses. The relative periods of application for these techniques are shown in Fig. 40.1. ‘.2 During Period AB, before any wells are drilled on the property, any recovery estimates will be of a very general nature based on experience from similar pools or wells in the same area. Thus, reserve estimates during this period are established by analogy to other production and usually are expressed in barrels per acre. The second period, Period BC, follows after one or more wells are drilled and found productive. The well logs provide subsurface information, which allows an acreage and thickness assignment or a geologic interpretation of the reservoir. The acre-foot volume considered to hold hydrocarbons, the calculated oil or gas in place per acre-foot, and a recovery factor allow closer limits for the recovery estimates than were possible by analogy alone. Data included in a volumetric analysis may include well logs, core analysis data, bottomhole sample information, and subsurface mapping. Interpretation of these

PETROLEUM

40-2

Fig. 4&l-Range in estimates of reservoir.

of ultimate recovery during life

data. along with observed pressure behavior during early production periods, may also indicate the type of producing mechanism to be expected for the reservoir. The third period, Period CD, represents the period after delineation of the reservoir. At this time, performance data usually are adequate to allow derivation of reserve estimates by use of numerical simulation model studies. Model studies can yield very useful reserve estimates for a spectrum of operating options if sufficient information is available to describe the geometry of the reservoir, any spatial distribution of the rock and fluid characteristics, and the reservoir producing mechanism. Because numerical simulators depend on matching history for calibration to ensure that the model is representative of the actual reservoir, numerical simulation models performed in the early life of a reservoir may not be considered to have high confidence. During Period DE, as performance data mature, the material-balance method may be implemented to check the previous estimates of hydrocarbons initially in place. The pressure behavior studied through the materialbalance calculations may also offer valuable clues regarding the type of production mechanism existent in the reservoir. Confidence in the material-balance calculations

ENGINEERING

HANDBOOK

depends on the precision of the reservoir pressures recorded for the reservoir and the engineer’s ability to determine the true average pressure at the dates of study. Frequent pressure surveys taken with precision instruments have enabled good calculations after no more than 5 or 6 % of the hydrocarbons in place have been produced. Reserve estimates based on extrapolation of established performance trends, such as during Period DEF, are considered the estimates of highest confidence. In reviewing the histories of reserve estimates over an extended period of time in many different fields, it seems to be a common experience that the very prolific fields (such as East Texas, Oklahoma City, Yates, or Redwater) have been generally underestimated during the early “barrels-per-acre-foot” period compared with their later performance, while the poorer ones (such as West Edmond and Spraberry) usually are overestimated during their early stages. It should be emphasized that, as in all estimates, the accuracy of the results cannot be expected to exceed the limitations imposed by inaccuracies in the available basic data. The better and more complete the available data, the more reliable will be the end result. In cases where property values are involved, additional investment in acquiring good basic data during the early stages pays off later. With good basic data available, the engineer making the estimate naturally feels more sure of his results and will be less inclined to the cautious conservatism that often creeps in when many of the basic parameters are based on guesswork only. Generally, all possible approaches should be explored in making reserve estimates and all applicable methods used. In doing this, the experience and judgment of the evaluator are an intangible quality, which is of great importance. The probable error in the total reserves estimated by experienced engineers for a number of properties diminishes rapidly as the number of individual properties increases. Whereas substantial differences between independent estimates made by different estimators for a single property are not uncommon, chances are that the total of such estimates for a large group of properties or an entire company will be surprisingly close.

Petroleum Reserves-Definitions and Nomenclature3 Definitions for three generally recognized reserve categories, “proved,” “probable,” and “possible,” which are used to reflect degrees of uncertainty in the reserve estimates, are listed as follows. The proved reserve definition was developed by a joint committee of the SPE, American Assn. of Petroleum Geologists (AAPG), and American Petroleum Inst. (API) members and is consistent with current DOE and SEC definitions. The joint committee’s proved reserve definitions, supporting discussion, and glossary of terms, are quoted as follows. The probable and possible reserve definitions enjoy no such official sanction at the present time but are believed to reflect current industry usage correctly. Proved Reserves Definitions3 The following is reprinted from the Journal of PetroleUM Technology (Nov. 1981, Pages 2113-14) proved reserve definitions, discussion, and glossary of terms.

ESTIMATION

OF OIL AND GAS RESERVES

40-3

Proved Reserves. Proved reserves of crude oil, natural gas, or natural gas liquids are estimated quantities that geological and engineering data demonstrate with reasonable certainty to be recoverable in the future from known reservoirs under existing economic conditions.* Discussion. Reservoirs are considered proved if economic producibility is supported by actual production or formation tests or if core analysis and/or log interpretation demonstrates economic producibility with reasonable certainty. The area of a reservoir considered proved includes (1) that portion delineated by drilling and defined by fluid contacts, if any, and (2) the adjoining portions not yet drilled that can be reasonably judged as economically productive on the basis of available geological and engineering data. In the absence of data on fluid contacts, the lowest known structural occurrence of hydrocarbons controls the lower proved limit of the reservoir. Proved reserves are estimates of hydrocarbons to be recovered from a given date forward. They are expected to be revised as hydrocarbons are produced and additional data become available. Proved natural gas reserves comprise nonassociated gas and associated/dissolved gas. An appropriate reduction in gas reserves is required for the expected removal of natural gas liquids and the exclusion of nonhydrocarbon gases if they occur in significant quantities. Reserves that can be produced economically through the application of established improved recovery techniques-are included in the proved classification when these qualifications are met: (1) successful testing by a pilot project or the operation of an installed program in that reservoir or one with similar rock and fluid properties provides support for the engineering analysis on which the project or program was based, and (2) it is reasonably certain the project will proceed. Reserves to be recovered by improved recovery techniques that have yet to be established through repeated economically successful applications will be included in the proved category only after successful testing by a pilot project or after-the operation of an installed-p&g&~ in the reservoir provides support for the engineering analysis on which the project or program was based. Estimates of proved reserves do not include crude oil, natural gas, or natural gas liquids being held in underground storage. Proved Developed Reserves. Proved developed reserves are a subcategory of proved reserves. They are those reserves that can be expected to be recovered through existing wells (including reserves behind pipe) with proved equipment and operating methods. Improved recovery reserves can be considered developed only after an improved recovery project has been installed. Proved Undeveloped Reserves. Proved undeveloped reserves are a subcategory of proved reserves. They are those additional proved reserves that are expected to be recovered from (I) future drilling of wells, (2) deepening of existing wells to a different reservoir, or (3) the installation of an improved recovery project. ‘Most reserve,, engmeers

add the expression

“considering

current technology.”

Glossary of Terms Crude Oil Crude oil is defined technically as a mixture of hydrocarbons that existed in the liquid phase in natural underground reservoirs and remains liquid at atmospheric pressure after passing through surface separating facilities. For statistical purposes, volumes reported as crude oil include: (1) liquids technically defined as crude oil; (2) small amounts of hydrocarbons that existed in the gaseous phase in natural underground reservoirs but are liquid at atmospheric pressure after being recovered from oilwell (casinghead) gas in lease separators*; and (3) small amounts of nonhydrocarbons produced with the oil. Natural Gas Natural gas is a mixture of hydrocarbons quantities of nonhydrocarbons that exists gaseous phase or in solution with crude underground reservoirs. Natural gas may fied as follows.

and varying either in the oil in natural be subclassi-

Associated Gas. Natural gas, commonly known as gascap gas, that overlies and is in contact with crude oil in the reservoir. ** Dissolved Gas. Natural gas that is in solution with crude oil in the reservoir. Nonassociated Gas. Natural gas in reservoirs that do not contain significant quantities of crude oil. Dissolved gas and associated gas may be produced concurrently from the same wellbore. In such situations, it is not feasible to measure the production of dissolved gas and associated gas separately; therefore, production is reported under the heading of associated/dissolved or casinghead gas. Reserves and productive capacity estimates for associated and dissolved gas also are reported as totals for associated/dissolved gas combined. Natural Gas Liquids Natural gas liquids (NGL’s) are those portions of reservoir gas that are liquefied at the surface in lease separators, field facilities, or gas processing plants. NGL’s include but are not limited to ethane, propane, butanes, pentanes, natural gasoline, and condensate. Reservoir A reservoir is a porous and permeable underground formation containing an individual and separate natural accumulation of producible hydrocarbons (oil and/or gas) that is confined by impermeable rock and/or water barriers and is characterized by a single natural pressure system.

‘From a technical standpoint, these hqulds are termed condensate”, however, they are commmgled wth Ihe crude stream and it IS impractical to meawe and report their volumes separately All other condensate IS reported as either “lease condensate” or “plant condensate” and Included I” natural gas l,q”,ds . ‘Where resewar cond,,,ons are such lhat the production of associated gas does not substantlallv affect the recwerv of crude 011 I” the reser~oll. such aas rnav be reclassitled’as nonassoclated gis by a regulatory agency In this w&t, res&es and producbon are reported I” accordance wth the classlficatw used by the regulatory agency

PETROLEUM

40-4

ENGINEERING

HANDBOOK

Probable Reserves

OIL-WATER CONTACT -7450

Probable reserves of crude oil, natural gas, or natural gas liquids are estimated quantities that geological and engineering data indicate are reasonably probable to be recovered in the future from known reservoirs under existing economic conditions. Probable reserves have a higher degree of uncertainty with regard to extent, recoverability, or economic viability than do proved reserves.

0

Possible Reserves

Fig. 40.2-Geological

map on

Possible reserves of crude oil, natural gas, or natural gas liquids are estimated quantities that geological and engineering data indicate are reasonably possible to be recovered in the future from known reservoirs under existing economic conditions. Possible reserves have a higher degree of uncertainty than do proved or probable reserves.

top (-) and base (-7) of reservoir.

Computation of Reservoir Volume4

In most situations, reservoirs are classified as oil reservoirs or as gas reservoirs by a regulatory agency. In the absence of a regulatory authority, the classification is based on the natural occurrence of the hydrocarbon in the reservoir as determined by the operator.

When sufficient subsurface control is available, the oilor gas-bearing net pay volume of a reservoir may be computed in several different ways. 1. From the subsurface data a geological map (Fig. 40.2) is prepared, contoured on the subsea depth of the top of the sand (solid lines), and on the subsea depth of the base of the sand (dashed lines). The total area enclosed by each contour is then planimetered and plotted as abscissa on an acre-feet diagram (Fig. 40.3) vs. the corresponding subsea depth as the ordinate. Gas/oil contacts (GOC’s) and water/oil contacts (WOC’s) as determined from core, log, or test data are shown as horizontal lines.* After the observed points are connected, the combined gross volume of oil- and gas-bearing sand may be determined by the following methods.

Improved Recovery Improved recovery includes all methods for supplementing natural reservoir forces and energy, or otherwise increasing ultimate recovery from a reservoir. Such recovery techniques include (1) pressure maintenance, (2) cycling, and (3) secondary recovery in its original sense (i.e., fluid injection applied relatively late in the productive history of a reservoir for the purpose of stimulating production after recovery by primary methods of flow or artificial lift has approached an economic limit). Improved recovery also includes the enhanced recovery methods of thermal, chemical flooding, and the use of miscible and immiscible displacement fluids.

