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Optimizing production from water drive gas reservoirs based on desirability concept
Journal of Natural Gas Science and Engineering 21 (2014) 260e269
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Optimizing production from water drive gas reservoirs based on desirability concept Meysam Naderi a, Behzad Rostami a, *, Maryam Khosravi a, b a b
Institute of Petroleum Engineering, School of Chemical Engineering, College of Engineering, University of Tehran, Tehran, Iran IOR Research Institute, National Iranian Oil Company, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 June 2014 Received in revised form 6 August 2014 Accepted 10 August 2014 Available online
There are various factors which determine the optimization and economic production from water drive gas reservoirs. These factors play an important role in designing an effective reservoir development plan. The present study, in the first step, investigates the relation between recovery factor, volumetric sweep efficiency and cumulative water production with six different engineering and geologic factors using design of experiments (DOE) and response surface methodology (RSM). Next, all derived response functions are optimized simultaneously based on the concept of desirability. In this manner, part of water drive gas reservoirs is simulated using BoxeBehnken design. Important factors that have been studied include reservoir horizontal permeability (Kh), permeability anisotropy (Kv/Kh), aquifer size (Vaq), gas production rate (Qg), perforated thickness (Hp) and tubing head pressure (THP). The results indicate that by combining various levels of factors and considering relative importance of each response function, optimized conditions could be raised in order to maximizing recovery factor, volumetric sweep efficiency and minimizing cumulative water production. Also high rates of gas production result poor volumetric sweep efficiency and early water breakthrough, hence ultimate recovery factor decreases by 3.2e8.4%. © 2014 Elsevier B.V. All rights reserved.
Keywords: Optimization Design of experiments Response surface methodology Desirability Recovery factor Volumetric sweep efficiency Cumulative water production
1. Introduction Prediction of gas production is an important part of reservoir development, management and economic evaluation. Today, with the increasing growth of need to use of fossil fuels and the high volume of trapped gas in reservoir, it is crucial to optimize production from water drive gas reservoirs. Without investigating and understanding the production sensitivities to parameters such as gas production rate, tubing head pressure, perforated length from these reservoirs, it is not possible to reach predetermined goals to increase profitability and reduce costs. Considering that a large part of gas reserves are recovered using the water drive process, it is very important to understand the mechanisms affecting the production from these types of reservoirs. The experimental study of Geffen et al. (1952) on core plugs revealed that the trapped gas saturation varied from 15 to 50 percent of the pore space for various porous media. Agarwal (1965) demonstrated that the ultimate recovery factor is a function of production rate, residual gas saturation, aquifer permeability and volumetric sweep efficiency. Gas recovery factor increases with increasing production rate and
* Corresponding author. Tel.: þ98 9125210473; fax: þ98 2188632976. E-mail addresses: [email protected], [email protected] (B. Rostami). http://dx.doi.org/10.1016/j.jngse.2014.08.007 1875-5100/© 2014 Elsevier B.V. All rights reserved.
decreasing aquifer permeability. Knapp et al. (1968) developed a two-phase two-dimensional model for predicting the gas recovery from aquifer storage fields as a function of production rate, aquifer strength and reservoir heterogeneity. Lutes et al. (1977) reported that the final blowdown of a Gulf Coast water drive gas reservoir at a reserves/production ratio of less than 2 provided an increase in gas recovery. Brinkman (1981) reported that accelerated gas withdrawals of up to 115 MMSCF/D from a U.S. gulf coast water drive gas reservoir resulted in a 20% increase in remaining gas recovery versus continued low-rate depletion. Al-Hashim (1998) studied the effect of aquifer size on partial water drive gas reservoirs. They concluded that if the ratio of aquifer external radius to reservoir external radius be less than two, the effect of aquifer on performance of the gas reservoir can be neglected. For ratios greater than two, gas recovery is sensitive to both initial reservoir pressure and aquifer size. Increasing aquifer size and initial reservoir pressure reduces gas recovery. A simulation study by Cohen (1989) determined that accelerating production rate and coproduction increases recovery by 2.3% and 5.6% respectively. A reservoir simulation study by Hower and Jones (1991) showed that to increase recovery, the production rate should be lowered rather than accelerated because of improved volumetric sweep efficiency. El-Ahmady et al. (2002) studied the effect of aquifer on estimation
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of the original gas in place. They showed that the unsteady state nature of aquifers can lead to overestimation of the original gas in place. Armenta and Wojtanowicz (2002), Armenta et al. (2003), Armenta and Wojtanowicz (2005) studied the effect of well completion on performance of gas wells. Sech et al. (2007) simulated the effect of production rate on recovery factor in horizontal wells. They found that recovery reduces by increasing the production rate and permeability anisotropy due to water cresting. Wang (2009) studied the impact of turbulence and the importance of hydraulic fracturing on well deliverability for both vertical and horizontal gas wells. Lee et al. (2010) presented a correlation for predicting recovery factor changes with aquifer size, ratio of residual to initial gas saturation, ultimate volumetric sweep efficiency, abandonment and average reservoir pressure. Wang (2012) studied the tubing limitation and turbulence effects on well deliverability of both vertical and horizontal wells with and without artificially induced hydraulic fractures. Sedaghatzadeh et al. (2013) investigated the optimum accelerating production rate from water drive gas reservoirs in laboratory scale systems. Rezaee et al. (2013) studied the effect of heterogeneity and aquifer to gas zone permeability on the gas phase trapping in water drive gas reservoir. Results of their study in laboratory scale show that heterogeneity is not always detrimental to gas recovery and it may be improved with increasing heterogeneity when ratio of aquifer to gas zone permeability is less than one. Review of previous studies shows that the simultaneous optimization of recovery factor, volumetric sweep efficiency and cumulative water production from water drive gas reservoirs has never been investigated. This objective is studied by applying design of experiments, response surface methodology and desirability concept. Experimental design and response surface methodology has been used in petroleum engineering applications including performance prediction (Chu, 1990), uncertainty modelling (Damsleth et al., 1991; Van Elk et al., 2000; Friedmann et al., 2001), sensitivity studies (White et al., 2000), upscaling (Narayanan et al., 1999), history matching (Anonsen et al., 1995) and development optimization (Dejean and Blanc, 1999). Also multiple response optimization based on desirability concept is widely used in numerous engineering fields (Harrington, 1965; Derringer and Suich, 1980). This study is one of the earliest ones taking advantages of design of experiments, response surface methodology and desirability concept for optimizing production from water drive gas reservoirs. Production optimization will be based on maximizing recovery factor and volumetric sweep efficiency and also minimizing cumulative water production.
261
Fig. 1. Simulated model. The reservoir is shown in red with 3300 ft external radius, and the aquifer in blue with 6600 ft. The reservoir has 100 layers with constant thickness of 300 ft and the aquifer with 20 layers of variable thickness. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
was set to 25%. The reservoir temperature and initial reservoir pressure at datum depth of 8530 feet were fixed to 220 F and 5290 psi, respectively. The gas viscosity was estimated using the correlation developed by Lee et al. (1966). The gas deviation factor was estimated using correlations presented by Dranchuk et al. (1974). Relative permeability and capillary pressure data measured for gasewater systems in the experimental work of Chierici et al. (1963) has been used in this survey. Production is via a single vertical well with 7 inch internal e tubing diameter. Reservoir horizontal permeability, permeability anisotropy, aquifer size, gas production rate, perforated thickness and tubing head pressure are varied simultaneously in simulation tests that are described in the following section. In this study, the well produces at constant tubing head pressure until production rate is greater than 10% of maximum initial gas production rate and bottom hole pressure is more than 500 psi. The Petalas and Aziz (2000) mechanistic model has been used to calculate vertical flow performance curves. This model is very accurate and it can be used for up and downhill flow, and for all pipe geometries.
