SPE Society of Petroleum Engineers

SPE 16484

GASWAT-PC: A Microcomputer Program for Gas Material Balance With Water Influx by B. Wang and T.S. Teasdale, Texaco Inc. SPE Members

Copyright 1987, Society of Petroleum Engineers

This paper was prepared for presentation at the Petroleum Industry Applications of Microcomputers held in Del Lago on Lake Conroe, Montgomery, Texas, June 23-26, 1987.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Write Publications Manager, SPE, P.O. Box 833836, Richardson, TX 75083-3836. Telex, 730989 SPEDAL.

ABSTRACT

GASWAT-PC is a microcomputer material balance program that calculates original gas in place (OGIP) from production history for gas reservoirs with or without water influx. Several well-known analytical tech-niques, and some extensions and improvements, are applied to a single input data set. The output for each method is given in both tabular and graphical form.

GASWAT-PC also has the capability to predict reservoir pressure and recovery efficiency for a specified production rate if OGIP and aquifer properties are known. An option is included to account for the effect of a large pressure gradient in the water-invaded region.

Case studies illustrate the application of GASWAT-PC to an abnormally pressured reservoir and to two water drive reservoirs. Finally, we compare the pressure performance predicted by a numerical simulator and by GASWAT-PC for a water drive reservoir with a high dip.

INTRODUCTION

Material balance methods have been employed for many years to calculate OGIP for gas reservoirs. Since reservoir data and aquifer properties are uncertain for most reservoirs, it is important to analyze the data using different techniques. By comparing the results of various methods, one can improve reservoir performance interpretation and can predict OGIP more accurately.

For ~nstance, several authors1,2 reported that it is difficult to distinguish natural depletion from partial water drive by using only the P/Z plot. Thus, extra-polating an apparently straight P/Z line may yield an

References and illustrations at end of paper.

25

incorrect OGIP. However, this problem may be avoided by drawing the Cole plot, 3 where a flat line or scattered distribution indicates natural depletion and an increasing trend of the data points indicates water influx.

Several widely accepted analytical methods include: P/Z plots for natural depletion reservoirs, Ramagost and Farshad4 (RF) P/Z plots for abnormally pressured reservoirs, Cole and Havlena and OdehS (HO) methods for water drive reservoirs, and pressure match methods6 for all types of reservoirs. For water drive reser-voirs, five aquifer models are usually considered: (1) small pot, 7 (2) Schilthuis steady state,S (3) Hurst simplified,9 (4) van Everdingen and Hurst (EH) infinite linear, and (5) EH finite/infinite radial aquifers.10 GASWAT, a Fortran program run on an IBM mainframe (IBM 3090) for two years, combines all the preceding tech-niques in a package where users input the required PVT and production data. We have found that the best analysis is achieved by a series of interactive runs where the user examines results in either graphical or tabular form, and then individual data points are selectively included in (or excluded from) each calcu-lation procedure.

This kind of trial and error work can very economically and easily be done on a microcomputer if computational times are adequate. GASWAT-PC gives results identical to the mainframe version with good response time. The advantages gained by using the microcomputer include portability, ease of operation, cost reduction, and time savings.

GAS MATERIAL BALANCE EQUATION The general form of the material balance equation for a condensate gas reservoir is7

F = G(Eg + Efw) +We .(1)

2 GASWAT-PC: A MICROCOMPUTER PROGRAM FOR GAS MATERIAL BALANCE WITH WATER INFLUX SPE 16484

where G is original wet gas in place, and

F = Gwgp Bg + Wp Bw (Gp + Kc GLp)Bg + Wp Bw (2)

Eg Bg - Bgi (3)

cw Swi + Cf Erw = Bgi Ce (Pi-P) Bgi 1 - Swi (Pi-P). (4)

We= U S(P,t) (5) Note that for a dry gas reservoir, wet gas production (Gwgp> equals dry gas production (Gp) since condensate production (GLp) is zero. The condensate conversion factor (Kc> is 132,790 rc/MLc and according to Cragoe,11 the molecular weight of condensate is given by MLc = 6084/(API gravity- 5.9). In the water influx term (We>, U is an aquifer constant and S(P,t) is defined separately for different aquifer types. The theoretical values for U and S(P,t) are given in Appendix A.

OGIP CALCULATIONS

GASWAT-PC requires schedule data (time, pressure, and gas, condensate, and water production) and PVT data. GASWAT-PC calculates the Z-factor by interpolating measured data, or alternatively, by the gas gravity or the gas composition corrtHation.12 In the correla-tion approach, impurities (i.e., N2, C02 and so2 )13 and the yield ratio are considered, and either Dranchuk et al. 14 or Hall and Yarborough15 methods can be selected for the Standing-Katz correlation16 calcula-tion. Radial water drive reservoirs require additional information, as discussed later.

To compare different calculation methods, we use a normal{zed standard deviation (%) defined as

a(Y) a = 1oo (Ymax - Ymin> where

a(y) =

n

2

B. Wang and T. S. Teasdale 3

3. HO Straight Line Method:

Havlena and Odeh5 proposed a straight line method for water drive reservoirs. The same technique can be used here by rearranging equation (10).

F

Eg G ( 1 + Bgi ce ) 0 0 0 0 0 0 (12)

A plot of F lEg versus (Pi-P) /Eg yields a straight line with intercept G at P = Pi, and slope (Bgi G ce>. Then Cf can be calculated from equation (4).

This approach can be extended to an abnormally pressured reservoir with some shale water or with a small pot aquifer. Then the water influx (We> is defined as We= U (Pi-P) ........ (13) and equation (1) becomes

F

Eg G + (Bgi G ce + U) 0 0 0 0 (14)

A plot of F/Eg versus (Pi-P)/Eg again yields a straight line with intercept G at P = Pi. The main advantage of this approach is that G can be obtained without having to specify Cf and U values.

C. Water Drive Reservoirs

For a water drive reservoir, Efw is neglected since it is small compared to Eg and We. Then equation (1) changes to

F = G Eg + We . . . . . . . . ( 1 5)

1. P/Z Methods:

Writing equation (15) in terms of P and Z p pi G - Gwgp z

- Zi G - y 0 (16)

where

pi Tsc y =

zi Psc T (We - Wp Bw) .(17)

The plot of P/Z versus Gwgp will not generally be linear since We varies with production and time.

However, this method may sometimes be used if we properly select the data points for regres-sion. Shagroni17 suggested using the early portion of the P/Z curve in a linear regres-sion since these points may not be strongly influenced by water influx. This approach has been tested in GASWAT-PC on many reservoirs, and often gives reasonable results.