GROSS

‘lf working I” Sl umls, the depths WIII be expressed in meters and the planlmetered areas enclosed by each contour w,ll be expressed I” hectares The resultant hectaremeter plot can be treated exactly llke the following acre-foot example to yield reserw~ ~oI!mes m cubic meters. (1 ha, m = 10,000 m3 )

GAS BEARING SAND VOLUME:

[(0+8&42lt4(24)]

GAS-OIL

~2367

ACRE FEET

CONTACT

GROSS OIL BEARING y

[W-42+

SAND VOLUME:

378 -242)+4(209-1061]=m

OIL-WATER

100

200 AREA

300

ENCLOSED

400

500

BY CONTOUR

Fig. 40.3-Acre-feet

diagram

CONTACT

600

ACRE FEET

ESTIMATION

40-5

OF OIL AND GAS RESERVES

Fig. 40.4-lsopachous

map-gas

a. Planimetered from the acre-feet diagram. b. If the number of contour intervals is even, computed by Simpson’s rule:

So/3[(0+136)+4(24+103)+2(46)]=

12,267 acre-ft.

(The separate calculations of the volume of gross gasbearing sand and gross oil-bearing sand by means of Simpson’s rule are shown in the diagram of Fig. 40.3.) r. With somewhat less accuracy, computed by the trapezoidal rule:

SO[%(O+ 136)+(24+46+

103)] = 12,050 acre-ft.

d. Computed by means of the somewhat more complicated pyramidal rule:

ss[(O+136)+2(24+46+

103)+J24x88

sand

available on many wells, it is sometimes justified to prepare an isopachous map of the number of porosity feet (porosity fraction times net pay in feet) and compute the total available void space in the net-pay section from such an isopachous map by the methods discussed under Item la, b, or c.

Computation of Oil or Gas in Place Volumetric Method If the size of the reservoir, its lithologic characteristics, and the properties of the reservoir fluids are known, the amount of oil or gas initially in place may be calculated with the following formulas: Free Gas in Gas Reservoir or Gas Cap (no residual oil present). For standard cubic feet of free gas, GFj =

= 11,963 acre-ft.

43,5601/,@(1 -Siw) *, ,

.

(1)

where

e. If the sand is ofuniform thickness, it will oftentimes suffice to multiply the average gross pay thickness h I by

the area enclosed by the contour 1/2Zfi above the WOC. J If the area within the top contour is circular (area A, height Z), then the top volume is QrZ+ %AZ if treated as a segment of a sphere, and %AZ if treated as a cone. From a study of the individual well logs or core data, it is then determined what fraction of the gross sand section is expected to carry and to produce hydrocarbons. Multiplication of this net-pay fraction by the gross sand volume yields the net-pay volume. If, for example, in the case illustrated with Figs. 40.2 and 40.3, it is found that 15% of the gross section consisted of evenly distributed shale or dense impervious streaks, the net gas- and oilbearing pay volumes may be computed as, respectively, net acre-ft of gas pay

and 0.85x9,900=8,415

map-oil

+m

+d5icEm-m-J]

0.85 x2,367=2,012

Fig. 40.5-lsopachous

sand

net acre-ft of oil pay.

2. From individual well-log data, separate isopachous maps may be prepared for the net gas pay (Fig. 40.4) or for the net pay (Fig. 40.5) and the total net acre-feet of oil- or gas-bearing pay computed as under It&m la, b, or c. 3. If the nature of the porosity varies substantially from well to well, and if good log and core-analysis data are

V, = net pay volume of the free-gas-bearing

4 S;, B, 43,560

= = = =

portion of a reservoir, acre-ft, effective porosity, fraction, interstitial water saturation, fraction, gas FVF, dimensionless, and number of cubic feet per acre-foot.

Values for the gas FVF or the reciprocal gas FVF, l/B,, may be estimated for various combinations of pressure, temperature, and gas gravity (see section on gas FVF).” Oil in Reservoir (no free gas present in oil-saturated portion). For stock-tank barrels of oil, N= 7,758V,4(1 -S,,) B,

,

....

. . . .

. (2)

where N = reservoir oil initially in place, STB, V, = net pay volume of the oil-bearing portion of a reservoir, acre-ft, B, = oil FVF, dimensionless, and 7,758 = number of barrels per acre-foot. ‘Refer ,oChaps. 20 through 25 for delaled properties. and correlalions.

coverage of 011.gas, condensate and watel

40-6

PETROLEUM

TABLE

40.1--BARRELS

OF STOCK-TANK

OIL IN PLACE

ENGINEERING

HANDBOOK

PER ACRE-FT

Porositv. d B

0

1.0

1.5

2.0

3.0

iwS 0.10 0.20 0.30 0.40 0.50 0.10 0.20 0 30 0.40 0.50 0.10 0 20 0.30 0.40 0.50 0.10 0.20 0.30 0.40 0.50

0.05

0.10

0.15

0.20

0.25

0.30

0.35

349 310 272 233 194 233 206 182 155 128 175 155 136 116 97 116 105 89 78 66

698 621 543 465 388 465 411 365 310 256 349 310 272 233 194 233 209 178 155 132

1,047 931 615 698 582 698 617 547 465 384 524 465 407 349 291 349 314 268 233 198

1,396 1,241 1,066 931 776 931 822 729 621 512 698 621 543 465 388 465 419 357 310 264

1,746 1,552 1.358 1.164 970 1,164 1,028 912 776 640 873 776 679 582 485 582 524 446 388 330

2,095 1,862 1,629 1,396 1,164 1,396 1,234 1,094 931 768 1,047 931 815 698 582 698 628 535 465 396

2,444 2,172 1,901 1,629 1,358 1,629 1,439 1,276 1,086 896 1,222 1,086 950 815 679 815 733 625 543 462

Table 40.1 shows the number of barrels of stock-tank oil per acre-foot for different values of porosity, 4, interstitial water saturation, S,,,., and the oil FVF, B,, Solution Gas in Oil Reservoir (no free gas present). For standard cubic feet of solution gas,

Gs =

7,7581/,@(1 -s,,.)R., Bo

.

.

(3)

to small changes in the two-phase FVF, B,, an adjustment procedure, called the Y method, may be used for the pressure range immediately below the bubblepoint. The method consists of plotting values of y= (Ph-PRPoi

pR(B,-B,,i)

Method5-8

In the absence of reliable volumetric data or as an independent check on volumetric estimates, the amount of oil or gas in place in a reservoir may sometimes be computed by the material-balance method.5 This method is based on the premise that the PV of a reservoir remains constant or changes in a predictable manner with the reservoir pressure when oil, gas, and/or water are produced. This makes it possible to equate the expansion of the reservoir fluids upon pressure drop to the reservoir voidage caused by the withdrawal of oil, gas, and water minus the water influx. Successful application of this method requires an accurate history of the average pressure of the reservoir, as well as reliable oil-, gas-, and waterproduction data and PVT data on the reservoir fluids. Generally, from 5 to 10% of the oil or gas originally in place must be withdrawn before significant results can be expected. Without very accurate performance and PVT data the results from such a computation may be quite erratic, 6 especially when there are unknowns other than the amount of oil in place, such as the size of a free-gas cap, or when a water drive is present. When the number of available equations exceeds the number of such unknowns, the solution should preferably be by means of the “method of least squares. “’ Because of the sensitivity of the material-balance equation

..........

.

. . . .(4)

where ph = pR = B, = Boi =

where G, is the solution gas in place, in standard cubic feet, and R,T is the solution GOR, in standard cubic feet per stock-tank barrel. Material-Balance

,

bubblepoint pressure, psia, reservoir pressure, psia, two-phase FVF for oil, dimensionless, initial oil FVF, dimensionless.

and

vs. reservoir pressure, PR, and bringing a straight line through the plotted points, with particular weight given to the more accurate values away from the bubblepoint. This straight-line relationship is then used to correct the previous values for Y, from which the adjusted values for B, are computed. Values of B, computed with this method for pressures substantially below the bubblepoint should not be used if differential liberation is assumed to represent reservoir producing conditions. When an active water drive is present, the cumulative water influx, W,, should be expressed in terms of the known pressure/time history and a water drive constant,’ thus reducing this term to one unknown. A completely worked-out example of the use of material balance that uses this conversion and in which the amount of oil in place is determined for a partial water drive reservoir where 36 pressure points and equations were available at a time when about 9 % of the oil in place had been produced is given in Ref. 7. The material-balance equation in its most general form reads N=

N,,[B,+O.l7XIB,(R,~-R,,)I-(W,,-~,,) B,q B, B,,, rnB + B- -(m+ ,q, 0,

I) I -

&RR(‘.,+S,,,“,!) 1 -s,,,

. . . . . . . . . . . . . . . . . . . . . . . ..~....

II (5)

’

ESTIMATION

40-7

OF OIL AND GAS RESERVES

TABLE Reservoir

40.2-CLASSIFICATION

OF MATERIAL-BALANCE Material-balance

Type

Oil reservoir with gas cap and active water drive

Np]B, +0.1781B,(R,

EQUATIONS

Equation’

-R,,)]-(W,

Unknowns

Equation

- WP)

N=

N, W,,

m

6

mB,,

Oil reservoir with gas cap; no active water drive (W, = 0)

Np[B, +0.1781B,(Rp

-I?,,)]+

w,

N=

N. m

7

N, W,

8

N, W,

9

N

10

N

11

ma,,

Initially undersaturated oil reservoir with active water drive (m =0): 1. Above bubblepoint

N,U N=

APl(C, +c, -S,&,

2. Below bubblepoint

N=

-S,,) 1(1

we-WP +APpRco) - ~ B,,

-c,)l

Npl~,+0.f781B,(R,-R,,)1-(W,-W,) 8, -60,

lnltially undersaturated oil reservoir: no active water drive (m = 0),(W, = 0): 1. Above bubblepoint

N,(l

+W&J-

F

1

(1 -St,)

01

N=

QJDR[c,+c,-S,,(c,-c,)l

NJ!3, 2. Below bubblemint

+ O.l781B,(R,

-R,,)]+

W,

N= 6, -go,

G=

G,B,

-5.615(W,

- WP)

Gas reservoir with active water drive

G

W,

12

B, --By

Gas reservoir; no active water drive we

=O)

where N,, R,, R.,, w,, w,, Aj?R

= = = = = =

B,pi = III = “f = c,, =

G,B,

+5.615W, G

G= 6,

13

-B,,

cumulative oil produced, STB, cumulative GOR, scf/STB. initial solution GOR. scf/STB, cumulative water influx, bbl, cumulative water produced, bbl, change in reservoir pressure, psi, initial gas FVF. res cu ftiscf, ratio of initial reservoir free-gas volume and initial reservoir-oil volume, compressibility of reservoir rock, change in PV per unit PV per psi, and compressibility of interstitial water, psi -’

When a free-gas cap is present, this equation may be simplified to Eq. 6 of Table 40.2 by neglecting the reservoir formation compressibility cf and the interstitial water compressibility c,,..