2. Model description 3. Methodology The model described here simulates gas reservoirs with an active bottom aquifer using a radial system (Fig. 1). The reservoir zone measures 3300 feet in the radial direction and has a maximum gas column thickness of 300 feet. A bottom water aquifer is connected to the base of reservoir. The aquifer measures 6600 feet in the radial direction, extending beyond the reservoir. The aquifer thickness is not constant and changes in order to modify aquifer size during simulation. Flow is simulated using a gridding scheme that is locally refined around the well, and coarsened away from the well in the radial direction. The width of cells increases exponentially and highest resolution cells are located close to the well. Grid layering in the vertical direction is considered 120 layers, 100 layers with thickness of 300 feet for reservoir and 20 layers for aquifer section. The first 10 layers of the aquifer has constant thickness of 100 feet and the second 10 layers has a variable thickness to be able to set the ratio of aquifer to reservoir volume namely aquifer size. Reservoir and aquifer porosity is constant in all simulation and it
3.1. Determination of the response functions The following study, inspect the effect of six engineering and geologic factors on gas recovery factor, volumetric sweep efficiency and cumulative water production. In this regard, design of experiment and response surface methodology was employed. Design of experiments (DOE) is a well known technique to get maximum information with simultaneous varying of all parameters and required less number of performing time consuming numerical tests. Response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables to obtain an optimal response. More details about design of experiments and response surface methodology are given in Box and Wilson (1951). Based on BoxeBehnken Design (1960), it is required to design and simulate 49 different models to extract necessary information such as reservoir pressure,
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pressure of the aquifer, and etc in order to be able to assess and quantify the impact of six factors on mentioned response functions. BoxeBehnken Design has been used because of the increased accuracy of the model by applying the quadratic terms of each factor and also reduced simulation runtime thereby reducing costs. After determining the response functions via response surface methodology, in the next step, it will be possible to optimize all functions simultaneously based on the desirability concept. Range of factors with their appropriate transfer functions are given in Table 1. In RSM, transfer functions are used to normalize range of data between 1 and þ1 and also ease of computation. In this table, X1, X2, X3, X4, X5 and X6 are coded variables between 1 and þ1 for Kh, Kv/Kh, Vaq, Qg, Hp and THP respectively. For example, range of horizontal permeability (Kh) from (10, 100, 1000) can be normalized to (1, 0, þ1) by using transfer function of Log (Kh) 2. All 49 different flow simulations were undertaken using the Eclipse® 100 black oil simulator and the following equations developed by RSM for gas recovery factor (RF), volumetric sweep efficiency (Ev) and cumulative water production (Wp).
RF ¼ 78:62 þ 8:33X1 1:8X2 þ 0:78X3 þ 0:5X4 2:1X5 10:07X6 5:68X12 2:22X52 þ 4:33X1 X2 þ 2:62X1 X3 þ 2:71X1 X5 þ 2:64X4 X5 (1) Ln Ev ¼ 1:83 þ 0:16X1 þ 0:02X2 þ 1:79X3 0:093X4 0:063X5 0:12X6 þ
0:09X32
þ
0:162X62
þ 0:063X1 X3
þ 0:241X2 X3 þ 0:123X2 X4 0:254X3 X4 0:024X3 X6 (2) Ln Wp ¼ 8:34 2:4X1 þ 0:78X2 þ 2:58X3 þ 0:75X4 þ 2:18X5 0:87X6 þ 0:8X22 þ 1:99X32 þ 1:14X52 þ 0:96X62 1:37X1 X2 þ 1:51X1 X3 þ 1:28X2 X3 þ 1:32X2 X4 X2 X6 1:35X3 X4 X3 X5 (3) where X1, X2, X3, X4, X5 and X6 are coded variables for mentioned factors in Table 1. The coefficient of determination (R2) and adjusted coefficient of determination for derived response functions are given in Table 2. The validity of three proxy equations is shown in Fig. 2 with a plot of predicted versus actual experiments. R2 and Adjusted R2 were used to check the quality of the given equations for prediction. The obtained R2 for three response functions shows that more than 89% of the variation in each response could be as a result of the selected factors. The derived proxies were also validated by 12 new simulation tests with the properties outside the group used to generate the proxies. Table 3 show the result of these
Table 1 Range of factors with transfer functions. Factor
Coded variable
Levels
Transfer function
Kh (md) Kv/Kh (fraction)
X1 X2
10 0.01
Vaq (fraction) Qg (MMSCF/D)
X3 X4
1 60
Hp (%)
X5
30
60
90
THP (psi)
X6
500
1000
1500
100 0.1
1000 1
LogðKh Þ 2 Log KKhv þ 1
10 90
100 120
LogðVaq Þ 1 Qg 90 30 Hp 60 30 THP1000 500
Table 2 Response functions statistics. Response
R2
Adjusted R2
RF Ln (Ev) Ln (Wp)
89.4 96.7 92.3
85.8 95.5 88.1
blind tests. The T-test for significance of all coefficients shows that all terms taken in the above models are significant at confidence level of 10% for prediction at 90% confidence interval. These equations were used to study the main effect, interaction and square power effect of parameters on the performance of water drive gas reservoirs. 3.2. Heterogeneity effect on productivity of water drive gas reservoirs In this section, the heterogeneity effect on gas recovery from water drive gas reservoirs was investigated by using DykstraeParsons coefficient (Dykstra and Parsons, 1950). The DykstraeParsons permeability variation is defined by the following expression:
V¼
K50% K84:1% K50%
where K50% and K84.1% are corresponding permeability values at 84.1% and 50% of cumulative sample thickness respectively. Variation of this coefficient is between zero and one. The zero value indicates the ideal homogeneous porous media. While it changes over a range of zero to half, experiments can be approximated by a homogeneous model in reservoir simulation with minimal error. If the coefficient varies over a range of half to one, the reservoir is very heterogeneous. We studied the effect of heterogeneity on water drive gas reservoirs performance by considering DykstraeParsons coefficients from 0.1 to 0.9. In this regard, eighteen heterogeneous models with six different DykstraeParsons coefficients and three different permeability anisotropy were simulated. Also six homogeneous models with the same properties and equivalent geometric mean of reservoir horizontal permeability (KG) were simulated. The result of these 24 simulations is given in Table 4. General regression Analysis of data in Table 4 for recovery factor shows that the DykstraeParson Coefficient is not significant at confidence level of 10%. P-value of DykstraeParson Coefficient in the final regression for recovery factor is 15.6%. This reveals that heterogeneity effect in terms of DykstraeParson Coefficient plays a weak role, but permeability anisotropy not. Also sensitivity analysis of recovery factor was done by considering log-normal distribution for permeability anisotropy and geometric mean of reservoir horizontal permeability and triangular distribution for DykstraeParson Coefficient. Results indicated that although recovery factor increases by increasing the level of heterogeneity, this effect in terms of contribution to total variance is less than 5% regardless of confidence level. This fact that heterogeneity in terms of DykstraeParsons Coefficient has little impact on performance of gas reservoirs has been reported already by Muladi and Pinczewski (1999). Therefore a homogeneous model with equivalent geometric mean of reservoir horizontal permeability can be used instead of heterogeneous one with the minimal error. 3.3. Optimization based on desirability concept The desirability function method has been used for many years in industry to optimize of multiple response functions. It is based