27

2. Cole Method:

Equation (15) can be written as

Gwgp Bg We - Wp Bw ( ) ---- = G + 18

Eg Eg

Cole3 proposed plotting GwgpBg/Eg versus Gwgp' and then extrapolating this curve to Gwgp = 0. The intercept shows the true value of G, since We and Wp are zero at zero Gwgp

Generally speaking, this method is most useful for very strong aquifers. However, Cole plots often show the presence of water influx much more clearly than P/Z plots.

When water influx declines in reservoirs with limited aquifers, Eg increases faster than (We - Wp Bw> and the term (We - Wp Bw)/Eg in equation (18) will decrease with time. Then the Cole plot curve starts bending down and this trend will cause a high and unrealistic prediction of G. However, this error can be minimized by excluding improper data points, which exhibit a negative slope on the Cole plot, from the linear regression.

3. HO Straight Line Method:

The HO method is the most popular technique used to estimate G for water drive reservoirs. They rearranged equation (15) by dividing both sides with Eg

_F_=G+U Eg

S(P.t) Eg

0 0 0 0 0 0 0 0 0 (19)

A plot of F /Eg versus S(P, t) /Eg yields a straight line with intercept Gat S(P,t) = 0, and slope U. Note that the definition of U and S(P,t) depends on the type of aquifer, as given in Appendix A.

For instance, assuming a Schil thuis steady state aquifer, then

S(P,t) = Pi - 0 0 0 0 0 0 0 (20)

A plot of F/Eg versus [Pi- (Pj + Pj-1)/2]/Eg will be made and, if this aqu~fer assumption is correct, the points will approximate a straight line with a small standard deviation between the fitting line and the observed points.

Since the computation time required on each aquifer calculation is small, GASWAT-PC makes an HO calculation for each aquifer geometry (small po't, Schil thuis, Hurst simplified, infinite linear, finite/infinite radial aquifers). For each method, the G, U, and standard deviations values are calculated. The results are listed in a summary table for comparison, and graphical output is also available.

4 GASWAT-PC: A MICROCOMPUTER PROGRAM FOR GAS MATERIAL BALANCE WITH WATER INFLUX SPE 16484

To analyze a radial aquifer situation, addi-tional data for aquifer permeability, porosity, water viscosity and compressibility, formation compressibility, and gas reservoir radius must be given to calculate the theoretical dimensionless time coefficient [ equation (A-10)]. Also, the aquifer radius is required.

Because of uncertainties in the dimensionless time coefficient and aquifer to reservoir radius ratio ( ra/ rg) estimates, up to 14 adjustment factors {TDF) may be applied to the theoretical dimensionless time, and 14 different ralrg values may be input. Each combination of TDF and ralrg is calculated, and the "best fit" may be selected.

We use two selection criteria for the radial aquifer case: (1) normalized standard deviation O'(F/Eg), and (2) curvature CURV(F/Eg). The curvature is a measure of how much the best parabola departs from a straight line over the range of the data (see Figure 8), and the calculation is described in Appendix B.

Theoretically, if the aquifer in the HO analysis is correctly specified, then the data points will lie on a straight line (a = 0, CURV = 0). A systematic upward (CURV > 0) or downward curvature (CURV < 0) in the data indicates that the specified aquifer is too small or too large, or it may be that the analysis is not appropriate for this data set. 5

In our experience, the minimum a and the minimum absolute curvature lcURVI will not occur for the same ralrg and TDF pair. It is recommended to pick the minimum a provided ICURVI is small enough (less than about 20%) so as to give credence to the validity of the analysis.

D. Pressure Match Method for All Kinds of Reservoirs

Most material balance methods calculate G and U directly from historical pressure, production, and time data. Another method is to assume values for G and U, and then calculate pressure as a function of time. One can find the G and U which minimize the difference between the calcu-lated and observed pressure data. This procedure is described next.

Modify equation (16) by including the formation compressibility term,

p

z G- Gwgp ..... (21)

G[1 - ce (Pi-P)] - Y

For a given G and U, the first pressure point P(1) at the first schedule data point [t(1), Gp(l), GLp(l) and Wp(l) 1 can be solved using an iteration technique. Then the second pressure point P(2) can be found with the second schedule data and P(l). The procedure continues until all the pressure points are found. Finally, the standard deviation of the observed and the calcu-lated pressure, and the water influx, are calcu-lated and saved for comparison.

28

The pressure match method is very accurate,6 and it can used for all kinds of reservoirs. The major drawback is that substantial computational time is required for the pressure calculation. In most cases, we only use this procedure to compare the results from other methods used in GASWAT-PC.

At the end of OGIP calculations, the calculated G and U of each method (P/Z, Cole, HO, etc.) are automatically input to the 'pressure match sec-tion.' A set of pressure and water influx his-tories, and the standard deviation of the pressure for each method, are calculated and listed in the summary table for comparison. Since the comparison criterion for all methods is the pressure standard deviation, the m~n~mum value should give the best estimates of G and U.

PRESSURE AND ULTIMATE RECOVERY PREDICTIONS

For many water drive reservoirs, production performance is very sensitive to gas rate. Field cases18,19 have shown that deliverabili ty and ultimate recovery can be greatly increased by properly controlling gas production.

GASWAT-PC has the capability to predict pressure and ultimate recovery for a gas reservoir produced at a specified rate if G and aquifer character are given. Thus, GASWAT-PC can both help reservoir engineers correctly estimate reserves, and also plan an optimum operational strategy.

The calculation procedure for pressure prediction is the same as that for the pressure match method men-tioned above, except that the future production data is manually generated based on a specified gas rate. Cumulative water influx at each pressure point is also calculated and listed in the output.

A. Reservoir Pressure Gradient

The pressure used in the aquifer equation (5), called P2, refers to the pressure at the original gas water contact. In a conventional material balance calculation, it is assumed that P2 equals P1 which is the average reservoir pressure in front of the current gas water contact. This assumption is reasonable early in the production life since only a small amount of influx has entered the reservoir.

But, later in the production life, a large pressure gradient may be created in the water invaded region between the original and the current water gas contact, which results in a lower P1 value. In other words, as production from a reservoir increases, conventional material balance methods may predict a higher reservoir pressure than the observed pressure. This difference is especially significant for reservoirs with large dip and strong aquifers.

Lutes et al.19 proposed a modified material balance method to account for the pressure gradient effect in a radial water drive reservoir. A similar approach was included in GASWAT-PC. In addition, the new option includes gravity20 and

SPE I (Q4 + (We2- Wp Bwl)(l- A- B) . . . (22)

where

Sgr Bgl - Bgt . (23) A 1 - Sgr - Swi Bgt

Swi ow + Of B = 1 - Sgr - Swi

(Pt - P1) . (24)

Here, P1 is the average reservoir pressure in front of the current gas water contact, P2 is the reservoir pressure at the original gas water contact, and Pt is the average reservoir pressure in the water-invaded region. Subscripts 1, 2 and t represent functions of P1, P2 and Pt.