When such a reservoir has no active water drive Eq. 7 results. For initially undersaturated reservoirs (m = 0) below the bubblepoint, Eqs. 6 and 7 reduce to Eqs. 9 and I I, depending on whether an active water drive is present. For initially undersaturated reservoirs (m=O) above the bubblepoint, no free gas is present (R,) -R,yi =O). while B, =Bo;+A~~c, (where c, is the compressibility of reservoir oil, volume per psi), so that general Eq. 5 reduces to Eqs. 8 and 10, depending on whether an active water drive is present. For gas reservoirs the material-balance equation takes the form of Eq. 12 or 13, depending on whether an active water drive is present. The numerator on the right side in each case represents the net reservoir voidage by production minus water influx, while the denominator is the gas-expansion factor (BR -B,;) for the reservoir. (W,,=O),

PETROLEUM

40-8

ENGINEERING

HANDBOOK

TABLE 40.3-CONDITIONS FOR UNIT-RECOVERY EQUATION. DEPLETION-TYPE RESERVOIR Initial Conditions’ Reservoir Interstitial Free gas, Reservoir

pressure water, @SW, bbllacre-ft &S,, bbllacre-ft oil, bbllacre-ft

$58 0

7pp58 7,758 7,758$~(l - S,, -S,,)

7,758$41 -s,,) 1-S

Stock-tank

oil, bbl/acre-ft

7,758

7,7584

d2 BO,

'SubstIMe

10 000 for the 7.758 constanf

11c"b,c melers per hectare.mefer

Saturated Depletion-Type Oil Reservoirs-Volumetric Methods General Discussion Pools without an active water drive that produce solely as the result of expansion of natural gas liberated from solution in the oil are said to produce under a depletion mechanism, also called an internal- or solution-gas drive. When a free-gas cap is present, this mechanism may be supplemented by an external or gas-cap drive (Page 40-13). When the reservoir permeability is sufficiently high and the oil viscosity low, and when the pay zone has sufficient dip or a high vertical permeability, the depletion mechanism may be followed or accompanied by gravity segregation (Page 40-14). When a depletion-type reservoir is first opened to production, its pores contain interstitial water and oil with gas in solution under pressure. No free gas is assumed to be present in the oil zone. The interstitial water is usually not produced, and its shrinkage upon pressure reduction is negligible compared with some of the other factors governing the depletion-type recovery. When this reservoir reaches the end of its primary producing life, and disregarding the possibility of gas-cap drive or gravity segregation, it will contain the same interstitial water as before, together with residual oil under low pressure. The void space vacated by the oil produced and by the shrinkage of the remaining oil is now filled with gas liberated from the oil. During the depletion process this gas space has increased gradually to a maximum value at abandonment time. The amount of gas space thus created is the key to the estimated ultimate recovery under a depletion mechanism. It is reached when the produced free GOR in the reservoir, which changes according to the relative permeability ratio relationship and the viscosities of oil and gas involved, causes exhaustion of the available supply of gas in solution. Unit-Recovery Equation The unit-recovery factor is the theoretically possible ultimate recovery in stock-tank barrels from a homogeneous unit volume of 1 acre-t? of pay produced by a given mechanism under ideal conditions. The unit-recovery equation for a saturated depletiontype reservoir is equal to the stock-tank oil initially in place in barrels per acre-foot at initial pressure pi minus the residual stock-tank oil under abandonment pressure pi,, as shown in Table 40.3.

Ultimate Conditions’

1 -s&v -s,, B w

IS used.

By difference, the unit recovery by depletion or solution-gas drive is, in stock-tank barrels per acre-foot, 1 - S,M - s,, B o(I

'

.'."

(14)

where S,, is the residual free-gas saturation under reservoir conditions at abandonment time, fraction, and B,, is the oil FVF at abandonment, dimensionless. The key to the computation of unit recovery by means of this equation is an estimate of the residual free-gas saturation S,, at the ultimate time. If a sufficiently large number of accurate determinations of the oil and water saturation on freshly recovered core samples is available, an approximation of S,, may be obtained by deducting the average total saturation of oil plus water from unity. This method is based on the assumption that the depletion process taking place within the core on reduction of pressure by bringing it to the surface is somewhat similar to the actual depletion process in the reservoir. Possible loss of liquids from the core before analysis may cause such a value for S,, to be too high. On the other hand, the smaller amount of gas in solution in the residual oil left after flushing by mud filtrate has a tendency to reduce the residual free-gas saturation. Those using this method hope that these two effects somewhat compensate for each other. A typical S,, value for average consolidated sand, a medium solution GOR of 400 to 500 cu ftibbl, and a crude-oil gravity of 30 to 4O”API is 0.25. Either a high degree of cementation, a high shale content of the sand, or a 50% reduction in solution GOR may cut this typical S,, value by about 0.05, while a complete lack of cementation or shaliness such as in clean, loose unconsolidated sands or a doubling of the solution GOR may increase the S,, value by as much as 0.10. At the same time, the crude-oil gravity generally increases or decreases the S,, value by about 0.01 for every 3”API gravity. Example Problem 1. A cemented sandstone reservoir has an interstitial water content a porosity $=0.13, S,,,.=O.35, a solution GOR at bubblepoint conditions, /?,I, =300 cu ftibbl, an initial oil FVF B,,; = 1.20, an oil FVF at abandonment B,, = I .07, and a stock-tank oil gravity of 40”API. Based on the above considerations, the higher-than-average oil gravity would just about offset the effect of the somewhat lower-than-average GOR. and the residual free-gas saturation S,, after a 0.05 reduction for the cementation can therefore be estimated at 0.20.

ESTIMATION

40-9

OF OIL AND GAS RESERVES

Solution. The unit recovery by depletion according to Eq. 14 would be

N,, =(7.758)(0.13)

l-0.35

l-O.35

-0.20

1.07 = 122 STBiacre-ft

This stepwise solution of the depletion equation yields the reservoir oil saturation S,, as a function of reservoir pressure pR. The results may be converted into cumulative recovery per acre-foot. In stock-tank barrels per acre-foot,

> (16)

[I57 m3/ha.mj.

where N,, is the unit recovery by depletion or solutiongas drive, STB. Muskat’s Method. 9 If the actual relationships between pressure and oil-FVF B,, gas-FVF B,, gas-solubility in oil (solution GOR) R, , oil viscosity p,), and gas viscosity ps are available from a PVT analysis of the reservoir fluids, and if the relationship between relative permeaand the total liquid saturation, S,, is bility ratio k,/k, known for the reservoir rock under consideration, the unit recovery by depletion can be arrived at by a stepwise computation of the desaturation history directly from the following depletion equation in differential form: As,, -1

The results may be converted into cumulative recovery as a fraction of the original oil in place (OOIP) by

L+L)

(?c), .,....,.......

N

(17)

while the GOR history, in standard cubic feet gasistocktank barrel, may be computed by

(18) where R is the instantaneous producing GOR, in standard cubic feet per stock-tank barrel, and the relative production rate in barrels per day by

APR

B, dR, d(liB,s) S,,‘+(I -s,, -s,,, )B,L!-+s,,--dl’R B,, ‘k’R

. ..t...

.I..........,.........

PL,,k,.,

dB,,

ko

I-‘,?k,,, Bdr’R

(15)

where S, = oil or condensate saturation under reservoir conditions, fraction, PLO= reservoir oil viscosity, cp, PLK= reservoir gas viscosity, cp, k, = relative permeability to gas as a fraction of absolute permeability, and k, = relative permeability to oil as a fraction of absolute permeability. The individual computations are greatly facilitated by computing and preparing in advance in graphical form the following groups of terms, which are a function of pressure only,

and the relative permeability ratio k,ik,,, which is a function of total liquid saturation S, only. The accuracy of this type of calculation on a desk calculator falls off rapidly if the pressure decrements chosen are too large, particularly during the final stages when the GOR is increasingly rapidly. With modern electronic computers, however, it is possible to use pressure decrements of IO psi or smaller, which makes a satisfactory accuracy possible.

Poi

PR .

where 90 kc, km Poi 40;

= = = = =

.

.

(19)

oil-production rate, B/D, effective permeability to oil. md, initial effective permeability to oil. md, initial reservoir oil viscosity, cp, and initial oil-production rate, B/D.

It should be stressed that this method is based on the assumption of uniform oil saturation in the whole reservoir and that the solution will therefore break down when there is appreciable gas segregation in the formation. It is therefore applicable only when permeabilities are relatively low. Another limitation of this method as well as of the Tarner method, discussed hereafter, is that no condensation of liquids from the produced gas is assumed to take place in the tubing or in the surface extraction equipment. It should therefore not be applied to the high-temperature, high-GOR, and high-FVF “volatile” oil reservoirs to be discussed later. Tarner’s Method. Babson ‘” and Tarner ” have advanced trial-and-error-type computation methods for the desaturation process that require a much smaller number of pressure increments and can therefore be more readily handled by a desk calculator. Both methods are based on a simultaneous solution of the material-balance equation (Eq. 11) and the instantaneous GOR (Eq. 18). Tarner’s method is the more straightforward of the two. The procedure for the stepwise calculation of the cumulative oil produced (N,,)I and the cumulative gas produced (Gp)* for a given pressure drop from p I to p, is as follows.

PETROLEUM

40-l 0

TABLE

40.4-COMPUTED

DEPLETION

RECOVERY

IN STBIACRE-FTIPERCENT

POROSITY

ENGINEERING

FOR TYPICAL

HANDBOOK

FORMATIONS

Solution GOR (cu ftlbbl)

Oil Gravity, (OAPI)

cRsb)

-70

Unconsolidated

Consolidated

Highly Cemented

Vugular

Fractured

;z 50 15 30 50 15 30 50 30 50 50

7.2 12.0 19.2 7.0 11.6 19.4 7.6 10.5 15.0 12.3 12.0 10.6

4.9 8.5 13.9 4.6 7.9 13.7 4.8 6.5 9.7 7.6 7.2 6.4

1.4 4.9 9.5 1.8 4.4 9.2 2.5 3.6 5.8 4.5 4.1 4.0

2.6 6.3 11.8 2.6 5.8 11.4 3.3

0.4 18 5.1 0.5 1.5 4.4 0.9

60 200

600

1,000 2,000

Sand or Sandstone (S,, = 0.25)

1. Assume that during the pressure drop from p , to pl the cumulative oil production increases from (N,) , to (N,,)* N, should be set equal to zero at bubblepoint. 2. Compute the cumulative gas produced (G,,)z at pressure p2 by means of the material-balance equation assuming (Eq. 111, which for this purpose-and Wp =0-is rewritten in the following form:

(G,,h =(N,h(R,,):!=N

(R.7,-R,\)-5.615

3. Compute the fractional total liquid saturation @,)I at pressure p2 by means of (s’);=S;~+(l-s;,,J~[l-~].

.., . ..(21)

4. Determine the k,lk,, ratio corresponding to the total liquid saturation (S,), and compute the instantaneous GOR at p2 by means of ....