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263
Pk w 1 i D ¼ d1 ðY1 Þw1 d2 ðY2 Þw2 / dk ðYk Þwk 1
(4)
where k and w denoting the number of responses and relative importance of each response respectively. The class of desirability functions that was proposed by Derringer and Suich (1980) is divided into three types, namely Nominal The Best (NTB), Larger The Better (LTB) and Smaller The Better (STB). Individual desirability function based on the optimization goals for types of NTB, LTB and STB are given in the following equations:
8 0 > > > > > > Y Li s i > > > < Ti Li DNTBi ðYi Þ ¼ > > Yi Ui t > > > > T Ui > > > i : 0
DLTBi ðYi Þ ¼
DSTBi ðYi Þ ¼
8 > > > > <
0
Yi Li > U > i Li > > : 1 8 > > > > <
s
1
Yi Ui > Li Ui > > > : 0
9 > > > > > > > if Li Yi ðxÞ Ti > > = if Yi ðxÞ < Li
> > > if Ti Yi ðxÞ Ui > > > > > > ; if Yi ðxÞ > Ui if Yi ðxÞ < Li if Li Yi ðxÞ Ui if Yi ðxÞ > Ui
s
if Yi ðxÞ < Li if Li Yi ðxÞ Ui if Yi ðxÞ > Ui
(5)
9 > > > > = > > > > ;
(6)
9 > > > > = > > > > ;
(7)
where Li, Ui and Ti are the lower, upper, and target values of response functions respectively, that are desired for response Yi (x), with Li < Ti < Ui. The exponents s and t determining how important it is to hit the target value. For s ¼ t ¼ 1, the desirability function increases linearly towards Ti; for s < 1, t < 1, the function is convex, and for s > 1, t > 1, the function is concave. Increasing exponents of desirability function, make the condition more difficult to reach an optimized solution. Optimization based on desirability concept consists of the following three steps: 1) Implement flow simulation based on DOE and determine all response functions by RSM. 2) Define individual desirability functions for each response. 3) Maximize the overall desirability D with respect to the controllable factors. 4. Results and discussion
Fig. 2. Predicted versus actual experiments for A) recovery factor, B) volumetric sweep efficiency and C) cumulative water production.
on the idea that performance of a product is generally characterized by many response variables. In many situations, these responses or quality characteristics are controlled by a set of independent factors. The desirability method finds operating conditions that provide simultaneously the most desirable response values. For each response Yi (x), a desirability function di (Yi) assigns numbers between 0 and 1 to the possible values of Yi (x), with di (Yi) ¼ 0 representing a completely undesirable value of Yi and di (Yi) ¼ 1 representing a completely desirable or ideal response value. The individual desirability's are then combined to give overall desirability (D) using the weighted geometric mean:
Equations (1)e(3) generally consist of three groups of expressions, the main factors, the interactions between factors and the square power of some factors. The constant in the first of equations implies that when factors are in their middle values, the mean value of the response is equal to that constant. The terms with negative coefficient cause reduction of response that is useful in reducing cumulative water production, while terms with positive sign tend to increase the recovery factor and volumetric sweep efficiency. The main effect of each factor is shown in Fig. 3. These factors have different effects on the three response functions trend. As indicated in the main effect plot for recovery factor, reservoir permeability and tubing head pressure have the most effect on recovery variation. This variation is very sharp for reservoir permeability and tubing head pressure of less than 100 md and 1000 psi respectively. After reservoir permeability and tubing head pressure;
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Table 3 Properties and results of 12 new blind simulation tests. Run#
X1
X2
X3
X4
X5
X6
Actual RF
Predicted RF
Actual Ln (Ev)
Predicted Ln (Ev)
Actual Ln (Wp)
Predicted Ln (Wp)
1 2 3 4 5 6 7 8 9 10 11 12
0.76 0.03 0.67 0.49 0.73 0.61 0.05 0.73 0.66 0.22 0.61 0.05
0.47 0.97 0.38 0.99 0.78 0.49 0.22 0.57 0.27 0.49 0.40 0.96
0.84 0.67 0.17 0.87 0.53 0.48 0.47 0.91 0.42 0.53 0.43 0.98
0.45 0.00 0.61 0.12 0.22 0.28 0.13 0.87 0.88 0.94 0.62 0.77
0.44 0.74 0.60 0.18 0.14 0.55 0.94 0.88 0.54 0.21 0.06 0.81
0.04 0.23 0.93 0.84 0.78 0.37 0.05 0.02 0.67 0.80 0.11 0.44
63.45 75.08 75.21 85.47 89.47 82.77 74.09 78.33 71.04 70.75 67.34 75.19
67.22 74.84 79.46 83.78 91.34 83.75 75.35 74.84 75.26 67.76 70.17 74.71
0.74 1.15 2.35 1.06 3.29 1.44 2.95 3.86 1.39 3.06 2.77 3.96
0.34 0.75 2.28 0.67 3.30 1.22 2.63 3.65 1.17 2.95 2.48 4.12
13.55 9.28 14.50 9.80 7.02 6.96 14.73 10.32 5.65 10.69 14.87 14.15
10.84 11.20 12.91 11.09 11.35 5.13 13.07 12.51 8.37 11.21 12.57 13.96
Table 4 The simulation result of 18 heterogeneous and 6 homogeneous models. Run
Kv/Kh
V
KG
(Kv/Kh)2
V2
(KG)2
Kv/Kh*V
Kv/Kh*KG
V*KG
Kv/Kh*V*KG
RF
Ev
Ln Wp
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.1 0.01 0.1 1 0.1 0.01 0.1 1 0.1 0.01 0.1 1 0.1 0.01 0.1 1 0.1 0.01 0.1 1 0.1 0.01 0.1 1
0 0.1 0.1 0.1 0 0.3 0.3 0.3 0 0.5 0.5 0.5 0 0.65 0.65 0.65 0 0.8 0.8 0.8 0 0.9 0.9 0.9
9.92 9.92 9.92 9.92 9.5 9.5 9.5 9.5 88.71 88.71 88.71 88.71 86.9 86.9 86.9 86.9 56.97 56.97 56.97 56.97 27 27 27 27
0.01 0.0001 0.01 1 0.01 0.0001 0.01 1 0.01 0.0001 0.01 1 0.01 0.0001 0.01 1 0.01 0.0001 0.01 1 0.01 0.0001 0.01 1
0 0.01 0.01 0.01 0 0.0841 0.0841 0.0841 0 0.2704 0.2704 0.2704 0 0.4225 0.4225 0.4225 0 0.64 0.64 0.64 0 0.7921 0.7921 0.7921
98.406 98.406 98.406 98.406 90.25 90.25 90.25 90.25 7869.5 7869.5 7869.5 7869.5 7551.6 7551.6 7551.6 7551.6 3245.6 3245.6 3245.6 3245.6 729 729 729 729
0 0.001 0.01 0.1 0 0.0029 0.029 0.29 0 0.0052 0.052 0.52 0 0.0065 0.065 0.65 0 0.008 0.08 0.8 0 0.0089 0.089 0.89
0.992 0.0992 0.992 9.92 0.95 0.095 0.95 9.5 8.871 0.8871 8.871 88.71 8.69 0.869 8.69 86.9 5.697 0.5697 5.697 56.97 2.7 0.27 2.7 27
0 0.992 0.992 0.992 0 2.755 2.755 2.755 0 46.1292 46.1292 46.1292 0 56.485 56.485 56.485 0 45.576 45.576 45.576 0 24.03 24.03 24.03
0 0.00992 0.0992 0.992 0 0.02755 0.2755 2.755 0 0.461292 4.61292 46.1292 0 0.56485 5.6485 56.485 0 0.45576 4.5576 45.576 0 0.2403 2.403 24.03