Equation (22) is derived in Appendix C. P1, P2 and Pt depend on reservoir geometry, and Appendix D shows the relations for linear and radial reservoirs.

An iterative technique is used to solve the modified material balance equation (22). This gives P1 and the current water gas contact (11 or r1) at any time step t. Ultimate Gas Recovery Prediction

1. Natural Depletion and Partial Water Reservoirs

For a natural depletion or partial water drive reservoir, the pressure drops rapidly as gas is produced, and production stops at the abandonment pressure (Pa), which depends on operating line pressure, depth of the reser-voir, and the productivity index. The recovery efficiency (ER) is expressed as

Gwgpa G = 1 -

. . . . . . (25)

where Gwgpa is the ultimate gas recovery at Pa

2. Watered-Out Reservoirs

For a strong water drive reservoir, water is continuously entering the reservoir, and all existing wells eventually water out. There-fore, the abandonment pressure (Paw> is higher than for a volumetric or partial water drive reservoir, and ER is lower since residual gas in the water invaded zone (and in unswept areas) is trapped at higher pressure. ER for a watered-out reservoir is defined as

Gwgpaw G 1 - Ft

. . . . . (26)

29

where Ft is the hydrocarbon pore volume of trapped gas, given by

Ft = (1 - Ev) + . . . . . (27)

Ev is the volumetric sweep efficiency, and Equation (26) can be rearranged as

(1 - Gwgpaw G ) (28)

Equation (28) is a line with intercept G at (P/Z) = 0, and slope[- Pi/(Ft G Zi)]. By plotting Equation (28) and P/Z versus Gwgp on the same graph (Figure 3), we can determine Gwgpaw and Paw from the intercept of the curves .

If the pressure gradient in the water invaded region is significant, the above relation is not correct since the reservoir pressure is not equal to the average pressure in the trapped area. However, the cumulative water influx at watered-out conditions is given by

Bgi G (1-Ft) + Wp Bw . . . (29)

Once the relations of Gwgp and P1 versus We are generated by solving equation (22), one can find Gwgpaw and Paw from Weaw

Since it is difficult to distinguish a partial water drive reservoir at early production times, GASWAT-PC will first calculate Paw and Gwgpaw for any water dri.ve reservoir. If the calculated Paw is larger than the input Pa which indicates that the reservoir will be watered-out, then the ultimate recovery and the abandonment pressure will be assigned to Gwgpaw and Paw respectively. Otherwise, the reservoir is a partial water drive type, and Gwgp and Pa will be assigned to Gwgpa and Pa respectively.

CASE STUDIES

To illustrate the various methods and their applica-tions, four cases were analyzed with GASWAT-PC: (1) an abnormally pressured reservoir with no influx, ( 2) a reservoir with a strong radial aquifer, ( 3) a reservoir with an infinite linear aquifer, and (4) a water drive reservoir with a large dip angle.

Case 1: An Abnormally Pressured Reservoir With No Influx

The Anderson "L" gas reservoir presented in the Ramagost and Farshad paper4 is used here for illus-tration. The reservoir has an initial pressure gradient of 0. 843 psi/ft [19. 07 kPa/m] and a forma-tion compressibility of 15 microsips (E-6 psi-1) [2.18 x lo-6 kPa-1]. G = 69 BCF [1.98 x 109 m3] by v ol umetrics. Reservoir information and production data are listed in Table 1.

The results are summarized in Table 1 and shown in Figures 4. a to 4. d. All the data points are used in the regression for P/Z method 1, and an incorrect

6 GASWAT-PC: A MICROCOMPUTER PROGRAM FOR GAS MATERIAL BALANCE WITH WATER INFLUX SPE 16484

G = 96.61 BCF [2.77 x 109m3] is obtained. Figure 4.a shows the result of P/Z method 2. Only the last three points on the second slope are used in this regression, and a more reasonable G = 79.86 BCF [2.29 x 109 m3] is calculated.

Figure 4. b demonstrates the result of the RF P/Z method. All of the points are included in the regres-sion and G = 74.04 BCF [2.12 x 109m3],

Figure 4.c shows the result of HO method. The calcu-lated G is 77.05 BCF [2.22 x 109m3], which is close to the RF P/Z value. A Cf of 12 microsips [1. 74 x 10-6 kPa-1] is calculated from the slope using equation (12), which is lower than the reported value of 15 microsips [2.18 x 10-6 kPa-1].

Note that six data points (in the first 10% gas production) are excluded from the HO calculation due to their scattered distribution. However, these points are included in P/Z method 1 and RF P/Z method, since P/Z plots are not as sensitive as HO calculations to the early life data.6

The G and Cf values from the RF P/Z and HO calculations are automatically input to the 'pressure match sec-tion, ' and then the reservoir pressure at each time step is calculated and plotted in Figure 4.d. Compar-ing the observed data and the calculated results, it is seen that both cases give excellent matches which indicate the correct G value is within the range of 74 to 78 BCF [2.12 x 109 to 2.22 x 109m3],

The calculation time required for this run on an IBM AT is 42 seconds.

Case 2: A Gas Reservoir With A Strong Radial Aquifer

This is a typical Louisiana offshore gas reservoir which has a strong radial water drive. G = 806 BCF [23.08 x 109 m3] is estimated by volumetrics. The reservoir data and the output summary are listed in Table 2. Figures S.a-S.h demonstrate the results of the various methods,

G = 1074.75 and 947.54 BCF [30.78 x 109 and 27.13 x 109 m3] are calculated by P/Z method 1 and method 2. Only the first four points, which are considered to be less influenced by the influx, are included in the regression for both methods. Figure S.a shows that the rest of the points depart from the regression line indicating the influx entering the reservoir.

Figure S.b is the Cole plot, which gives G = 916.65 BCF [26.25 x 109 m3] and U = 1.74. The first two points, and those points after 400 BCF [11.45 x 109 m3] gas production (which have negative slope) are excluded from the calculation.

Six data points are excluded from the HO regressions (see 'data select' Table 2). Except for the small pot aquifer, these excluded points are not shown on HO plots since they change the value of the S(P,t) term [see S(P,t) calculation in Appendix A].

Figure S.c shows the HO plot of small pot aquifer. The calculated G = 1430.23 [40.96 x 109 m3] is incor-rect due to the large deviation. Figures S.d and S.e are the HO plots of Schil thuis and Hurst Simplified

30

aquifers. The calculated G and U are 961.39 BCF and 15143.31 RB/psi/yr [27.53 x 109 m3 and 349.19 m3/kPa/yr] for the Schilthuis aquifer, and 892.683 BCF and 81380.50 RB/psi [25.56 x 109 m3 and 1876.57 m3/kPa] for the Hurst simplifiedaquifer, respectively. The Hurst aquifer yields a a smaller deviation of 6.94%.