R* =R,$ +ui15$+. RPK

..

(22)

ro

5. Compute the cumulative gas produced at pressure p2 by means of (G,)2=(Gp)1+

RI +R, ---[VP)2 2

-VP) 11,

. (23)

in which RI represents the instantaneous GOR computed previously at pressure p, . Usually three judicious guesses are made for the value (N,) 2 and the corresponding values of (G,,) 2 computed by both Steps 2 and 5. When the values thus obtained for (G,) 2 are plotted vs. the assumed values for (N,) 2 , the intersection of the curve representing the results of Step 2 and the one representing Step 5 then indicates the cumulative gas and oil production that will satisfy both equations. In actual application, the method is usually simplified further by equating the incremental gas production (Gp)z -(G,) I) rather than (G,)Z itself. This

Limestone, Dolomite or Chert (S,, =0.15)

4.7 7.2 5.4 4.8 (4.3)

(1.2) (2.1) (1.6) (1.2) (1.5)

equality signifies that at each pressure step the cumulative gas, as determined by the volumetric balance, is the same as the quantity of gas produced from the reservoir, as controlled by the relative permeability ratio of the rock, which in turn depends on the total liquid saturation. Although the Tamer method was originally designed for graphical interpolation, it also lends itself well to automatic digital computers. The machine then calculates the quantity of gas produced for increasing oil withdrawals by both equations and subtracts the results of one from the other. When the difference becomes negative, the machine stops and the answer lies between the last and next to last oil withdrawals. Tarner’s method has been used occasionally to compute recoveries of reservoirs with a free-gas cap or to evaluate the possible results from injection of all or part of the produced gas. When a free-gas cap is present, or when produced gas is being reinjected, breakthrough of free gas into the oil-producing section of the reservoir is likely to occur sooner or later, thus invalidating the assumption of uniform oil saturation throughout the producing portion of the reservoir, on which the method is based. Since such a breakthrough of free gas causes the instantaneous GOR (Eq. 18) as well as the entire computation method to break down, the use of Tamer’s method in its original form for this type of work is not recommended. It should also be used with caution when appreciable gas segregation in an otherwise uniform reservoir is expected. Computed Depletion-Recovery Factors. Several investigators9, 12-14 have used the Muskat and Tarner methods to determine the effects of different variables on the ultimate recovery under a depletion mechanism. In one such attempt I2 the k,lk, relationships for five different types of reservoir rock representing a range of conditions for sands and sandstones and for limestones, dolomites, and cherts were developed. These five types of reservoir rock were assumed to be saturated under reservoir conditions with 25 % interstitial water for sands and sandstones and 15 % for the limestone group and with 12 synthetic crudeoil/gas mixtures representing a range of crude-oil gravities from 15 to 5O”API and gas solubilities from 60 to 2,000 cu ft/STB. Their production performance and recovery factors to an abandonment pressure equal to 10% of the bubblepoint pressure were then computed by means of depletion (Eq. 15).

ESTIMATION

OF OIL AND GAS RESERVES

10.0 z 2

1.0

e = P

0.1 0.01 5 TOT PER

Notes: interstitial water is assumed to be 30% of pore space and deadoil viscosity at reservoir temperature to be 2 cp. Equilibrium gas saturation is assumed to be 5% of pore space. As here used “ultimate oil recovery” is realized when the reservoir pressure has declined from the bubblepoint pressure to atmospheric pressure. FVF units are reservoir barrels per barrel of residual oil. Solution GOR units are standard cubic feet per barrel of residual oil. Example 1: Required: Ultimate recovery from a system -having a bubblepoint pressure = 2,250 psia, FVF = 1.6, and a solution GOR. Procedure: Starting at the left side of the chart, proceed horizontally along the 2,250-psi line to FVF = 1.6. Now rise vertically 10 the 1,300-scflbbl line. Then go horizontally and read an ultimate recovery of 23.8%. Example 2: F)eqoired:Convert the recovery figure determined in Example 1 to tank oil recovered. Data requirements: Differential liberation data given in Example 1. Flash liberation data: bubblepoint pressure = 2,250 psia, FVF = 1.485, FVF at atmospheric pressure = 1.080 for both flash and differential liberation.

FORMATIONVOLUME FACTOR Procedure: Calculate the oil saturation at atmospheric pressure by substituting differential liberation data in the equation as follows:

Oil saturation at atmospheric pressure = 0.360. Next, substitute the calculated value of oil saturation and the flash liberation data into the previous equation and calculate the ultimate oil recovery as a percentage of tank oil originally in place.

N,, (ultimate place.

oil recovery)=29.3%

of tank oil originally

in

Fig. 40.6-Chart for estimating ultimate recovery from solution gas-drive reservoirs.

These theoretical depletion-recovery factors, expressed as barrels of stock-tank oil per percent porosity, will be found in Table 40.4 for the different types of reservoir rocks, oil gravities, and solution GOR’s assumed. In cases where no detailed data are available concerning the physical characteristics of the reservoir rock and its fluid content, Table 40.4 has been found helpful in estimating the possible range of depletion-recovery factors. It may be noted that the k,lk, relationship of the reservoir rock is apparently the most important single factor governing the recovery factor. Unconsolidated intergranular material seems to be the most favorable, while increased cementation or consolidation tends to affect recoveries unfavorably. Next in importance is crude-oil gravity with viscosity as its corollary. Higher oil gravi-

ties and lower viscosities appear to improve the recovery. The effect of GOR on recovery is less pronounced and shows no consistent pattern. Apparently the beneficial effects of lower viscosity and more effective gas sweep with higher GOR is in most cases offset by the higher oil FVF’s. In general, these data seem to indicate a recovery range from the poorest combinations of 1 to 2 bbl/acre-fi for each percent porosity to the best combinations of 19 to 20 bbllacre-Mpercent porosity. An overall average seems to be around 10 bbliacre-ftlpercent porosity. It is also of interest to note that when the reservoir is about two-thirds depleted, the pressure has usually dropped to about one-half the value at bubblepoint.

PETROLEUM

40-12

In another attempt ” nine nomographs were developed, each for a given combination of the k, lk ,.(,curve, “deadoil” viscosity, and interstitial water content. The nomograph for an average k,lk, relationship, an interstitial water content of 0.30. and a dead-oil viscosity of 2 cp is reproduced as Fig. 40.6. Instructions for its use are shown opposite the figure. The authors ” also introduced an interesting empirical relationship between the relative permeability ratio k,/k,, the equilibrium gas saturation S,,., the interstitial water saturation S,,., and the oil saturation S,: k ri: = i(O.0435 +0.4556E), k t-0

. (24)

where t;=(l -S,,.-S,, -S,)/(S, -0.25). A similar correlation I5 for sandstones that show a linear relationship between lip,’ (where p,.=critical pressure) and saturation is k rg -=

(1 -S*)I[

1 -@*)I]

(s*)4

k ro

,

.

(25)

where effective saturation S*=S,I(l -Si,). This tion represents a useful expression for calculating tive permeability ratios in sandstone reservoirs for an average water saturation has been obtained by electrical log or core analysis.

equarelawhich either

In a statistical study of the actual performance of 80 solution gas-drive reservoirs, the API Subcommittee on Recovery Efficiency I6 developed the following equation for unit recovery (N,,) below the bubblepoint for solution gas-drive reservoirs, in stock-tank barrels per acre-foot*: N,, =3,*44 [ 44;,y

1.‘6” x (2-J

0.1741

x(s,

,)O.3722x !k IM ( >

.

With progressively deeper drilling, a number of oil reservoirs have been encountered that, while lacking an active water drive, are in undersaturated condition. Because of the expansion of the reservoir fluids and the compaction of the reservoir rock upon pressure reduction, substantial recoveries may sometimes be obtained before the bubblepoint pressure pb is reached and normal depletion sets in. Such recoveries may be computed as follows. The oil initially in place in stock-tank barrels per acrefoot at pressure pi is according to Eq. 2, .. ‘.

73758x4i(1-Siw)

’

where 4; is initial porosity. By combining this expression with the material-balance equation (Eq. 10). the recovery factor above the bubblepoint in stock-tank barrels per acre-foot may be expressed as

Np=

7375Wi(Pi-Pb)[Co

+Cf-Siw(cc~-~w)l

I (27)

Boi[lfco(Pi-Pb)l

where c,,, is the compressibility volume per volume per psi.

.. .

Example Problem 2. Zone D-7 in the Ventura Avenue field, described by E.V. Watts,” is an example of an undersaturated oil reservoir without water drive. Its reservoir characteristics are

(26)

where k = absolute permeability,

P,~ = Pa = pb =

Undersaturated Oil Reservoirs Without Water Drive Above the BubblepointVolumetric Method t7-19

o.0979

Pa

B ob =

HANDBOOK

be made for each permeability bank that is known to be continuous and the results converted into rate/time curves for each by combining Eqs. 16 and 19. The estimated ultimate recovery will then be based on a superposition of such rate/time curves for the different zones. If there is a wide divergence in permeabilities, one may find that at a time when the combined rate for all zones has reached the economic limit the more permeable banks will be depleted and have yielded their full unit recovery while the pressure depletion and the recovery from the tighter zones are still incomplete.

Boi

API Estimation of Oil and Gas Reserves

ENGINEERING

darcies, oil FVF at bubblepoint, RBLSTB, oil viscosity at bubblepoint, cp, abandonment pressure, psig, and bubblepoint pressure, psig.

The permeability distribution in most reservoirs is usually sufficiently nonuniform in vertical and horizontal directions to cause the foregoing depletion calculations on average material to be fairly representative. However, when distinct layers of high and low permeability, separated by impervious strata, are known to be present, the depletion process may advance more rapidly in high-permeability strata than in low-permeability zones. In such cases separate performance calculations should

of interstitial water in

pi = 8,300 psig at 9,200 ft,

pb = #Ii = s 1M’ = B oh = B o(1 = 70 = CO = cw = Cf = S,, = Rsb =

3,500 psig, 0.17, 0.40,

1.45, 1.15, 32 to 33”API, 13x10-6, 2.7~10-~, 1.4x10-6, 0.22, and 900 cu ft/bbl.

Solution. On the basis of these data, Watts computes the recovery by expansion above the bubblepoint at 47 bbliacre-ft and by a depletion mechanism below the bubblepoint at 110 bbl/acre-ft (see Ref. 19 for details).