65.13 76.04 65.37 55.88 64.53 75.66 64.18 53.72 79.82 81.11 80.42 78.28 79.74 80.79 79.82 77.4 78.04 79.93 79.16 76.86 73.48 78.6 76.82 76.08
42.47 36.46 42.54 35.74 42.04 36.69 42.15 34.1 57.1 55.78 57.81 55.04 57.03 55.47 57.16 54.07 54.85 50.54 56.35 53.51 51.15 46.22 51.96 53.67
14.55 9.47 14.53 16.27 14.51 9.55 14.55 16.30 14.91 13.62 14.72 15.49 14.91 13.71 14.83 15.62 14.97 13.14 14.93 15.67 14.91 9.94 13.90 14.97
permeability anisotropy, production rate, aquifer size and perforated interval are the most effective factors respectively. Aquifer size has the highest effect on volumetric sweep efficiency and cumulative water production. For aquifer sizes greater than 10, volumetric sweep efficiency and cumulative water production increases sharply. As revealed in Fig. 4, the interaction plot for recovery factor; increasing tubing head pressure reduces recovery factor regardless of interaction with other factors. Production rate has different interaction effect with different factors. Increasing length of perforated interval has negative effect on recovery factor when reservoir permeability is low. Among all interaction terms, interaction between perforated interval and production rate is very important because of its economical effect on production and development of water drive gas reservoirs. Based on the perforated interval length, the recovery factor can be maximized or minimized with increasing production rate. Therefore, this important interaction effect will be explained more after optimization section. In order to determine the optimum condition to produce from water drive gas reservoirs in efficient manner, it is required to consider the relative importance of each response function simultaneously. This objective can be well developed by the application of the desirability concept. In the first step of simultaneous optimization of all response functions, the lower (Li) and upper (Ui) bounds of desirability equations (6) and (7) should be determined.
Values of Li and Ui for individual response function are calculated by considering the probability distribution functions for the independent variables and performing Monte Carlo simulation. In this regard, based on previous studies and available information (Peake et al., 2005), we assigned log-normal distribution for the three factors: reservoir horizontal permeability, permeability anisotropy, aquifer size and triangular distribution for gas production rate, perforated thickness, and tubing head pressure. Table 5 shows the values of Li and Ui for all three response functions. We optimized all three functions simultaneously by considering three values for exponent of desirability function (s in equations (6) and (7)) and four values of relative importance (wi in equation (4)) for recovery factor. Optimization results are given in Tables 6 through 8 for s ¼ 0.1, 1 and 10 respectively. In the referred tables, when relative importance of recovery factor (wRF) is equal to 100%, it means the recovery factor will be optimized and the other two responses are not important. In this research, the relative importance of volumetric sweep efficiency and cumulative water production are assumed equal. For instance, in Table 6, when relative importance of recovery factor is 75%, the relative importance for each of the volumetric sweep efficiency and cumulative water production will be 12.5%. It is required to assign relative importance for individual responses to optimize all three functions simultaneously based on overall desirability (D). As indicated in Tables 6 through 8, overall
Fig. 3. Main effects plot for A) recovery factor, B) volumetric sweep efficiency and C) cumulative water production. A main effect is present when different levels of a factor affect the response differently. The steeper the slope of the line, the greater the magnitude of the main effect.
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Fig. 4. Interaction plot for recovery factor. An interaction is present when the effect of one factor depends on the level of the other factor. Parallel lines in an interaction plot indicate no interaction. The greater the difference in slope between the lines, the higher the degree of interaction.
factors, optimum criteria are different. Regardless of the desirability function weight and relative importance of functions, this study shows that reducing the gas production rate from water drive gas reservoirs can optimize the recovery factor, volumetric sweep efficiency and cumulative water production. Variation of recovery factor with normalized aquifer size and reservoir horizontal permeability for different values of Kv/Kh is given in Fig. 5. While reservoir horizontal permeability is less than 70.8 md, recovery factor decreases by increasing aquifer size because of early breakthrough time and increasing water production. However, recovery factor increases by increasing aquifer size (<14%) for horizontal permeability greater than 70.8 md due to decreased rate of reservoir pressure decline. This interaction between reservoir and aquifer allows further gas to be produced prior to abandonment at the bottom hole pressure limit. Although this behaviour is not
Table 5 Values of Li and Ui for recovery factor, volumetric sweep efficiency and cumulative water production. Response function
Li
Ui
Optimization type
RF (%) Ln (Ev) Ln (Wp)
46 0.23 0.6
93 4.17 19
Maximum Maximum Minimum
desirability decreases with increasing exponent of desirability function. This means that by increasing value of s, it will be more difficult to reach optimum condition. However; increased accuracy of the simultaneous optimization solution compensates this reduction of overall desirability. The results of the simultaneous optimization of response functions (Tables 6 through 8) show that by combining various levels of
Table 6 Optimization with s ¼ 0.1 for desirability function of the STB and LTB type. Run#
wRF
X1
X2
X3
X4
X5
X6
RF
Ln (Ev)
Ln (Wp)
D
1 2 3 4
100% 75% 50% 25%
1 1 1 1
1 1 0.374 0.758
1 0.515 0.333 0.394
1 1 1 1
0.737 0.717 0.859 0.778
1 0.758 1 0.616
98.96 93 93 90.47
e 3.453 3.132 3.112
e 9.416 8.231 7.694
1 0.99 0.98 0.97
Table 7 Optimization with s ¼ 1 for desirability function of the STB and LTB type. Run#
wRF
X1
X2
X3
X4
X5
X6
RF
Ln (Ev)
Ln (Wp)
D
1 2 3 4
100% 75% 50% 25%
1 1 1 1
1 0.657 0.256 0.758
1 0.273 0.377 0.394
1 1 1 1
0.737 0.939 0.778 0.778
1 0.964 1 0.616
98.96 93 93 90.47
e 2.972 3.228 3.112
e 7.709 8.573 7.694
1 0.91 0.82 0.74
M. Naderi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 260e269
267
Table 8 Optimization with s ¼ 10 for desirability function of the STB and LTB type. Run#
wRF
X1
X2
X3
X4
X5
X6
RF
Ln (Ev)
Ln (Wp)
D
1 2 3 4
100% 75% 50% 25%
1 0.939 1 1
1 0.313 0.879 0.758
1 0.293 0.414 0.394
1 1 1 1
0.737 0.798 0.697 0.778
1 1 0.819 0.616
98.96 93 92.98 90.47
e 3.027 3.228 3.112
e 8.051 8.573 7.694
1 0.359 0.132 0.05
Fig. 5. Contour plot for recovery factor versus normalized values of aquifer size (Vaq) and reservoir horizontal permeability (Kres) for A) Kv/Kh ¼ 0.01, B) Kv/Kh ¼ 0.1 and C) Kv/Kh ¼ 1.