For the radial aquifer calculation, a range of ralrg ratios (5 to 20) and TDF values (0 .S to 2. 0) are input due to the uncertainties of aquifer size and reservoir parameters. Each pair of ralrg ratio and TDF is used to calculate G and U. So a total of 132 runs (12*11) are made, and then the two "best fit" runs (minimum a and minimum absolute CURV) are automa-tically selected and listed in the summary table.

The minimum a run (ralrg=10 and TDF=0.6) shows a small CURV of -11.82%, so we choose it as the best representation for a radial aquifer. G = 821.88 BCF [23 .54 x 109 m3], and U = 6330.95 RB/psi [145.99 m3 /kPa] which is smaller than the theoretical value of 10740 RB/psi [247.66 m3/kPa]. The correct dimen-sionless time coefficient (Ktn> can be calculated by

KtD = KtD TDF = (12.78) (0.6) 7. 668 yr-1 (30)

Figure S .f demonstrates the HO plot of the radial aquifer of ralrg = 10 and TDF = 0.6. Note that the points moves up from Y value of 1210 to 1630, then drop back to 1340 along a straight line. This indi-cates that the assumed radial geometry is correct.

After the OGIP calculations, the pressure match is automatically checked for each method as shown in Figure S.g. In general, the match is good for all methods up to 400 BCF [11.45 x 109 m3] production. However, the radial aquifer gives a better pressure match during the 400 to 600 BCF [11.45 x 109 to 17.18 x 109 m3] production period.

Now we can predict the future pressure performance and ultimate gas recovery by using the select~d G and U values. G = 821.88 BCF [23.54 X 109m3], U = 6330.95 RB/psi [145.99 m3/kPa], ralrg = 10, and tn/t = 7.668 1/yr are input in the pressure prediction section. A constant rate of 30 BCF/yr is assumed, and the pressure gradient in the water invaded region is neglected since the dip angle is small. Sgr and Ev are assigned 0.25 and 0.8, respectively.

The result is shown in Figure S.h. Under the assumed conditions, the reservoir will water out at an abandon-ment pressure of 2138 psi a [14, 741 kPa] and an ultimate wet gas recovery of 631.68 BCF [18.09 x 109m3],

The calculation time required for this study is 333 seconds.

Case 3: A Gas Reservoir With An Infinite Linear Aquifer

This is an infinite linear water drive reservoir illustrated in reference 21. The reservoir has an initial pressure of 2333 psia and G = 337.7 BCF by volumetrics. The reservoir information and the output summary are listed in Table 3. Figures 6.a to 6.d show the results of various methods. All the data points are included in regression for all methods.

SPE I {Q4cg4- B. Wang and T. S. Teasdale 7

Figure 6. a is the result of P /Z method 1, which yields a G of 383.27 BCF [10.98 x 109m3] and a small a of 0.91%. Figure 6.b shows the result of the Cole plot, and a lower G = 347.83 BCF [9.96 x 109 m3] is found. The increasing trend of the data points reveals influx entering the reservoir even though the P/Z plot (Figure 6.a) appears quite linear. The result of the HO method for small pot aquifer is incorrect since the slope of the regression line is negative. The calculated G and U for Schilthuis, Hurst simplified and infinite linear aquifers are 339.69 BCF and 5960 RB/psi/yr [9. 73 x 109 m3 and 137.43 m3/kPa/yr], 332.13 BCF and 34315.29 RB/psi [9.51 x 109 m3 and 791.28 m3/kPa], and 309.03 and 28424.93 RB/psi/yr**O.S [8.85 x 109 m3 and 655.46 m3/kPa/yr**0.5], respectively. The infinite linear aquifer yields the best straight line with a 3.09% deviation (see Figure 6.c). Just comparing the P/Z and the linear aquifer plots, it is difficult to decide which is the better represen-tation. Here, the pressure match method can help assist in this decision.

The CPU time required for Run1 was 10 seconds.

CONCLUSIONS

GASWAT-PC is a powerful, low cost microcomputer program for material balance analysis of gas reser-voirs. Several commonly accepted techniques and aquifer models (and extensions) are included for interactive examination of condensate, volumetric, abnormally pressured, and water drive situations.

This program also can predict reservoir pressure performance and ultimate gas recovery. Some other important features of this system include:

1. Gas gravity and gas composition correlations for Z-factor calculation,

2. User selection of the data points included in each method,

3. Automatic selection for the best radial aquifer,

4. Calculation of pressure gradient in the water Figure 6. d illustrates the matches of the observed invaded region, P/Z and the calculated P/Z values for both methods. It is evident that the linear aquifer gives the better pressure (or P/Z) match. Note that this straight line behavior of the P/Z plot for water drive gas reservoirs has been reported in many references.1,2 Bruns et al. indicated that the inflection of this curve must be considered as a sign of water drive.

The calculation time required in this case is 40 seconds.

Case 4: A Water Drive Reservoir With A High Dip Angle

To illustrate the effect of a pressure gradient in the water invaded region, a hypothetical water drive reservoir with a high dip angle was constructed. A radial aquifer with ralrg = S was assigned, and G = 20 BCF [0. 57 x 109 m3 J. The dip angle is 30 degrees and Sgr is 0.25. The other reservoir param-eters are listed in Table 4.

GASWAT-PC is used to predict the pressure performance at constant gas production rate, with and without the pressure gradient option ( Run1 and Run2, respectively). The results are listed in Table 4 and shown in Figure 7.

Note that P1 in Table 4 indicates the average pressure in front of the current water gas contact. It is found that the pressure difference between the two runs is 742 psi [5,116 kPa] at the end of production. This shows that the pressure gradient effect is significant in a strong or medium water drive reservoir with a high dip angle.

The reservoir was modeled with a commercial numerical simulator. RZ coordinates of 134*6 grid blocks were used in this study, and the results are listed in Table 4 and shown in Figure 7. GASWAT-PC and the simulator give .very comparable pressure performance predictions.