ESTIMATION

40-13

OF OIL AND GAS RESERVES

Volatile Oil ReservoirsVolumetric Methods20-25 Deeper drilling, with accompanying increases in reservoir temperatures and pressures, has also revealed a class of reservoir fluids with a phase behavior between that of ordinary “black” oil and that of gas or gas condensate. These intermediate fluids are referred to as “highshrinkage” or “volatile” crude oils because of their relatively large percentage of ethane through decane components and resultant high volatility. Volatile-oil reservoirs are characterized by high formation temperatures (above 200°F) and abnormally high solution GOR and FVF (above 2). The stock-tank gravity of these volatile crudes generally exceeds 45 “API. The inherent differences in phase behavior of volatile oils are sufficiently significant to invalidate certain premises implicit in the conventional material-balance methods. In such conventional material-balance work it is assumed that all produced gas, whether solution gas or free gas, will remain in the vapor phase during the depletion process, with no liquid condensation on passage through the surface separation facilities. Furthermore, the produced oil and gas are treated as separate independent fluids, even though they are at all times in compositional equilibrium. Although these basic assumptions simplify the conventional material-balance calculations, highly inaccurate predictions of reservoir performance may result if they are applied to volatile-oil reservoirs. In highly volatile reservoirs, the stock-tank liquids recovered by condensation from the gaseous phase may actually equal or even exceed those from the associated liquid phase. This rather surprising occurrence is exemplified in a paper by Woods,24 in which the case history of an almost depleted volatile-oil reservoir is presented. Example Problem 3. Woods’ reservoir data for this volatile-oil reservoir were pi = 5,000 psig, pb = 3,940 psig, TR = 250”F, c$ = 0.198. k = 75 md, Sib,, = 0.25, R,,, = 3,200 scf/bbl, yoi = 44”API, You = 62”API, and B oh = 3.23. Solution. At 80% depletion when pR = 1,450 psig and R =23,000 scf/bbl, the percentage recovery was 2 1% of which 5% was from expansion above the bubblepoint, 9% from the depletion mechanism, and 7% from liquids condensed out of the gas phase by conventional field separation equipment (see Ref. 24 for details). In view of the increasing number and importance of volatile-oil reservoirs in recent years, appropriate techniques have been developed to provide realistic predictions of the anticipated production performance of these reservoirs. 2o-z5 The depletion processes are simulated by an incremental computation method, using multicomponent flash calculations and relative-permeability data, as indicated in the following stepwise sequence for a chosen pressure decrement:

1. The change in composition of the in-place oil and gas is determined by a flash calculation. 2. The total volume of fluids produced at bottomhole conditions is determined by a volumetric material balance. 3. The relative volumes of oil and gas produced at bottomhole conditions are determined by a trial-and-error procedure that involves simultaneously satisfying the volumetric material balance and the relative-permeability relationship. 4. This total well-stream fluid is then flashed to actual surface conditions to obtain the producing GOR and the volume of stock-tank liquid corresponding to the selected pressure decrement. When this calculation procedure is repeated for successive pressure decrements, the resultant tabulations represent the entire reservoir depletion and recovery processes. Since these stepwise calculations are rather tedious and time-consuming, the use of digital computers is recommended. This method of reservoir analysis provides compositional data on all fluid phases, including the total wellstream. This information is then readily available for separator, crude-stabilization, gasoline-plant, or related studies at any desired stage of depletion. In the case of small reservoirs with relatively limited reserves, such lengthy laboratory work and phasebehavior calculations may not be justified. An empirical correlation was developed24 for prediction of the ultimate recovery in such cases, based only on the initial producing GOR, R, the reservoir temperature, TR, and the initial stock-tank oil gravity, yO;. N,, = -0.070719+-

+O.O011807y~i,

143.50 +O.O001208OT,

R

.

. .

.

(28)

where N,, =ultimate oil production from saturation pressure ph to 500 psi, in stock-tank volume per reservoir volume of hydrocarbon pore space. It is claimed that this correlation will give values within 10% of those calculated by the more rigorous procedure previously outlined.

Oil Reservoirs With Gas-Cap DriveVolumetric Unit Recovery Computed by Frontal-Drive MethodZ628 The Buckley-Leverett frontal-drive method may be used in calculating oil recovery when the pressure is kept constant by injection of gas in a gas cap but is also applicable to a gas-cap drive mechanism without gas injection when the pressure variation is relatively small so that changes in gas density, solubility, or the reservoir volume factor may be neglected. A reservoir with a very large gas-cap volume as compared with the oil volume can sometimes be considered to meet these qualifications even though no gas is being injected. The two basic equations, Eqs. 29a and b, refer to a linear reservoir under constant pressure with a constant cross-sectional area exposed to fluid flow and with the free gas moving in at one end of the reservoir and fluids being produced at a constant rate at the other end. Interstitial water is considered as an immobile phase.

PETROLEUM

40-14

s?

I

I

I

0

I

I

I

lbfil -Al

VE A

!I

HANDBOOK

Note: Sk as used in this section is gas saturation as a fraction of the hydrocarbon-filled pore space. When N is in cubic meters, q1 is in cubic meters per day. The calculation procedure is first to calculate the fractional-flow curve (Fig. 40.7, Curve A). The average gas saturation in the swept area at breakthrough, which is equivalent to the fraction of oil in place recovered, may then be obtained from the fractional-flow curve by constructing a straight line tangent to the curve through the origin and reading Sk at fR = 1.O. The time of breakthrough at the outlet face may be computed from the slope of the curve at the point of tangency. The subsequent performance history after breakthrough may then be calculated by constructing tangents at successively higher values of Sk and obtaining Sh in a similar manner.

’

I

ENGINEERING

--i

Example Problem 4. Welge2s presents a typical calculation of gas-cap drive performance for the Mile Six Pool in Peru. Given: 0=

I

0

0.10 0.20 0.30

I

&O

I 0.50 0.60

Reservoir volume= 1,902 X lo6 cu ft, distance from original GOC to average withdrawal point = 1,540 ft,

0.70

S&GAS SATURATION, FRACTION OF HYDROCARBON FILLED PORE SPACE Fig. 40.7-Frontal-drive

method in gas-cap

1,902x IO6 average cross-sectional

drive

=1.235x106 If the capillary-pressure forces are neglected. fractional-flow equation of gas is

the

(294

E=

k sin @A@,--pR)

.

..

36%.,qr

(29b)

where fX = E = 8 = A =

fractional flow of gas, parameter, dip angle, degrees, area of cross-section normal to bedding plane, sq ft, PO = density of reservoir oil, g/cm3, ph’ = density of reservoir gas, g/cm3. and q, = total flow rate, reservoir cu ft/D.

5.615NB, q,(df,,dS;)

.

.

1,540

sq ft,

k, = 300 md, 8 = 17.50, ps = 0.0134 cp,

P”o = 1.32 cp, q, = 64,000 res cu ft!D [I8 125 res m”/d], B,, = 1.25, B, = 0.0141 N = 44~ lo6 STB [6.996x106 m”], R,, = 400 cu ft/bbl [71.245 m’/m’J, PO = 0.78 g/cm”, and Ph’ = 0.08 g/cm 3 Solution. The performance history calculations given in Table 40.5 in a slightly simplified form.

are

Oil Reservoirs Under Gravity Drainage 29-37 Occurrence of Gravity Drainage

Since the ratio of k,lk, is a function of gas saturation, and all other factors are constant, j$ can be determined by Eq. 29a as a function of gas saturation (see Fig. 40.7, Curve A). The rate-of-frontal-advance equation may be rearranged to give the time in days for a given displacing-phase saturation to reach the outlet face of the linear sand body as a function of the slope of the fractional flow vs. saturation curve (Fig. 40.7, Curve B) as follows: t=

area =

(30)

Gravity drainage is the self-propulsion of oil downward in the reservoir rock. Under favorable conditions it has been found to effect recoveries of 60% of the oil in place, which is comparable with or exceeding the recoveries normally obtained by water drive. Gravity is an ever-present force in oil fields that will drain oil from reservoir rock from higher to lower levels wherever it is not overcome by encroaching edge water or expanding gas. Gravity drainage will be most effective if a reservoir is produced under conditions that allow flow of oil only or counterflow of oil and gas. This may be attained under pressure maintenance by crestal-gas injection, which keeps the gas in solution, or it may be attained by a gradual reduction in pressure, so that the oil and gas can segregate continuously by counterflow. It also may be obtained by

ESTIMATION

OF OIL AND GAS RESERVES

40-15

first producing the reservoir under a depletion-type mechanism until the gas has been practically exhausted, then by gravity drainage. A thorough discussion of the many aspects of gravity drainage will be found in the classic paper by Lewis.32 Several investigators 33m36have attempted to formulate gravity drainage analytically, but the relationships are quite complicated and not readily adaptable to practical field problems. Most studies agree, however, that the occurrence of gravity drainage of oil will be promoted by low viscosities, p,, , high relative permeability to oil, k,, high formation dips or lack of stratification, and high density gradients (p, -p,). Thick sections of unconsolidated sand with minimal surface area, large pore sizes, low interstitial water saturation, and consequently high k, appear to be especially favorable. These factors usually are combined in a rate-of-flow equation. which states that such flow must be proportional to (k,,lp,)(p,, -p,) sin 8, in which 8 represents the angle of dip of the stratum. Smithj7 compared the values of this term for a dozen reservoirs, some of which had strong gravity-drainage characteristics and some of which lacked such characteristics. When expressing k,,, in millidarcies, p,, in centipoises, and p,, and pI: in g/cm”, it was found that for reservoirs exhibiting strong gravity-drainage characteristics the value of the term (k,,ip,)(p, -P,~) sin 0 ranged from 10 to 203 and that in reservoirs where gravity-drainage effects were not apparent, this function showed values between 0.15 and 3.4.

y(, =22.5”API, N,, for Jan. 1, 1957=44.6 million bbl of oil; estimated ultimate 47 million bbl or I, 124 bbliacreft, corresponding to 63% of the initial oil in place. During the first 20 years the oil level in the field receded almost exactly in proportion to the amount of oil produced, just as in a tank. 2. Okluhoma City Wilcox Reservoir, OK. 29~32The discovery well, Mary Sudik No. I, blew out in March 1930, and flowed wild for 11 days. The segregation of gas and development of gravity drainage began to be important in 1934, when the average pressure became less than 750 psig, and was virtually complete by 1936, when the average pressure had dropped to 50 psig. Water influx played an effective role until 1936, when it came to a halt after invading the bottom 40% of the reservoir. Gravity has been the dominant mechanism since. The Wilcox sand consists of typical round frosted sand grains, clean and poorly cemented. The average depth is 6,500 ft; the formation dip is 5 to 15”; 884 wells have been drilled on a total area of 7,080 acres. The net pay thickness is 220 ft. The 890,000 net acre-ft of Wilcox pay contained originally 1,083 million bbl of stock-tank oil, as confirmed by material balance. Reservoir data for this reservoir are pi =ph = 2,670 psi at minus 5,260 ft, TR= 132”F, $=0.22, k ranges from 200 to 3,000 md, S;,.=O.O3 (oil wet), Rt,, =735 cu ft/bbl, B,;=l.361, y,i=40”APl, yoci=38 tO 39”API. According to Katz, z9 oil saturations found in the gas zone were between 1 and 26%, while saturations between 53 and 93% were found in the oil-saturated zone below the GOC. The oil saturation below the WOC has been estimated at 43%, showing gravity to be more effective than water displacement in this reservoir. Cumulative production, N,, for Jan. 1, 1958, is estimated at 525 million bbl and the ultimate recovery at 550 million bbl. After an estimated 189 million bbl displaced by the water influx is deducted, the upper 60% of the Wilcox reservoir will yield under gravity drainage ultimately 361 million bbl or 696 bbliacre-ft, corresponding to 57% of the oil in place.