Fig. 6. Contour plot for recovery factor versus normalized values of gas production rate (Qg) and perforated thickness (Hp) for A) Kv/Kh ¼ 0.01, B) Kv/Kh ¼ 0.1 and C) Kv/Kh ¼ 1.
268
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predicted by material balance equation, it has been observed in Columbus Basin gas fields and also by simulation study for the Mahogany 20 Gas Sand (McMullan and Bassiouni, 2000; Hallam and Maria, 2003; Ali-Nandalal and Gunter, 2003). Contour plot of recovery factor versus normalized production rate and perforated interval for different values of Kv/Kh is shown in Fig. 6. In this figure, gas recovery factor increases with increasing production rate for perforated interval values greater than 60% of total pay zone thickness. In other words, when perforated interval is less than 60% of the total gas column thickness, recovery decreases with high values of production rate due to decreased gasewater ratio. Therefore, the well will be shut in because of the economical limit of gasewater ratio. Considering relative importance of volumetric sweep efficiency and the cumulative water production, recovery factors greater than 80% can be obtained for perforation length of less than 44% and production rate between 60 and 70 MMSCF/D. In other words, although accelerating production decreases ultimate recovery factor by 3.2e8.4%, it may be economically favourable over much shorter timescales. Results of the optimization reveal that for tubing head pressures between 500 psi to 1000 psi, all three responses will be optimized.
Kh reservoir horizontal permeability KG geometric mean of reservoir horizontal permeability Kv/Kh permeability anisotropy Li lower bound for the response function MMSCF/D million standard cubic feet per day Qg gas production rate RF gas recovery factor S&t weight of the desirability function THP tubing head pressure Ti target value for the response function Ui upper bound for the response function Vaq aquifer size V DykstraeParsons coefficient wi relative importance of response function Wp cumulative water production X1 coded variable for reservoir horizontal permeability X2 coded variable for permeability anisotropy X3 coded variable for aquifer size X4 coded variable for gas production rate X5 coded variable for perforated thickness X6 coded variable for tubing head pressure Yi (X) value of response function at X
5. Conclusions References In the present study, optimization of production rate from water drive gas reservoirs based on the concept of desirability has been investigated. Part of reservoir and aquifer with properties described in Section 2 and Table 1 was simulated and the effect of six factors on the recovery factor, volumetric sweep efficiency and cumulative water production was established by the experimental design techniques and response surface methodology. It is very important to note that the results and conclusions brought as an insight of this study are just valid under the specific model and assumed properties and its ranges. In addition to review the most important mechanisms and effective parameters, we aimed to introduce the useful methodology for analysis of the water drive gas reservoirs. The optimum conditions calculated using desirability functions and the main conclusions are as follow: 1. Accelerating production decreases ultimate recovery factor because of poor volumetric sweep efficiency and early water breakthrough. 2. Considering economic strategies, although accelerating production decreases ultimate recovery factor by 3.2e8.4%, it may be economically favourable over much shorter timescales. 3. Increasing production rate from 60 to 120 MMSCF/D increases gas recovery factor at least by 8% for perforated intervals greater than 60% of the gas column thickness. 4. Recovery factors greater than 80% can be obtained with production rates between 60 and 70 MMSCFD for perforated intervals less than 44% of gas column thickness. 5. Recovery factor increases by increasing aquifer size (<14%) for reservoir permeability greater than 70.8 md because of slow rate of reservoir pressure decline and increased amount of produced gas prior to abandonment at the bottom hole pressure limit. 6. Decreasing tubing head pressure below 1000 psi optimizes the production from water drive gas reservoirs. Nomenclature Di (Yi (X))desirability value for Yi (X) Ev volumetric sweep efficiency Hp perforated thickness k number of response functions
Agarwal, R.G., 1965. The Importance of Water Influx in Gas Reservoirs. Paper SPE 1244 presented at the SPE Annual Fall Meeting held in Denver, Colorado, 3e6 October. Al-Hashim, H.S., 1998. Effect of aquifer size on the performance of partial water drive gas reservoirs. SPE J. 3 (2), 380e386. SPE-13233-PA. Ali-Nandalal, J., Gunter, G., 2003. Characterizing Reservoir Performance for the Mahogany 20 Gas Sand Based on Petrophysical and Rock Typing Methods. Paper SPE-81048-MS presented at SPE Latin American and Caribbean Petroleum Engineering Conference, Port-of-Spain, Trinidad and Tobago, 27e30 April. Anonsen, S.I., Eide, A.L., Holden, L., Aasen, J.O., 1995. Optimizing Reservoir Performance under Uncertainty with Application to Well Location. Paper SPE 30710 presented at the Annual Technical Conference & Exhibition held in Dallas, U. S. A., 22e25 October. Armenta, M., Wojtanowicz, A.K., 2002. Severity of Water Coning in Gas Wells. Paper SPE 75720 presented at the SPE Gas Technology Symposium held in Calgary, Alberta, Canada, 30 Aprile2 May. Armenta, M., Wojtanowicz, A.K., 2005. Incremental recovery using dual-completed wells in gas reservoirs with bottom water drive: a feasibility study. J. Can. Pet. Technol. 44 (6). PETSOC-05-06-04. Armenta, M., White, C.D., Wojtanowicz, A.K., 2003. Completion length optimization in gas wells. In: Cdn. Intl. Pet. Conf. Calgary, Alberta, 10e12 June. Box, G.E.P., Behnken, D.W., 1960. Some new three level designs for the study of quantitative variables. Technometrics 2 (4), 455e475. Box, G.E.P., Wilson, K.B., 1951. On the experimental attainment of optimum conditions. J. R. Stat. Soc. Ser. B 13, 1e45. Brinkman, F.P., 1981. Increased gas recovery from a moderate water drive reservoir. J. Pet. Technol. 33 (12), 2475e2480. SPE-9473-PA. Chierici, G.L., Ciucci, G.M., Long, G., 1963. Experimental research on gas saturation behind the water front in gas reservoirs subjected to water drive. WPC-10134. In: Proceeding of the 6th World Petroleum Congress. Frankfurt am Main, Germany, 25 June. Chu, C., 1990. Prediction of Steamflood Performance in Heavy Oil Reservoirs Using Correlations Developed by Factorial Design Method. Paper SPE 20020 presented at the 60th California Regional Meeting held in Ventura, California, 4e6 April. Cohen, M.F., 1989. Recovery Optimization in a Bottom/Edge Water-Drive Gas Reservoir, Soehlingen Schneverdingen. Paper SPE 19068 presented at the SPE Gas Technology Symposium held in Dallas, TX, June 7e9. Damsleth, E., Hage, A., Volden, R., 1991. Maximum Information at Minimum Cost: a North Sea Field Development Study Using Experimental Design. Paper SPE 23139 presented at the Offshore Europe Conference held in Aberdeen, 3e6 September. Dejean, J.P., Blanc, G., 1999. Managing Uncertainties on Production Predictions Using Integrated Statistical Methods. Paper SPE 56696 presented at the Annual Technical Conference and Exhibition held in Houston, Texas, 3e6 October. Derringer, G., Suich, R., 1980. Simultaneous optimization of several response variables. J. Qual. Technol. 12, 214e219. Dranchuk, P.M., Purvis, R.A., Robison, D.B., 1974. Computer Calculations of Natural Gas Compressibility Factors Using the Standing and Katz Correlation. Institute of Petroleum Technical Series, No. IP 74-008. Dykstra, H., Parsons, R.L., 1950. The prediction of oil recovery by water flood. In: Secondary Recovery of Oil in the United States, second ed. API, pp. 160e174.