31

S. Pressure match checking for each method,

6. Input panel to assist users in creating or editing dataset, and

7. Summary table and graphic output to help users in analyzing results.

NOMENCLATURE

cw CGWC CURV

gas formation volume factor, RB/MCF [res m3/stock-tank m3] water formation volume factor, RB/STB [res m3/stock-tank m3] effective compressibility, psi-1 [kPa-1] (cwSwi + Cf)/(1-Swi> formation compressibility, psi-1 [kPa-1] water compressibility, psi-1 [kPa-1] current gas water contact curvature, defined in equation (B-4) expansion of water and reduction in pore volume, RB/STB [res m3/stock-tank m3], defined in Equation (4) expansion of gas, RB/MCF [res m3 /stock-tank m3], defined in Equation (3) recovery efficiency, fraction volumetric sweep efficiency, fraction underground withdrawal, RB [res m3], defined in Equation (1) total trapped gas volume in HCPV, fraction original wet gas in place, BCF [m3] cumulative condensate produced, MMSTB [stock-tank m3] cumulative dry gas produced, BCF [m3] trapped wet gas, BCF [m3] cumulative wet gas produced, BCF [m3] net thickness, ft. [m]

HCPV = hydrocarbon pore volume, MMRB [res m3] Kc = condensate conversion factor, MCF/STB [m3Jm3] ~tD dimensionless time coefficient, yr-1 KtD = theoretical dimensionless time coefficient,

yr-1, defined in Equation (A-10)

8 GASWAT-PC: A MICROCOMPUTER PROGRAM FOR GAS MATERIAL BALANCE WITH WATER INFLUX SPE 16484

kw

kwrg

L1

L2

MLc OGWC

p p1

p2

Pt

qw Qd ra rg

r1

r2 Sgr Swi

S(P,t) t

tD TDF

T u u

vaa

We Wp

y z

~ 1-lw e 'II=

Yc Yw

a

0'

effective permeability to water in the aquifer, md effective permeability to water at residual gas saturation, md distance of linear gas reservoir at current gas water contact, ft [m] distance of linear gas reservoir at original gas water contact, ft [m] molecular weight of condensate original gas water contact pressure, psia [kPa] average pressure in front of current gas water contact, psia [kPa] pressure at original gas water contact, psia [kPa] average pressure in the water invaded region, psia [kPa] water influx rate, RB/D [res m3/D] dimensionless water influx aquifer radius, ft [m] radius of gas reservoir at original gas water contact, ft [m] radius of gas reservoir at current gas water contact, ft [m] rg f't [m] residual gas saturation, fraction initial water saturation, fraction aquifer function, defined in Appendix A time, years dimensionless time dimensionless time adjusting factor reservoir temperature, F [C] aquifer constant theoretical aquifer constant, defined in Appendix A pore volume of aquifer, MMRB [res m3] width of linear reservoir, ft [m] cumulative water influx, MMRB [res m3] cumulative water produced, MMSTB [stock-tank m3] function defined in equation (17) gas deviation factor porosity, fraction water viscosity, Cp [Pas] dip angle, degree influx enchroachment angle, degree specific gravity of condensate specific gravity of formation water standard deviation, defined in Equation (7) normalized standard deviation, defined in Equation ( 6)

Subscriots

a = minimum abandonment pressure condition aw = watered-out abandonment condition

i initial condition j index of loops

sc standard condition 1 location at current gas water contact 2 location at original gas water contact t trapped gas in water invaded region

ACKNOWLEDGMENTS

The authors thank Texaco Inc. for permission to publish this paper. They also thank Dr. Joe P. Vogt for his assistance in making the numerical simulation run.

32

REFERENCES

1. Bruns, v. c.:

J. R., Fetkovich, M. J. and Meitzen, "The Effect of Water Influx on P/Z-

Cumulative Gas Production Curves," J. Pet. Tech. (March 1965) 287-291.

2. Chierici, G. L., Pizzi, G. and Ciucci, G. M.: "Water Drive Gas Reservoirs: Uncertainty in Reserves Evaluation From Past History," J. Pet. Tech. (February 1967) 237-244.

3. Cole F. W.: Reservoir Engineering Manual, Houton, Gulf Publishing Co., 1969,

4. Ramagost, B. P. and Farshad, F. F.: "P/Z Abnor-mally Pressured Gas Reservoirs," paper SPE 10125, presented at the 1981 SPE Annual Technical Conference and Exhibition, San Antonio, Oct. 1981.

S. Havlena, D. and Odeh, A. S.: "The Material Balance as an Equation of a Straight Line," Trans., AIME. Part 1:228 (1963). Part 2:231 (1964) I-815.

6. Tehrani, D. H.: "An Analysis of Volumetric Balance Equation for Calculation of Oil in Place and Water Influx," J. Pet. Tech. (September 1985) 1664-1670.

7 Dake, L. P. : Fundamentals of" Reservoir Engineer-ing, Elservier Scientific Pubishing Company, 1978.

8. Schilthuis, R. J.: "Active Oil and Reservoir Energy," Trans., AIME, 118:37.

9. Hurst, W.: "Water Influx Into a Reservoir and Its Application to the Equation of Volumetric Balance," Trans., AIME, 151. 1943, 57.

10. van Everdingen, A. F. and Hurst, W.: "Application of the Laplace Transform to Flow Problems in Reservoirs," Trans., AIME, 186. 1949. 305-324B.

11. Cragoe, C. S.: "Thermodynamic Properties of Petro-leum Product," Bureau of Standards, U. S. Depart-ment of Commerce Misc. Pub., No. 7 (1929) 26.

12. Standing M. B.: "Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems," SPE AIME, Dallas, 1977.

13. Wichert, E. and Aziz, K.: "Calculation of Z 's for Sour Gases," Hydrocarbon Processing, 51(5) 1972, 119-122.

14. Dranchuk, P. M., Purvis, R. A. and Robinson, D. B.: "Computer Calculation of Natural Gas Compressibility Factors Using the Standing and Katz Correlation," Institute of Petroleum, IP 74-008 (1974).

15. Hall, K. R. and Yarborough, L.: "A New Equation of State for Z-factor Calculations," Oil and Gas J. (June 1973) 82-92.

16. Standing M. B. and Katz, D. L.: "Density of Natural Gases," Trans., AIME, 146. 1942, 64-66.

17. Shagroni, M. A.: "Effect of Formation Compres-sibility and Edge Water on Gas Field Performance," master thesis, Colorado School of Mines, 1977.

SPE Jv4~~-----------------------------B_._w_a_n~g_a_n~d_T_. __ s_._T_e_a_sd_a_l_e _______________________________ 9-. 18. Agarwal. R. G., Al-Hussainy, R. and Ramey, H. J.,

Jr.: "The Importance of Water Influx in Gas Reservoirs," J. Pet. Tech. (November 196S) 1336-1342.

19. Lutes, J. L. et al: "Accelerated Blowdown of a Strong Water-Drive Gas Reservoir," J. Pet. Tech. (December 1977) 1S33-1S38.

20. Dumore, J. M. : "Material Balance for a Bottom-Water Drive Gas Reservoir," Soc. Pet. Eng. J. (December 1973) 328-334.