Case Histories of Gravity Drainage After Pressure Depletion The most spectacular cases of gravity drainage have been of this kind. Following are the two best known. 1. Lukeview Pool in Kern County, CA. 3’~32 The discovery well in the Lakewood gusher area blew out in March 1910, flowed wild for 544 days, and ultimately produced 8% million bbl of oil, depleting the reservoir pressure. Gravity drainage thereafter controlled this reservoir. There was no appreciable water influx. The sand is relatively clean and poorly cemented. The average depth is 2,875 ft. The formation dip is IS to 45”. There are I26 producing wells on 588 acres. The net sand thickness averages 7 1 ft, the height of the oil column is 1,285 ft. and there are 41,798 net acre-ft of pay. Reservoir data for this reservoir are pi =P/, = 1,285 psi& PR on Jan. I, l957=35 psig, r,= 115°F. 4=0.33, k ranges up to 4,800 md and averages 3.600 md (70% of samples above 100 md, 37% above 1,000 md), S,,, =0.235, R,,,=200 cu ftibbl, Boi= 1.106,

TABLE

Oil Reservoirs With Water DriveVolumetric Method9 General Discussion Natural-water influx into oil reservoirs is usually from the edge inward parallel to the bedding planes (edgewater drive) or upward from below (bottomwater drive). Bottomwater drive occurs only when the reservoir thickness exceeds the thickness of the oil column, so that the oil/water interface underlies the entire oil reservoir. It is

40.5~PERFORMANCE-HISTORY

CALCULATION

s: = S’ near Outget Face 0.30 a 35

ro k 0.197 0.140

kro’k,, 0.715 0.364

0.496 0.642

0.395 0.40 0.45 0.50

0.102 0.097 0.067 0.045

0.210 0.200 0.118 0.0715

0.739 0.752 0.829 0.885

f,

df,lds;

1 .a7 1.81 1 .25 0.94

Recover; Fraction of Oil in Place

k

Flowing GOR =

If,41 -01(&/Q x5. I?? l+R,

-

-

-

7.1 7.3 10.6 14.1

0.534 0.535 0.586 0.622

1.808 1.908 2.811 4.227

PETROLEUM

40-16

TABLE 40X-CONDITIONS FOR UNIT-RECOVERY EQUATION, WATER-DRIVE RESERVOIR

Reservoir pressure Interstitial water, bbllacre-ft Reservoir oil, bbllacre-ft Stock-tank oil, bbllacre-ft

Initial Conditions

Ultimate Conditions

Pi

Pa

7,75848,,

7,75&S,,

7.756@(1 -S,,)

7,758@,,

Recovery-Efficiency

further possible only when vertical permeabilities are high and there is little or no horizontal stratification with impervious shale laminations. In either case, water as the displacing medium moves into the oil-bearing section and replaces part of the oil originally present. The key to a volumetric estimate of recovery by water drive is in the amount of oil that is not removed by the displacing medium. This residual oil saturation (ROS) after water drive, S,,, plays a role similar to the final (residual) gas saturation, S,, , in the depletiontype reservoirs. To determine the unit-recovery factor, which is the theoretically possible ultimate recovery in stock-tank barrels from a homogeneous unit volume of 1 acre-ft of pay produced by complete waterflooding, the amount of interstitial water and oil with dissolved gas initially present will be compared with the condition at abandonment time, when the same interstitial water is still present but only the residual or nonfloodable oil is left. The remainder of the original oil has at that time been removed by water displacement. Unit-Recovery

Equation

The unit recovery for a water-drive reservoir is equal to the stock-tank oil originally in place in barrels per acrefoot minus the residual stock-tank oil at abandonment time (Table 40.6). By difference, the unit recovery by water drive, in stock-tank barrels per acre-foot, is .(31)

where N,,. is the unit recovery by water drive, in stocktank barrels, and S,, is the residual oil saturation, fraction. The ROS at abandonment time may be found by actually submitting cores in the laboratory under simulated reservoir conditions to flooding by water (flood-pot tests). Another method commonly used is to consider the oil satuTABLE

40.7-RECOVERY-EFFICIENCY

Reservoir Number 1 2 3 4 5

$I 0.179 0.170 0.153 0.192 0.196

Factor

The unit recovery should be multiplied by a permeabilitydistribution factor and a lateral-sweep factor before it may be applied to the computation of the ultimate recovery for an entire water-drive reservoir. These two factors usually are combined in a recoveryefficiency factor. Baucum and Steinle3’ have determined this recovery-efficiency factor for five water-drive reservoirs in Illinois. Table 40.7 lists the recovery efficiencies for these reservoirs, together with some other pertinent data. Average Recovery Factor From Correlation of Statistical Data In 1945, Craze and Buckley,39,40 in connection with a special API study on well spacing, collected a large amount of statistical data on the performance of 103 oil reservoirs in the U.S. Some 70 of these reservoirs produced wholly or partially under water-drive conditions. Fig, 40.8 shows the correlation between the calculated ROS under reservoir conditions and the reservoir oil viscosities for these water-drive reservoirs. The deviation of the ROS from the average trend in Fig. 40.8, vs. permeability, is given by the average trend in Fig. 40.9. The deviation of the ROS from the average trend in Fig. 40.8, vs. reservoir pressure decline, is given by the average trend in Fig. 40.10. Example Problem 5. In a case where the porosity, 4=0.20, the average permeability, k=400 md, the interstitial water content, Si,=O.25, the initial oil FVF, B,, = 1.30, the oil FVF under abandonment conditions, B, = 1.25, the initial reservoir oil viscosity, pLo= 1.O cp, and the abandonment pressure, pu =90% of the initial pressure, pi, determine the average ROS. Solution. S,, may be estimated as 0.35+0.03-0.04= 0.34 and the average water-drive recovery factor from Eq. 31 is l-O.25

N,,.=(7,758)(0.20)

0.34 >

=473 STBlacre-ft FOR WATER-DRIVE

S,,

B,

S,,’

Unit-Recovery Factor (bbl/acre-ft)

0.400 0.340 0.265 0.370 0.360

1.036 1.017 1.176 1.176 1.017

0.20 0.20 0.20 0.20 0.20

526 592 504 500 653

RESERVOIR

Actual Recovery’ (bbllacre-ft) 429 430 428 400 482

Recovery Efficiency (O/o) 82 73 85 80 74 Average = 79

‘From

flood-pot tests

HANDBOOK

ration as found by ordinary core analysis after multiplying with the oil FVF at abandonment, B,)O, as the residual oil saturation in the reservoir to be expected from flooding with water. This is based on the assumption that water from the drilling mud invades the pay section just ahead of the core bit in a manner similar to the water displacement process in the reservoir itself.

- S,,)IB,, 7,75&S~B,,

7,7584(1

ENGINEERING

ESTIMATION

OF OIL AND GAS RESERVES

40-17

lo.30 .. 5, F :: *a20 Lsk 3a LiL 1 8 lO.‘O 02 2’0 ?I+ 0 OIL h i0 g 6 -o .,o & L 4 EE -0.20 2 g 0 0.2

-0.30 0.4 06

I

2

4

6

IO

20

40

60

100

EC0

20

40

OIL VISCOSITY AT RESERVOIR CONDITIONS; CENTIPOISES

Fig. 40.8-Effect

of oil viscosity on ROS water-drive sand fields.

In another statistical study of the Craze and Buckley data and other actual water-drive recovery data on a total of 70 sand and sandstone reservoirs, the API Subcommittee on Recovery Efficiency t6 developed Eq. 32 for unit recovery for water-drive reservoirs, N,,. In stocktank barrels per acre-foot,*

-0.2159

,

.

..

(32)

where symbols and units are as previously defined except permeability, k, is in darcies, and pressure, p, is in psig. Example Problem 6. For the same water-drive reservoir used previously and assuming pwi =O.S cp, the API statistical equation yields the following unit recovery factor: (0.20)(1-0.25) N,, =4,259 1.30

1.0

x-

100

200

400

AVERAGE PERMEABILITY

lcco

EOW

4oM)

Io.ow

OF RESERVOIR; MILLIDARCIES

Fig. 40.9-Relation between deviation of ROS from average trend in Fig. 40.8 and permeability water-drive sand fields.

Water-Drive Unit Recovery Computed by Frontal-Drive Method26-28 The advance of a linear flood front can be calculated by two equations derived by Buckley and Leverettz6 and simplified by Welge** and by Pirson. ” These are known as the fractional-flow equation and the rate-of-frontaladvance equation. This method assumes that (1) a flood bank exists, (2) no water moves ahead of this front, (3) oil and water move behind the front, and (4) the relative movement of oil and water behind the front is a function of the relative permeability of the two phases. If the throughput is constant and the capillary-pressure gradient and gravity effects are neglected, the fractionalflow equation can be written as follows:

fw=

1 1 +(k,lk,,,,)(pJp,)

’

.

. (33)

1

‘.0422

-0.2159

( > 0.9

= 504 STB/acre-ft Because data were arrived at by comparing indicated recoveries from various reservoirs with the known parameters from each reservoir, the estimated residual oil and the average recovery factor based on these correlations allows for a recovery-efficiency factor (permeabilitydistribution factor times lateral-sweep factor) that is not present in the unit-recovery factor based on actual residual oil as found by flood-pot tests or in the cores. ‘because Eq 32 IS empirlcally darned, conversion to metric units jmJ/ha.m) mulbpl~cark?m of Nup by 1.2899

requires

0

20 RESERWR

40 60 SO PRESSURE DECLINE: PER CENT

100

Fig. 40.10--Relation between deviation of ROS from average trend in Fig. 40.8 and pressure-decline water-drive sand flelds.

PETROLEUM

40-18

ENGINEERING

HANDBOOK

3 1.0 5 0.9 2

k-~0.8 d 5 0.7 Iz - 0.6 ii? :

0.5

1.05 Iv..

.2

I

0.3

0.4

0.5

WATER

0.6

0.7

SATURATION,

FRACTION

0.8

0.9

S,, TIME

OF PORE SPACE

YEARS

+

Fig. 40.11-Fraction of water flowing in total stream f, and slope off, curve df,/dS,, vs. water saturation S,, (example: frontal-water-drive problem).

Fig.

wheref,, is the fraction of water flowing in the reservoir at a given point, k,. is the water relative permeability, fraction, and pn, is the reservoir water viscosity, cp. Since k,,lk,. is a function of water saturation, f,+, can be determined by Eq. 33 as a function of water saturation for a given water/oil viscosity ratio (see Fig. 40.11, Curve A). The Buckley-Leverett rate-of-frontal-advance equation may be rearranged to give the time in days for a given displacing phase saturation to reach the outlet face of the linear sand body as a function of the slope of the fractional flow vs. saturation curve (Fig. 40.1 I, Curve B) as follows:

Fig. 40.12 is a plot of the results of the performancehistory calculation from Table 40.8. If the economic limit is taken to be a WOR of 50, then it can be noted from Fig. 40.12 that the unit-recovery factor will be 575 bbllacre-ft to be recovered in 20.7 years.