M. Naderi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 260e269 El-Ahmady, M.H., Wattenbarger, R.A., Pham, T.T., 2002. Overestimation of original gas in place in water-drive gas reservoirs due to a misleading linear p/z plot. J. Can. Pet. Technol. 41 (11). PETSOC-02-11-01. Friedmann, F., Chawathe, A., Larue, D.K., 2001. Assessing Uncertainty in Channelized Reservoirs Using Experimental Designs. Paper SPE 71622 presented at the Annual Technical Conference and Exhibition held in New Orleans, Louisiana, 30 Septembere3 October. Geffen, T.M., Parrish, D.R., Haynes, G.W., Morse, R.A., 1952. Efficiency of gas displacement from porous media by liquid flooding. J. Pet. Technol. 4 (2), 29e38. SPE-952029-G. Hallam, R., Maria, L.K., 2003. Performance of Trinidad Gas Reservoirs (Cassia, Immortelle, Flamboyant, Mahogany, Amherstia and Teak). Paper SPE-81010-MS presented at SPE Latin American and Caribbean Petroleum Engineering Conference, Port-of-Spain, Trinidad and Tobago, 27e30 April. Harrington Jr., E.C., 1965. The desirability function. J. Ind. Qual. Control 21 (10), 494e498. Hower, T.L., Jones, R.E., 1991. Predicting Recovery of Gas Reservoirs under Water Drive Conditions. Paper SPE 22937 presented at the SPE 66th Annual Conference and Exhibition held in Dallas, TX, 6e9 October. Knapp, R.M., Henderson, J.H., Dempsey, J.R., Coats, K.H., 1968. Calculation of gas recovery upon ultimate depletion of aquifer storage. J. Pet. Technol. 20 (10), 1129e1132. SPE-1815-PA. Lee, A.L., Gonzalez, M.H., Eakin, B.E., 1966. The viscosity of natural gases. J. Pet. Technol. 18 (8), 997e1000. SPE-1340-PA. Lee, M., Yang, W.J., Xiao, Q.Y., 2010. Determination of the Aquifer Activity Level and the Recovery of Water Drive Gas Reservoir. Paper SPE 127497 presented at the North Africa Technical Conference and Exhibition, Cairo, Egypt, 14e17 February. Lutes, J.L., Chiang, C.P., Rossen, R.H., Brady, M.M., 1977. Accelerated blowdown of a strong water-drive gas reservoir. SPE J. 29 (12), 1533e1538. SPE-6166-PA. McMullan, J.H., Bassiouni, Z., 2000. Optimization of Gas-Well Completion and Production Practices. Paper SPE-58983-MS presented at SPE International Petroleum Conference and Exhibition in Mexico, Villahermosa, Mexico, 1e3 February.
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Muladi, A., Pinczewski, W.V., 1999. Application of Horizontal Well in Heterogeneity Gas Condensate Reservoir. SPE 54351 presented at the Oil and Gas Conference and Exhibition, Jakarta, Indonesia, 20e22 April. Narayanan, K., White, C.D., Lake, L., Willis, B.J., 1999. Response Surface Methods for Upscaling Heterogeneous Geologic Models. Paper SPE 51923 presented at the Reservoir Simulation Symposium held in Houston, Texas, 14e17 February. Peake, W.T., Abadah, M., Skander, L., 2005. Uncertainty Assessment Using Experimental Design: Minagish Oolite Reservoir. Paper SPE 91820 presented at the Reservoir Simulation Symposium, Houston, Texas U.S.A., 31 Januarye2 February. Petalas, N., Aziz, K., 2000. A mechanistic model for multiphase flow in pipes. J. Can. Pet. Technol. 39 (6), 43e55. Rezaee, M., Rostami, B., Pourafshary, P., 2013. Heterogeneity effect on non-wetting phase trapping in strong water drive gas reservoirs. J. Nat. Gas Sci. Eng. 14, 185e191. Sech, R.P., Jackson, M.D., Gary, H., 2007. Controls on Water Cresting in High Productivity Horizontal Gas Wells. Paper SPE 107169-MS presented at the EUROPEC/EAGE Conference and Exhibition, London, U.K., 11e14 June. Sedaghatzadeh, M., Rezaee, M., Rostami, B., Mojarrad, M., 2013. Experimental determination of optimized production rate and its upscaling analysis in strong water drive gas reservoirs. IPTC-16938. In: International Petroleum Technology Conference. Beijing, China, 26e28 March. Van Elk, J.F., Guerrera, L., Gupta, R., 2000. Improved Uncertainty Management in Field Development Studies through the Application of the Experimental Design Method to the Multiple Realizations Approach. Paper SPE 64462 presented at the Annual Technical Conference and Exhibition held in Dallas, Texas, 1e4 October. Wang, X., 2009. J. Nat. Gas Sci. Eng. 1 (1e2), 14e24. Wang, X., 2012. Key issues in high-performance natural gas wells. J. Nat. Gas Sci. Eng. 6, 37e42. White, C.D., Willis, B.J., Narayanan, K., Dutton, S.P., 2000. Identifying Controls on Reservoir Behavior Using Designed Simulations. Paper SPE 62791 presented at the SPE Annual Technical Conference and Exhibition held in Dallas, Texas, 1e4 October.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
Optimizing production from water drive gas reservoirs based on desirability concept Meysam Naderi a, Behzad Rostami a, *, Maryam Khosravi a, b a b
Institute of Petroleum Engineering, School of Chemical Engineering, College of Engineering, University of Tehran, Tehran, Iran IOR Research Institute, National Iranian Oil Company, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 June 2014 Received in revised form 6 August 2014 Accepted 10 August 2014 Available online
There are various factors which determine the optimization and economic production from water drive gas reservoirs. These factors play an important role in designing an effective reservoir development plan. The present study, in the first step, investigates the relation between recovery factor, volumetric sweep efficiency and cumulative water production with six different engineering and geologic factors using design of experiments (DOE) and response surface methodology (RSM). Next, all derived response functions are optimized simultaneously based on the concept of desirability. In this manner, part of water drive gas reservoirs is simulated using BoxeBehnken design. Important factors that have been studied include reservoir horizontal permeability (Kh), permeability anisotropy (Kv/Kh), aquifer size (Vaq), gas production rate (Qg), perforated thickness (Hp) and tubing head pressure (THP). The results indicate that by combining various levels of factors and considering relative importance of each response function, optimized conditions could be raised in order to maximizing recovery factor, volumetric sweep efficiency and minimizing cumulative water production. Also high rates of gas production result poor volumetric sweep efficiency and early water breakthrough, hence ultimate recovery factor decreases by 3.2e8.4%. © 2014 Elsevier B.V. All rights reserved.