21. Ikoku, C. U.: "Natural Gas Engineering," PennWell Publishing Co. (1980).

APPENDIX A

DEFINITIONS OF AQUIFER FUNCTIONS AND AQUIFER CONSTANTS

The following equations define the aquifer functions [S(P, t)] for various aquifers, and the theoretical aquifer constants (U) for the small pot, infinite linear, and finite/infinite radial aquifers. Note that the correct U for various aquifers are determined from matching production history, using HO or other material balance methods.

1. Small Pot AquiferS

S(P,t) = (Pi - Pn) . (A-1)

where subscr~pt n refers to time step n, and the theoretical U is defined as

U is defined as follows: 1. Normalize the data set (xiYi) by setting

xi - Xmin Xmax - Xmin

Yi - y(xmin> Xmax - Xmin

. (B-1)

. (B-2)

where max and mix represent the maximum and minimum values, and the range of Xi is 0 to 1.

2. Find the best quadratic fit (2nd order polynomial regression) to the normal;ized data points (Xi, Yi).

(B-3)

10 GASWAT-PC: A MICROCOMPUTER PROGRAM FOR GAS MATERIAL BALANCE WITH WATER INFLUX SPE 16484

3. The curvature CURV (%) is then defined as

CURV = 100 (C-4)

CURV represents a measure of how much the parabola (B-3) departs from a straight line over the range X = 0 to 1 (Y = bo + b1 X is not necessarily the "best" straight line for the data set). Figl,lre 8 shows 100(Y - bo) /b1 for different values of CURV. Note that CURV > 0 yields an upward curvature while CURV < 0 gives a downward curvature.

APPENDIX C

MODIFIED MATERIAL BALANCE EQUATION As water influx enters into the reservoir, the conven-tional material balance equation has to be modified by considering the pressure gradient in the water invaded region (see Figure 2). original HCPV = current HCPV in front of CGWC

th\,lS,

G Bgi

+ current HCPV between CGWC and OGWC + HCPV reduction in front of CGWC + HCPV reduction between CGWC and OGWC + water influx - underground water withdrawal

(G - Gwgp - Gt)Bg1 + Gt Bgt

[aBgi-1

- Swi ] + ce (Pi-P1) Gt Bgt Sgr [at Bgt

1 - Swi ] + Ce (Pi-Pt) Sgr

+ We2

- Wp Bw1 . . . . . (C-1) and since

(We2 - Wp Bw1> Sgr Gt Bgt = 1 - Sgr - Swi

. . . . (C-2)

equation (C-1) can be rearranged as

Gwgp Bg1 G(Eg1 + Efw1> + (We2 - Wp Bw1> (1 - A - B) .. (C-3)

where

Sgr Bg1 - Bgt (C-4) A 1 - Sgr - Swi Bgt Swi cw + Cf

B = 1 Sgr - Swi (Pt-P1) (C-5)

-

(C-6)

34

If the pressure gradient is negligible, P1 equals to P2 and A and B will be zero. Then the modified material balance equation will be identical to the conventional equation (1).

APPENDIX D

P1 AND Pt CALCULATIONS FOR LIBEAR AND RADIAL RESERVOIRS

Linear Reservoir:

P2 - ( .001127 kwrg h W

+ 0.433 rw tane) (L2-L1).

Radial Reservoir:

(D-1)

.(D-2)

.(D-3)

5.6146 x 106 (We2-WpBw1> 0.5 (1-Sgr-Swi> ~nh W/360) ] .. (D-4)

qwuwln(r2/r1) P2- - 0.433rwtane(r2-r1) (D-5) 0.00708kwrgh

qwllw [ P2 - 0 5 -0.00708kwrgh

2

3 r22+r1r2+r12

r1+r2 ] . (D-6)

The qw in equations (D-2), (D-5) and (D-6) is defined as

365.25 We2- Wp Bw1 ... (D-7)

tn - tn-1

SI METRIC CONVERSION FACTORS

OAPI 141.5/(131.5 + 0 API) bbl X 1.589 873

cp x 1.0*

ft X 3.048*

ft3 X 2.831 685

(F-32) /1.8 psi x 6.894 757

* Conversion factor is exact.

= g/cm3

E-01 m3

E-03 Pas

E-01 = m

oc

E+OO kPa

TABLE 1

INPUT_AND OUTPUT OF CASE 1

This Example Chosen from SPE paper 10125:4 Anderson "L" Abnormally Pressured Reservoir. South Texas.

water compressibility formation compressibility porosity initial water saturation reservoir temperature initial pressure initial gas formation factor depth original gas in place

3 microsips 15 microsips 0.2 0.35 267 F 9507 psia 0.5537 RB/MCF -11167 ft 69 BCF by volumetrics

SP'E 1 6 4 8 4,

.............. Production Data ...............

PZ1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 1

Data Selection t p Gp PZ2 RF HO {da:i:sl {J2sia} z {BCF~

0 1 1 0.0 9507.0 1.4400 o.o 0 1 0 69.00 9292.0 1.4180 0.393 0 1 0 182.00 8970.0 1.3870 1. 642 0 1 0 280.00 8595.0 1.3440 3.226 0 1 0 340.00 8332.0 1.3160 4.260 0 1 0 372.00 8009.0 1.2820 5.504 0 1 0 455.00 7603.0 1.2390 7.538 0 1 1 507.00 7406.0 1.2180 8.749 0 1 1 583.00 7002.0 1.1760 10.509 0 1 1 628.00 6721.0 1.1470 11.758 0 1 1 663.00 6535.0 1.1270 12.789 0 1 1 804.00 5764.0 1.0480 17.262 0 1 1 987.00 4766.0 0.9770 22.890 1 1 1 1183.00 4295.0 0.9280 28.144 1 1 1 1373.00 3750.0 0.8910 32.567 1 1 1 1556.00 3247.0 0.8540 36.820

excluded in regression included in regression

................ Output Summary ................ . Method Used P/Z method 1 P/Z method 2 Ramagost-Farshad Havlena-Odeh *

Gl BCF 96.61 79.86 74.04 77.05

~~. % 3.18 0.22 2.12 9.94

GLp {MSTB}

o.o 29.90

122.90 240.90 317.10 406.90 561.20 650.80 776.70 864.30 939.50

1255.30 1615.80 1913.40 2136.00 2307.80

* Formation compressibility of 12 microsips is calculated from the slope

35

TABLE 2

INPUT AND OUTPUT OF CASE 2

A Typical Louisana Offshore Gas Reservoir with Strong water Influx

water compressibility formation compressibility water viscosity porosity initial water saturation reservoir temperature initial pressure initial gas formation factor gas radius aquifer permeability aquifer thickness