5.615 NB, t= qr(df,,,dSi,*,)

(

.

..

.

(34)

where df,ldS,,. is the slope of thef, vs. Si, curve; the time, t, is in days; and the total liquid flow rate, qr, is in reservoir cubic feet per day. The average water saturation behind the flood front at breakthrough, and therefore the oil recovery, may be obtained from the fractional-flow curve by constructing a straight line tangent to the curve through S;, atf,=O, and reading S ;,, at f, = 1.O. The time of breakthrough at the producing well may be computed from the slope of the curve at the point of tangency. The subsequent performance history after breakthrough may be calculated by constructing tangents at successively higher values of S;, and obtaining Si, in a similar manner. Table 40.8 illustrates the calculation procedure for a water drive at constant pressure in a homogeneous reservoir and with a water-influx rate equal to the production rate.

40.12-Example of frontal-drive problem, unit-recovery factor, and WOR vs. time.

Effect of Permeability

Distribution ‘t41-44

In some reservoirs there may be distinct layers of higher and lower permeabilities separated by impervious strata. which appear to be more or less continuous across the reservoir. In such a case, water and oil will advance much more rapidly through the higher-permeability streaks than through the tighter zones, and therefore the recovery at the economic limit will be less than that indicated by the unit-recovery factor. Methods for computing waterflood recoveries that take into account the permeability distribution were proposed by Dykstra and Parsons,4’ Muskat. and Stiles.43 In the Dykstra-Parsons paper4’ it is assumed that individual zones of permeability are continuous from well to well, and a computation procedure as well as charts are presented for the coverage or fraction of the total volume of a linear system flooded with water for given values of (1) the mobility ratio knvpolkropw, (2) the produced WOR, and (3) the permeability variance. This permeability variance is a statistical parameter that characterizes the type of permeability distribution. It is obtained by plotting the percentage of samples “larger than” the sample being plotted vs. the logarithm of permeability for that sample on log-probability graph paper and then dividing the difference between the median or 50% permeability and the 84. I % permeability by the median permeability. Although the Dykstra-Parsons method

ESTIMATION

OF OIL AND GAS RESERVES

TABLE

40-l 9

40.8-WATER-DRIVE

PERFORMANCE-HISTORY

Time

s

1w

~ S,,

0.545 0.581 0.605 0.634 0.673 0.718

0.619 0.655 0.675 0.697 0.720 0.748

fw

df,JdS,w

(years)

0.800 0.875 0.910 0.940 0.970 0.990

2.70 1.69 1.29 0.95 0.64 0.33

3.94 6.29 8.24 11.19 16.61 32.21

Residual Oil Saturation (1 -S,,) 0.381 0.345 0.325 0.303 0.280 0.252

CALCULATION* Unit-Recovery Factor (bbl/acre-ft)

WOR = f,/l -f,

441 484 507 534 561 594

4.0 7.0 10.1 15.7 32.3 99.0

‘N = 597,000 STB, ao, = 1 30, o=o 20. S,, =0 25, and qr = 200 E/D x 5 615 cu ftlbbl = > ,222 ,esewow cu fl/D

does not allow for variations in porosity, interstitial water. and floodable oil in the different permeability groups, it has apparently been used extensively and successfully on close-spaced waterfloods. mainly in California. Johnson’4 in 1956 published a simplification of this method and presented a series of charts showing the fractional recovery of oil in place at a given produced WOR for a given permeability variance, mobility ratio, and water saturation. Reznik er al. 4s published an extension to the Dykstra-Parsons method that provides a discrete analytical solution to the permeability stratification problem on a real-time basis. In the Stiles method4” it again is assumed that individual zones of permeability are continuous from well to well and that the distance of penetration of the flood front in a linear system is proportional to the average permeability of each layer. Instead of representing the entire permeability distribution by one statistical parameter, Stiles tabulates the available samples in descending order of permeability and plots the results in terms of dimensionless permeability and cumulative capacity fraction as a function of cumulative thickness. From these data, Stiles computes the produced water cut of the entire system as the watering out progresses through the various layers, starting with those of the highest permeability. Stiles then assumes that at a given time each layer that has not had breakthrough will have been flooded out in proportion to the ratio of its average permeability to the permeability of the last zone that had just had breakthrough, and then constructs a recovery vs. thickness relationship. This then is combined with previous results to yield a recovery vs. water-cut graph. The Stiles method is used extensively and successfully, mainly in the midcontinent and Texas, for close-spaced waterfloods. It does not make allowance for the difference in mobility existing in the formation ahead of and behind the flood front. which the DykstraParsons method allows for. It also does not provide for differences in porosity, interstitial water, and floodable oil in the various permeable layers. Arps ’ introduced in 1956 a variation of the Stiles method, called the “permeability-block method.” This method handles the computations by means of a straightforward tabulation and does make allowance for the differences in porosity, interstitial water, and floodable oil existing in the various permeable layers. Since it is designed primarily for the computation of recoveries from waterdrive fields above their bubblepoint. no free-gas satura-

tion is assumed. The method further assumes that (I) no oil moves behind the front, (2) no water moves ahead of the front, (3) watering out progresses in order from zones of higher to zones of lower permeability. and (4) the advance of the flood front in a particular permeability streak is proportional to the average permeability. This method, applied to a hypothetical pay section 100 ft thick, is illustrated in Table 40.9, which is based on data from a Tensleep sand reservoir in Wyoming where good statistical averages of more than 3,000 core analyses were available. Part of these cores were taken with water-base mud that yielded the residual-oil figures on Line 6. Another portion was taken with oil-base mud and yielded the interstitial-water figures of Line 7. An oil/water viscosity ratio of 12.5 was used in calculating the WOR of Line 13. In Group I the recovery of 61.7 bbliacre-ft for WOR= 15.5 is the product of the fraction of samples in the group and the unit-recovery factor. In all other groups for WOR = 15.5 the full recovery is reduced in the proportion of its average permeability to 100 md. The total recovery at WOR= 15.5 is shown as 175.6 bbliacre-ft. The cumulative recoveries for WOR’s of 35.9, 76.5, 307.7, and infinity are calculated in a similar manner. Fig. 40.13 is a plot of WOR vs. recovery factor. From Fig. 40.13 it can be seen that, if the economic limit is taken to be a WOR of 50, the recovery factor would be 297 bbliacre-ft. It should be stressed that the permeability-block method is applicable only when the zones of different permeability are continuous across the reservoir, or between the source of the water and the producing wells. When the waterfront has to travel over large distances, nonuniformity of permeability distribution in lateral directions begins to dominate, and recoveries will approach those obtainable if the formation were entirely uniform (permeability distribution factor= 1). In such a case, an estimate based on the permeability-block method may be considered as conservative, except for the fact that one of the basic assumptions of this method is that the WOC, or front, moves in pistonlike fashion through each permeability streak, sweeping clean all recoverable oil. In reality, part of this oil will be recovered over an extended period after the initial breakthrough, which may tend to make the estimate optimistic. Those using the permeability-block method hope that these two effects are more or less compensating.

PETROLEUM

40-20

TABLE

40.9-WATER

DRIVE

PERMEABILITY-BLOCK

(15) (16) (17) (18)

Permeability range, mud Percent of samples in group Average permeability, md Capacity, darcy-ft (2) x (3) + 1,000 Average porosity fraction $ Average residual-oil fraction Sgr Average interstitial-water fractron S,, Relative water permeabrlity behind front k Relative oil permeability ahead of front k,, Unit-recovery factor (B,, = 1.07) Cumulative “wet” capacity, E(4) Cumulative “clean oil” capacity, 3.241 - (11) Water-oil ratio WOR= (~00~c)(8/9)(1 l/12) Cumulative recovery at WOR = 15.5 bbllacre-ft Min k wei =I00 md Cumulative recovery at WOR = 35.9 bbllacre-ft Min k,,, =50 md Cumulative recovery at WOR = 76.5 bbl/acre-ft Min k we, = 25 md Cumulative recovery at WOR = 307.7 bbllacre-ft Min k we, =lO md Cumulative recovery at WOR = mbbllacre-ft Min k wer=0 md

Effect of Buoyancy and Imbibition In limestone pools producing under a bottomwater drive, such as certain of the vugular D-3 reef reservoirs in Alberta, one finds an extreme range in the permeabilities, often running from microdarcies on up into the darcy range. Under those conditions the modified Stiles method heretofore described yields results that are decidedly too

400,

I

I

I

I

r

f

n /

1

200. 0 G.--

I I

~100 g

80-

5

40

I I 1

I I I

- ECONOMIC , .9 WOR=5Ojmi ’ 60kIMIT 5 50 -“T---q---

I I I

!

RECOVERY FACTOR =297 BBL/ACRE, FT@ WOR =50

20

lOI 0 RECOVERY

31 200 FACTOR,

, 400

,

I 600

BBL/ACRE-FT

Fig. 40.13-Example of modified Stiles permeability-block method WOR vs. recovery factor.

HANDBOOK

CALCULATIONS Total

2

3

4

5

>lOO 8.5 181.3 1.541 0.159 0.173 0.185 0.65 0.475 726 1.541 1.700 15.5 61.7

50 to 100 10.9 69.0 0.752 0.150 0.195 0.154 0.63 0.53 693 2.293 0.948 35.9 52.1

25 to 50 14.5 34.4 0.499 0.152 0.200 0.131 0.60 0.61 722 2.792 0.449 76.5 36.0

10 to 25 21.2 16.1 0.341 0.130 0.217 0.107 0.56 0.66 623 3.133 0.108 307.7 21.3

0 to 10 44.9 2.4 0.108 0.099 0.222 0.185 0.54 0.47 415 3.241 0 4op5

175.6

61.7

75.5

72.0

42.5

8.9

260.6

61.7

75.5

104.7

85.1

17.9

344.9

61.7

75.5

104.7

132.1

44.7

418.7

61.7

75.5

104.7

132.1

186.3

560.3

Group (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

ENGINEERING

100.0 3.241

low. The reason is that, in pools like the Redwater D-3, there is a substantial density difference between the rising salt water and the oil. While the water rises and advances through the highly permeable vugular material, it may at first bypass the low-permeability matrix material, leaving oil trapped therein. However, as soon as such bypassing occurs, a buoyancy gradient is set up across this tight material, which tends to drive the trapped oil out vertically into the vugular material and fractures. In the case of Redwater D-3, where the density difference between salt water and oil is 0.26, while the vertical permeabilities for matrix material are only a fraction of the horizontal permeabilities, a simple calculation based on Darcy’s law applied to a vertical tube shows that during the anticipated lifetime of the field very substantial additional oil recovery may be obtained because of this socalled buoyancy effect. To calculate the recovery under a buoyancy mechanism it is necessary first to determine by statistical analysis of a large number of cores the average interval between highpermeability zones or fractures. A separate computation is then made for each of the permeability ranges to determine what percentage of the matrix oil contained in a theoretical tube of such average length may be driven out during the producing life of the reservoir under the effect of the buoyancy phenomenon. Surprisingly improved recoveries are sometimes indicated by this method over what one would expect from a Stiles type of calculation, and the results from recent studies of the rise in water table of the Redwater D-3 seem to confirm the validity of this concept. In addition to this buoyancy phenomenon the effect of capillarity and preferential wetting of the reservoir rock by water also should be considered. Imbibition of water from fractures and vugular material into the lowpermeability matrix as the water advances may materially aid the buoyancy mechanism but is much more difficult to evaluate quantitatively.