Keywords: Optimization Design of experiments Response surface methodology Desirability Recovery factor Volumetric sweep efficiency Cumulative water production
1. Introduction Prediction of gas production is an important part of reservoir development, management and economic evaluation. Today, with the increasing growth of need to use of fossil fuels and the high volume of trapped gas in reservoir, it is crucial to optimize production from water drive gas reservoirs. Without investigating and understanding the production sensitivities to parameters such as gas production rate, tubing head pressure, perforated length from these reservoirs, it is not possible to reach predetermined goals to increase profitability and reduce costs. Considering that a large part of gas reserves are recovered using the water drive process, it is very important to understand the mechanisms affecting the production from these types of reservoirs. The experimental study of Geffen et al. (1952) on core plugs revealed that the trapped gas saturation varied from 15 to 50 percent of the pore space for various porous media. Agarwal (1965) demonstrated that the ultimate recovery factor is a function of production rate, residual gas saturation, aquifer permeability and volumetric sweep efficiency. Gas recovery factor increases with increasing production rate and
* Corresponding author. Tel.: þ98 9125210473; fax: þ98 2188632976. E-mail addresses: [email protected], [email protected] (B. Rostami). http://dx.doi.org/10.1016/j.jngse.2014.08.007 1875-5100/© 2014 Elsevier B.V. All rights reserved.
decreasing aquifer permeability. Knapp et al. (1968) developed a two-phase two-dimensional model for predicting the gas recovery from aquifer storage fields as a function of production rate, aquifer strength and reservoir heterogeneity. Lutes et al. (1977) reported that the final blowdown of a Gulf Coast water drive gas reservoir at a reserves/production ratio of less than 2 provided an increase in gas recovery. Brinkman (1981) reported that accelerated gas withdrawals of up to 115 MMSCF/D from a U.S. gulf coast water drive gas reservoir resulted in a 20% increase in remaining gas recovery versus continued low-rate depletion. Al-Hashim (1998) studied the effect of aquifer size on partial water drive gas reservoirs. They concluded that if the ratio of aquifer external radius to reservoir external radius be less than two, the effect of aquifer on performance of the gas reservoir can be neglected. For ratios greater than two, gas recovery is sensitive to both initial reservoir pressure and aquifer size. Increasing aquifer size and initial reservoir pressure reduces gas recovery. A simulation study by Cohen (1989) determined that accelerating production rate and coproduction increases recovery by 2.3% and 5.6% respectively. A reservoir simulation study by Hower and Jones (1991) showed that to increase recovery, the production rate should be lowered rather than accelerated because of improved volumetric sweep efficiency. El-Ahmady et al. (2002) studied the effect of aquifer on estimation
M. Naderi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 260e269
of the original gas in place. They showed that the unsteady state nature of aquifers can lead to overestimation of the original gas in place. Armenta and Wojtanowicz (2002), Armenta et al. (2003), Armenta and Wojtanowicz (2005) studied the effect of well completion on performance of gas wells. Sech et al. (2007) simulated the effect of production rate on recovery factor in horizontal wells. They found that recovery reduces by increasing the production rate and permeability anisotropy due to water cresting. Wang (2009) studied the impact of turbulence and the importance of hydraulic fracturing on well deliverability for both vertical and horizontal gas wells. Lee et al. (2010) presented a correlation for predicting recovery factor changes with aquifer size, ratio of residual to initial gas saturation, ultimate volumetric sweep efficiency, abandonment and average reservoir pressure. Wang (2012) studied the tubing limitation and turbulence effects on well deliverability of both vertical and horizontal wells with and without artificially induced hydraulic fractures. Sedaghatzadeh et al. (2013) investigated the optimum accelerating production rate from water drive gas reservoirs in laboratory scale systems. Rezaee et al. (2013) studied the effect of heterogeneity and aquifer to gas zone permeability on the gas phase trapping in water drive gas reservoir. Results of their study in laboratory scale show that heterogeneity is not always detrimental to gas recovery and it may be improved with increasing heterogeneity when ratio of aquifer to gas zone permeability is less than one. Review of previous studies shows that the simultaneous optimization of recovery factor, volumetric sweep efficiency and cumulative water production from water drive gas reservoirs has never been investigated. This objective is studied by applying design of experiments, response surface methodology and desirability concept. Experimental design and response surface methodology has been used in petroleum engineering applications including performance prediction (Chu, 1990), uncertainty modelling (Damsleth et al., 1991; Van Elk et al., 2000; Friedmann et al., 2001), sensitivity studies (White et al., 2000), upscaling (Narayanan et al., 1999), history matching (Anonsen et al., 1995) and development optimization (Dejean and Blanc, 1999). Also multiple response optimization based on desirability concept is widely used in numerous engineering fields (Harrington, 1965; Derringer and Suich, 1980). This study is one of the earliest ones taking advantages of design of experiments, response surface methodology and desirability concept for optimizing production from water drive gas reservoirs. Production optimization will be based on maximizing recovery factor and volumetric sweep efficiency and also minimizing cumulative water production.
261
Fig. 1. Simulated model. The reservoir is shown in red with 3300 ft external radius, and the aquifer in blue with 6600 ft. The reservoir has 100 layers with constant thickness of 300 ft and the aquifer with 20 layers of variable thickness. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
was set to 25%. The reservoir temperature and initial reservoir pressure at datum depth of 8530 feet were fixed to 220 F and 5290 psi, respectively. The gas viscosity was estimated using the correlation developed by Lee et al. (1966). The gas deviation factor was estimated using correlations presented by Dranchuk et al. (1974). Relative permeability and capillary pressure data measured for gasewater systems in the experimental work of Chierici et al. (1963) has been used in this survey. Production is via a single vertical well with 7 inch internal e tubing diameter. Reservoir horizontal permeability, permeability anisotropy, aquifer size, gas production rate, perforated thickness and tubing head pressure are varied simultaneously in simulation tests that are described in the following section. In this study, the well produces at constant tubing head pressure until production rate is greater than 10% of maximum initial gas production rate and bottom hole pressure is more than 500 psi. The Petalas and Aziz (2000) mechanistic model has been used to calculate vertical flow performance curves. This model is very accurate and it can be used for up and downhill flow, and for all pipe geometries.