3.1 microsips 3.2 microsips 0.33 cp 0.306 0.139 201 F 4243 psia 0.7673 RB/MCF 13000 ft 571 md 59 ft 180 degree water enchroachment angle

original gas in place theoretical dimensionless time

806 BCF by volumetrics

coefficient KtD theoretical radial aquifer constant U

12.78 yr-1 10740 RB/psi

.... Production Data ............... Data Select t p Gp GLp Wp ~ c..QI& tlQ ~ ~ __ z_ ~ CMSTB) _.J.('1..S_'rn.l

1 1 1 o.o 4243.0 0.9718 o.o o.o o.o 1 0 0 283.0 4165.0 0.9676 23.451 129.58 o.o 1 0 0 452,0 4108.0 0.9651 29.324 150.52 0.72 1 1 0 799.0 3935.0 0.9566 57.343 214.96 9.42 0 1 0 1296.0 3760.0 0.9486 102.318 338.86 27.54 0 1 1 2172.0 3538.0 0.9395 165.984 534.18 56,95 0 1 0 2607.0 3433.0 0.9357 190.122 603.17 62.65 0 1 1 2966.0 3408.0 0.9348 226.208 698.68 78.22 0 1 1 3298.0 3331.0 0.9322 263.241 808.73 93.05 0 1 0 3663.0 3178.0 0.9275 297.909 902.81 118.55 0 1 1 4028.0 3138.0 0. 92 64 327.622 1001.46 173.58 0 1 1 4390.0 3100.0 0.9253 352.843 1070.02 201.13 0 1 1 4767.0 3054.0 0.9242 370.186 1114.92 207.46 0 1 1 5156.0 3078.0 0.9248 387.236 1165.00 240.77 0 0 1 5414.0 2933.0 0.9214 417.052 1251.62 307.01 0 0 1 5504.0 2917.0 0.9211 424.861 1272.07 317.04 0 0 1 5579.0 2897.0 0.9207 433.398 1291.01 343.42 0 0 1 5809.0 2875.0 0.9202 449.844 1330.60 439.21 0 1 5940,0 2807.0 0.9190 465.379 1359.84 457.99 0 1 62 9 3 . 0 2496.0 0.9146 521.950 1457.16 667.32 0 0 1 6501.0 2355.0 0.9142 549.323 1493.04 861.46 0 0 1 6807.0 2300.0 0.9143 575.786 1524.69 1153.86

0 1 6875.0 2291.0 0.9143 581.104 1531.62 1245.00

0 excluded in regression 1 included in regression

TABLE 2 (Cont.) INPUT AND OUTPUT OF CAS.IL2

...................... Output Summary ........................

___ M_et,\}~q .. P.~-~g ____ G. BCF u _L_%

P/Z method 1 1074.75 9.19 P/Z method 2 947.54 7.85 Cole 916.65 1. 74 6.97 Havlena-Odeh:

Small Pot 1430.23 7681.15 26.69 Schilthuis 961.39 15143.31 10.31 Hurst Simplified 892.68 81380.50 6.94 Radial (best fit)

ralrg=10 TDF=0.6 821.88 6330.95 3. 04 ralrg=18 TDF=0.6 862.67 5270.78 5.70

units of aquifer constant U for different types:

Cole method Small Pot Schilthuis Hurst simplified Radial

dimensionless RB/psi RB/psi/yr RB/psi RB/psi

minimal absolute normalized stanoa.rcl deviation a(F/EG) minimal absolute curvature ICURVI

36

QQ.RY ..... %

-11.82 - 0,08

SPE 1 b 4 84.

w -...j

TABLE 3

INPUT AND OUTPJ!~ .. Of .. CASE 3

A Gas Reservoir with Infinite Linear Water Influx This example chosen from "Natural Gas Engineering" by C. U. Ikoku21

water compressibility formation compressibility porosity initial water saturation reservoir temperature initial pressure initial gas formation factor aquifer permeability aquifer thickness

3 microsips 4 microsips 0. 308 0.425 155 F 2333 psia 1.172 RB/MCF 100 md 32.5 ft

original gas in place 337.7 BCF by volumetric

............ Production Data ........... . t P G U?~r:..tl ~ __ z_ ___rn_&:L_

0.0 2333.0 0.8829 0.0 2.00 2321.0 0.8843 2.305 4.00 2203.0 0.8849 20.257 6.00 2028.0 0.8893 49.719 8.00 1854.0 0.8956 80.134

10.00 1711.0 0. 9 005 105.930 12.00 1531.0 0.8997 135.350 14.00 1418.0 0.9134 157.110 16.00 1306.0 0.9223 178.300 18.00 1227.0 0.9231 192.089 20.00 1153.0 0.9289 205.744

All data points are included in regression for all methods.

................... Output Summary ................... . Method Used G. BCF U * ~

P/Z method 1 Cole Havlena-Odeh:

Small Pot Schilthuis Hurst Simplified Infinite Linear

383.27 347.83

429.56 339.69 332.13 309.03

0.21

-44339.40 5960.92

34315.29 28424.93

Units of aquifer constant U for different types: Cole method = dimensionless Small Pot = RB/psi Schilthuis = RB/psi/yr Hurst simplified = RB/psi Infinite Linear = RB/psi/yr0.5

0.91 9.77

18.37 10.17

9.36 3. 09

TABLE 4

INPUT AND OUTPUT OF CASE 4

Pressure Prediction for a Hypothetical Water Drive Reservoir With A High Dip Angle.

formation compressibility water viscosity porosity initial water saturation residual gas saturation reservoir temperature gas gravity gas radius aquifer permeability, kw effective permeability to water at

residual gas saturation, kwrg aquifer thickness water enchroachment angle dip angle original gas in place aquifer radius/gas radius ratio, ralrg dimensionless time coefficient KtD radial aquifer constant u

4 microsips 0.25 cp 0.2 0.25 0.25 250 F 0.6 3786 ft 500 md

125 md 100 ft 30 degree 30 degree 20 BCF 5 230.1 yr-1 187.2 RB/psi

...................... Output Summary .................... t Gp Calculated P1, psia

.lQgy_.l ~ Simulation ..._...R]m!_ ~ 0.0 o.o 10173.0 10173.0 10173.0

182.5 0.730 10009. 0 10010.9 10027.4 3 65.0 1.460 9866.0 9859.4 9893.8 547.5 2.190 9709.0 9704.6 9757.9 730.0 2.920 9562.0 9547.8 9620.9 912.5 3.650 9403.0 9388.5 9482.7

1095.0 4.380 9236.0 9226.4 9343.2 1277.5 5.110 9068.0 9061.5 9202.1 1460.0 5.840 8908.0 8893.9 9059.5 1642.5 6.570 8747.0 8723.4 8915.7 1825.0 7.300 8567.0 8549.0 8770.8 2007.5 8.030 8392.0 8370.2 8623.9 2190.0 8.760 8221.0 8186.4 8475.3 2372.5 9.490 8030.0 7996.6 8324.4 2555.0 10.220 7840.0 7801.2 8171.5 2737.5 10.950 7643.0 7597.8 8016.1 2920.0 11.680 7435.0 7384.3 7859.2 3102.5 12.410 7214.0 7156.9 7699.6 3285.0 13 .140 6972.0 6909.9 7537.1 3467.5 13.870 6684.0 6629.0 7371.3

Simulation = numerical simulation result Run1 GASWAT-PC result, included pressure gradient effect in

water invaded region Run2 GASWAT-PC result, excluded pressure gradient effect

en -o rn

a---a 0

~ ex; ~ ....