ESTIMATION

OF OIL AND GAS RESERVES

TABLE

40-21

40.10-PSEUDOCRITICAL

CALCULATIONS Volume % or MO&

Component (11 Methane Ethane Propane lsobutane Normal butane

lsopentane Normal penlane Hexanes

86.02 7.70 4.26 0.57 0.87 0.11 0.14 0.33

343.5 550.1 666.2 733.2 765.6 630.0 847.0 914.6

Factor

The compressibility factor z is a dimensionless factor which, when multiplied by the reservoir volume of gas, as computed by the ideal-gas laws, yields the true reservoir volume. The reservoir volume occupied by 1 lbmmole of gas (gas weight in pounds equal to molecular weight), in cubic feet, is G=

(10.73)z(460+TR) PR

(

.

. . . (35)

where G is the total initial gas in place in reservoir, in standard cubic feet, and TR is the reservoir temperature, “F. For example, 1 lbm-mole of methane (molecular

Critical Pressure

2x3 100

(;:)

Volumetric Recovery Estimates for Nonassociated Gas Reservoirs46-53 Compressibility

Critical Temperature

(2)

100.00

FROM GAS ANALYSIS

(77 673 708 617 530 551 482 485 434

(5) 296-42.4 26.4 4.2 6.7 0.9 1.2 3.0 362.6

2x4 100

(‘3) 572 54.5 26.3 3.0 4.8 0.5 0.7 1.4 663.2

weight 16.04) under standard conditions (PR = 14.7 psia, TR=~O’F) occupies 379.4 cu ft. The compressibility factor may be determined in the following ways. 1. Experimentally by PVT analysis of a gas sample. 2. By computation from an analysis of the gas expressed in mol% or volume %. With this method a weightedaverage or pseudocritical pressure and temperature are obtained for the gas by multiplying the individual critical pressure and temperature for each component, with the corresponding mol% of such component as shown in Table 40.10. The gas whose composition is given in Table 40.10 has a pseudocritical temperature of 382.8”R and a pseudocritical pressure of 663.2 psia. The pseudoreduced temperature then is found at a temperature of 150°F as (460 + 150)/382.8 = 1.59 and its pseudoreduced pressure

PSEUOO REDUCED PRESSURE

Fig. 40.14B-Compressibility factors for natural sures of 10,000 to 20,000 psia.

PSEUDO REDUCED PRESSURE

Fig.

40.14A-Compressibility

factors for natural gases.

gases

et

pres.

40-22

PETROLEUM

RESERVOIR

PRESSURE

Fig. 40.15-Gas

FVF 8,

1 B&l

460+ T, --------Z 460+60

14.17 = ~ p,+14.7

and reciprocal

RESERVOIR

(pR] IN PSI GAUGE

Fig. 40.16-Gas

p,+14.7

460+60

1

14.7

460+T,

z

8,

40.1 I-PSEUDOCRITICAL CALCULATIONS FROM SPECIFIC GRAVITY

Specific gravity of Gas (Air=l.O)

Pseudocritical Temperature (OR) (460+ OF)

Pseudocritical Pressure (psia) (14.7+ psig)

0.55 0.60 0.70 0.80 0.90 1 .oo 1.10 1.20 1.30 1.40 1.50 1.60 1.65

348 363 392 422 451 480 510 540 570 600 629 658 673

674 672 669 665 660 654 648 641 632 623 612 600 593

460+ T, p-z 460+60

gas FVF

pR+14.7 14.7

460+60 ~460+T,

1 z

vs. pressure,

psig, and temperature, Gas gravity 0.7 (air 1 .O).

OF

at 750 psia as 7501663.2 = 1.13. These ratios are entered into the chart of Fig. 40.14A to read z=O.91. This correlation chart46 and an extended correlation chartj7 for higher-pressure gas reservoirs up to 20,000 psia, Fig. 40.14B, are designed for gaseous mixtures containing methane and other natural gases but substantially free of nitrogen. For hydrocarbon gases containing substantial amounts of hydrogen sulfide or CO1 , these correlations do not apply, and additional corrections are necessary as described in Ref. 48. (See Chap. 20 for complete coverage of gas properties and gas property correlations, some of which are specific to computer applications.)

HANDBOOK

(pR) IN PSI GAUGE

14.17 = ~ p,+14.7

FVF 8,

1 -=

vs. pressure, psig, and temperature, Gas gravity 0.6 (air 1 .O).

TABLE

PRESSURE

and reciprocal

gas FVF

ENGINEERING

OF.

3. By computation from the specific gravity of the gas. If only the specific gravity of the gas (air= 1.O) is known, another approximate correlation can be used, based on California natural gases,49 which is expressed by Table 40.11. For example, if the specific gravity of a gas is 0.66. the pseudocritical temperature can be estimated by interpolation as 381 “R and pseudocritical pressure as 670 psia. The pseudoreduced values then are found as before and the z factor read from Fig. 40.14A. Gas FVF The gas FVF, B,, is a dimensionless factor representing the volume of free gas at a reservoir temperature of T”F and a pressure of p psia per unit volume of free gas under standard conditions of 60°F and 14.7 psia. If the compressibility factor, z, is known, B,? may be computed by 14.7 460+7-, B,=pR

460+60

z=O.O2827(46O+T,)i. PR

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (36) Typical values of the gas FVF, B, , and the reciprocal gas FVF, l/B,, for different temperatures and pressures and for gases of specific gravities between 0.6 and 1.0 will be found in Figs. 40.15 through 40.19. In estimating gas reserves, the estimator should be careful to indicate clearly the pressure base at which the reserves are stated. Reserves at a base pressure of 14.4 psia will be approximately 16% greater than the same reserves stated at a base pressure of 16.7 psia.

ESTIMATION

40-23

OF OIL AND GAS RESERVES

400

0.4

as

xx)

0.2

02

01 ma

2" g

0.06

= L

a04

Y 3

a02

6 s H e

QOl o.cca 3 0.006 a004

l.ow RESERVilR

PRESSURE

Fig. 40.17-Gas

&)

14.17 = ___ p,+14.7

FVF 6,

and reciprocal 1 S,, may be estimated from a material-balance calculation on the condensate present in the reservoir gas under initial conditions, and the condensate to be recovered during the depletion of the reservoir in the surface separation equipment. Effect of Permeability

Distribution

Unless a gas reservoir is known to be permeable and homogeneous, the unit-recovery factor should be corrected for the fact that depletion may progress more rapidly in the high-permeability strata than in the low-permeability zones, particularly if these zones are separated by impervious strata. An uneconomic rate of production may be reached before the tighter zones are drained down to abandonment pressure. In many cases, nonuniformity of permeability in lateral directions provides a compensating influence. In very hard and tight formations, extensive fracturing may have the same result. A computation based on the assumption that the strata of different permeabilities are uniform and continuous across the reservoir is therefore in most cases too pessimistic. Such a computation does provide a means, however, to indicate the minimum recoverable reserves while the assumption of a completely homogeneous reservoir and the direct use of the unit-recovery factor indicate a maximum figure for the recoverable reserves. A permeability-block method to compute such minimum reserves for a nonassociated dry gas reservoir is as follows.

ESTIMATION

40-25

OF OIL AND GAS RESERVES

According to Eqs. 13 and 14 of Sec. 11.15 in Ref. 50, the boundary pressure in a closed cylindrical gas reservoir, drained by a well in the center with zero pressure against the sandface, may be approximated as

TABLE

40.13-CONDITIONS FOR UNIT RECOVERY EQUATION IN A DRY-GAS RESERVOIR

Reservoir pressure

,

.(39)

while the gas production rate is .

ql: =C*k,h,p*,

Free gas, scf/acre-ft

.

.

(41,)I Cl(k#)lP,

PI (--> PI

Ultimate Conditions

PI

P.3

43,56OQS,,

43,56O~S,,

43,5604(1 -S,,)

B,l

.(40)

in which Ct and Cz are constants and $h and h, are effective hydrocarbon porosity and effective thickness, respectively It will be assumed that a large number of core analyses are available on a gas reservoir, which are divided in permeability groups as shown on Table 40.13. The average permeability, k , for each group is then corrected to the relative gas permeability,J,, , at the given Si,V saturation. The average porosity. 4. for each group is corrected also to the effective hydrocarbon-bearing porosity, +/, =4(1 -s,,, 1. It will further be assumed that each permeability group represents a separate and distinct homogeneous layer having a relative gas permeability k, and a hydrocarbonfilled porosity 4h equal to the average for each group. Each layer is sealed off from the others and feeding into a common wellbore that is exposed to zero pressure. To keep the computations as simple as possible it will further be assumed that the ideal-gas laws are applicable. The same method may be applied by taking the deviation from the ideal-gas laws into consideration, by assuming other than zero wellbore pressure, and by taking into account liquid condensation in gas-condensate reservoirs, but such computations soon become rather unwieldy. By the time Group I, comprising the highest permeability, is bled down to a pressure p t , a time 1 has expired, which according to Eq. 39 is equal to

t=

Interstitial water, cu ftl acre-ft

Initial Conditions

43,56Oc$(i-S,) B !F

while the cumulative production from all layers, G,,, , at this time is

(44)

in which C3 and Cd are constants. The fractional production rate from all layers. fsn, with respect to the initial production rate from all layers is, therefore,

while the cumulative production from all layers as a fraction of the total gas in place in all layers is ,1

~(~~),(h,),,[l-(P,ipi)l 1

(41)

G P”

.,....... G,Z

The fractional pressure Pn/p; in any layer n at this same time t is found by substituting the t value of Eq. 41 into Eq. 39.

-I

..(42)

The combined production rate from all layers, y,, , at this time is, according to Eq. 40.

(43)

_

(46)

II

Thus a rate-cumulative relationship may be established based on Eqs. 45 and 46, whereby the rate is expressed as a fraction or percentage of the initial rate, and the cumulative as a fraction or percentage of the gas in place. By selecting an appropriate economic limit rate the recovery factor can then be found. The computation procedure is illustrated with the example in Table 40.14. Usually only three or four assumptions for the ratio p,/p t are necessary to delineate the curve, which may then be plotted on semilog paper as shown in Fig. 40.20. In this particular case, it could be estimated that the minimum recovery factor for this reservoir at a time when the production rate has declined to 1% of its initial value would be on the order of 74% of the gas in place.

PETROLEUM

40-26

TABLE

40.14-PERMEABILITY-BLOCK

METHOD

2

1

Group (n) (I) Permeability,

lO