5000

a:S 4000 .....

en p.. N' P:;- 3000

2000

SPE 1 6 4 8~ ~

OBS P-~!_ ____ _

PZ2 1000~----~--~-----r----~----~--~-----r--~

0 20 40 Gwgp,BCF

60 80

Fig. 1-Sample plots of P/Z Method 1 and Method 2.

vr 1.o z ~ 1-Sor ~ oC Ill

ffi Swl ....

oC

~

GAS

0

INFLUX

I

Lz ,. 1 ~"z

LENGTH OR RADIUS

(C)

AQUIFER

La ~"a

Fig. 2-Schematic of water-drive reservoirs: (a) linear flow model, (b) radial flow model, and (c) saturation profiles.

0.8

0.6

0.4

0.2

WATER DRIVE PERFORMANCE

/ \ \

\\WATER-OUT LINE

', \\ (Gwgp aw, Paw)

\ \ G Zaw

',

0~------~----~--~-------r------~~----~ 0 0.2 0.4 0.6 0.8 1

Gwgp/G, fraction

Fig. 3-Uitimate gas recovery calculations for depletion .and water-drive reservoirs.

38

0 0) 0.

N .........

(a) P/Z METHOD 2. G = .79.86 BCF

8000T-----------------------------~

USED IN FIT

NOT USED

Q. 4000

lL (.) m

2000,+-------~------~------~------~ 0 20

Gwgp, BCF

(c) HO METHOD. G = 77.05 BCF

40

150~----~----------~----------~

100

USED IN FIT .............

NOT USED ~' ....

. ~ v--............

50 ~o----~---2-oTo,-o-o--~---4o-oro-o---?---8-0~ooo (PI-P)/Eg, psi-MCF/RB

SP'E 1 6 4 8 4.

(b) RF P/Z' PLOT I G = 74:.04: B CF

: .... I 0. ..,.

~ 8000 v

0

l '"""--"0 .,..,.. ............_

i:, 4 0 0 0 .'""' ~ .,. ~

0

2000+-------~------~------~----~ 0 20 40

Gwgp,BCF

(4) COJIP.ARISON OF PRESSURE JI.ATCBIS

10000T-----------------------------~

-~~ ... ~ .....

.,.

..............

HO

--RF

111. 1000 ---...

................

-- --.

0+-------~------T-------~----~ 0 20

Gwgp, BCF 40

Fig. 4-Piots of Case 1.

39

(a) P/Z METHOD 1, G = 107-&.'75 BCF

SOOOr----------------------------,

" 4000

~ ~ 0.. 3000

~!!~P.l!tf.l.~ NOT USED

20oo+---~----r---~--~r-------~ 0 20 0 40 0 800

Gwgp, BCf

(o) SIIALL POT AQUIFIR, G '" U80.88 BCF

2000r----------------------------,

,. . ~ 1800 1--------:::'""''--, . _ m ,.... ~ "' w

-;;:. 1000 ~!!~~.l!4.f.l.~ NOT USED

soo+-----~--~--~------r-~---4

2000

.... 1800 0 m

j -;;:. 1000

500

2000 4000 8000 (PI-P)/Eg, psi-MCf/RB

1000

(e) BURST AQUIFER, G::: 8112.118 BCF

...

. :~: ___.___. ...

0 5000 10000 S(P,T)/Eg, pei-MCf/RB

(a) COMPARISON OF PRISSURIIIATCBIS

SOOOr----------------------------, -----COLE ---- SCHIL THUIS

4000 ....

-----HURST ....

" .......

:. "'~ ---RADIAL a: ..... ~ 3000 ......... .,,... ____

~ ~..;:~ 2000

0 200 40 0 eoo Gwgp,BCf

SPE 1 6 4 84;

(b) COLE PLOT, G = 916.85 BCF

2000r----------------------------,

.... . -~ 0 .-" .... ~ ~~ ... :: . . ......... ~ 1000 ........ .-! E ~~m~

NOT USED o+---~----r-------~r-------~

0 100 400 100 Gwgp,BCf

(4) SCBILTBUIS AQUIFIR, G '" tel.at BCF

2000r----------------------------,

~ 1100 -~~ : _.,...,.. ~1000/'

soo+--------2 o-o~o-o------.-o~oo-o------,-4oooo S(P,T)/Eg, pii-YR-MCf/RB

2000

.... uoo 0 m .;. w

-;;:. 1000

sao

(f) ~A.~~"\1~11\u,~v n/ra=10 TDF=O.II

~ ....... ~ ~--

0 50000 100000 150000 S(P,T)/Eg, pal-MCf/RB

(b) ULTIIIATI GAB RICOYIRY CALCULATION Pa'" 11811 pla OWapa'" 881,811 BCF

8000~---------------------------,

. ...... ' "

4000 '

'ii ... ' a. .... ,.~ \ ~ ..... ;, Q. 1000

"" ---RADIAL ' ' ' ----WATER-OUT ' ' LINE ' 0 \

0 500 1000 Gwgp,BCf

Fig. 5-Piots of Case 2.

40

(a) P/Z METHOD 2, G::: !183.2'7 BCF (b) COLE PLOT, G = 3f'7.83 BCF

3000,.---"'---'~---------, 400,.--~~----------,

0 ;;

CD .;. . w .

": 2000

~. ~.A~~ ~ J!IO ~~ ~

II... 0 m .;. w ~

< ~ (/] Il-l

i

If ~ 0. 0>

~ IOOO+-----,--~---r--------1

100 200 300 300~-~-~--~--r--~-~

100 zoo 300 Gwgp, BCF Cwgp, BCF

(o) llffiMIT. LIMIAa .lCIUIJIII, Q " ad.OII ICr {4) COMPARISON or PRISSURI 114TCBIS

400 40

....

.t

,.....

~ 0

,.0 I ~

SPE 1 b 4 8 4~

CURV=% 1.5------------------------ 60

1 ... ..... ........ ......................... a

40 30 20 10 0 -10 -20 -30 -40 ~~ .......... -, -60

0~--------~--------+-----------------~ 0 0.5

X

42

1