2. Model description 3. Methodology The model described here simulates gas reservoirs with an active bottom aquifer using a radial system (Fig. 1). The reservoir zone measures 3300 feet in the radial direction and has a maximum gas column thickness of 300 feet. A bottom water aquifer is connected to the base of reservoir. The aquifer measures 6600 feet in the radial direction, extending beyond the reservoir. The aquifer thickness is not constant and changes in order to modify aquifer size during simulation. Flow is simulated using a gridding scheme that is locally refined around the well, and coarsened away from the well in the radial direction. The width of cells increases exponentially and highest resolution cells are located close to the well. Grid layering in the vertical direction is considered 120 layers, 100 layers with thickness of 300 feet for reservoir and 20 layers for aquifer section. The first 10 layers of the aquifer has constant thickness of 100 feet and the second 10 layers has a variable thickness to be able to set the ratio of aquifer to reservoir volume namely aquifer size. Reservoir and aquifer porosity is constant in all simulation and it
3.1. Determination of the response functions The following study, inspect the effect of six engineering and geologic factors on gas recovery factor, volumetric sweep efficiency and cumulative water production. In this regard, design of experiment and response surface methodology was employed. Design of experiments (DOE) is a well known technique to get maximum information with simultaneous varying of all parameters and required less number of performing time consuming numerical tests. Response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables to obtain an optimal response. More details about design of experiments and response surface methodology are given in Box and Wilson (1951). Based on BoxeBehnken Design (1960), it is required to design and simulate 49 different models to extract necessary information such as reservoir pressure,
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pressure of the aquifer, and etc in order to be able to assess and quantify the impact of six factors on mentioned response functions. BoxeBehnken Design has been used because of the increased accuracy of the model by applying the quadratic terms of each factor and also reduced simulation runtime thereby reducing costs. After determining the response functions via response surface methodology, in the next step, it will be possible to optimize all functions simultaneously based on the desirability concept. Range of factors with their appropriate transfer functions are given in Table 1. In RSM, transfer functions are used to normalize range of data between 1 and þ1 and also ease of computation. In this table, X1, X2, X3, X4, X5 and X6 are coded variables between 1 and þ1 for Kh, Kv/Kh, Vaq, Qg, Hp and THP respectively. For example, range of horizontal permeability (Kh) from (10, 100, 1000) can be normalized to (1, 0, þ1) by using transfer function of Log (Kh) 2. All 49 different flow simulations were undertaken using the Eclipse® 100 black oil simulator and the following equations developed by RSM for gas recovery factor (RF), volumetric sweep efficiency (Ev) and cumulative water production (Wp).
RF ¼ 78:62 þ 8:33X1 1:8X2 þ 0:78X3 þ 0:5X4 2:1X5 10:07X6 5:68X12 2:22X52 þ 4:33X1 X2 þ 2:62X1 X3 þ 2:71X1 X5 þ 2:64X4 X5 (1) Ln Ev ¼ 1:83 þ 0:16X1 þ 0:02X2 þ 1:79X3 0:093X4 0:063X5 0:12X6 þ
0:09X32
þ
0:162X62
þ 0:063X1 X3
þ 0:241X2 X3 þ 0:123X2 X4 0:254X3 X4 0:024X3 X6 (2) Ln Wp ¼ 8:34 2:4X1 þ 0:78X2 þ 2:58X3 þ 0:75X4 þ 2:18X5 0:87X6 þ 0:8X22 þ 1:99X32 þ 1:14X52 þ 0:96X62 1:37X1 X2 þ 1:51X1 X3 þ 1:28X2 X3 þ 1:32X2 X4 X2 X6 1:35X3 X4 X3 X5 (3) where X1, X2, X3, X4, X5 and X6 are coded variables for mentioned factors in Table 1. The coefficient of determination (R2) and adjusted coefficient of determination for derived response functions are given in Table 2. The validity of three proxy equations is shown in Fig. 2 with a plot of predicted versus actual experiments. R2 and Adjusted R2 were used to check the quality of the given equations for prediction. The obtained R2 for three response functions shows that more than 89% of the variation in each response could be as a result of the selected factors. The derived proxies were also validated by 12 new simulation tests with the properties outside the group used to generate the proxies. Table 3 show the result of these
Table 1 Range of factors with transfer functions. Factor
Coded variable
Levels
Transfer function
Kh (md) Kv/Kh (fraction)
X1 X2
10 0.01
Vaq (fraction) Qg (MMSCF/D)
X3 X4
1 60
Hp (%)
X5
30
60
90
THP (psi)
X6
500
1000
1500
100 0.1
1000 1
LogðKh Þ 2 Log KKhv þ 1
10 90
100 120
LogðVaq Þ 1 Qg 90 30 Hp 60 30 THP1000 500
Table 2 Response functions statistics. Response
R2
Adjusted R2
RF Ln (Ev) Ln (Wp)
89.4 96.7 92.3
85.8 95.5 88.1
blind tests. The T-test for significance of all coefficients shows that all terms taken in the above models are significant at confidence level of 10% for prediction at 90% confidence interval. These equations were used to study the main effect, interaction and square power effect of parameters on the performance of water drive gas reservoirs. 3.2. Heterogeneity effect on productivity of water drive gas reservoirs In this section, the heterogeneity effect on gas recovery from water drive gas reservoirs was investigated by using DykstraeParsons coefficient (Dykstra and Parsons, 1950). The DykstraeParsons permeability variation is defined by the following expression:
V¼
K50% K84:1% K50%
where K50% and K84.1% are corresponding permeability values at 84.1% and 50% of cumulative sample thickness respectively. Variation of this coefficient is between zero and one. The zero value indicates the ideal homogeneous porous media. While it changes over a range of zero to half, experiments can be approximated by a homogeneous model in reservoir simulation with minimal error. If the coefficient varies over a range of half to one, the reservoir is very heterogeneous. We studied the effect of heterogeneity on water drive gas reservoirs performance by considering DykstraeParsons coefficients from 0.1 to 0.9. In this regard, eighteen heterogeneous models with six different DykstraeParsons coefficients and three different permeability anisotropy were simulated. Also six homogeneous models with the same properties and equivalent geometric mean of reservoir horizontal permeability (KG) were simulated. The result of these 24 simulations is given in Table 4. General regression Analysis of data in Table 4 for recovery factor shows that the DykstraeParson Coefficient is not significant at confidence level of 10%. P-value of DykstraeParson Coefficient in the final regression for recovery factor is 15.6%. This reveals that heterogeneity effect in terms of DykstraeParson Coefficient plays a weak role, but permeability anisotropy not. Also sensitivity analysis of recovery factor was done by considering log-normal distribution for permeability anisotropy and geometric mean of reservoir horizontal permeability and triangular distribution for DykstraeParson Coefficient. Results indicated that although recovery factor increases by increasing the level of heterogeneity, this effect in terms of contribution to total variance is less than 5% regardless of confidence level. This fact that heterogeneity in terms of DykstraeParsons Coefficient has little impact on performance of gas reservoirs has been reported already by Muladi and Pinczewski (1999). Therefore a homogeneous model with equivalent geometric mean of reservoir horizontal permeability can be used instead of heterogeneous one with the minimal error. 3.3. Optimization based on desirability concept The desirability function method has been used for many years in industry to optimize of multiple response functions. It is based
M. Naderi et al. / Journal of Natural Gas Science and Engineering 21 (2014) 260e269
263
Pk w 1 i D ¼ d1 ðY1 Þw1 d2 ðY2 Þw2 / dk ðYk Þwk 1
(4)
where k and w denoting the number of responses and relative importance of each response respectively. The class of desirability functions that was proposed by Derringer and Suich (1980) is divided into three types, namely Nominal The Best (NTB), Larger The Better (LTB) and Smaller The Better (STB). Individual desirability function based on the optimization goals for types of NTB, LTB and STB are given in the following equations:
8 0 > > > > > > Y Li s i > > > < Ti Li DNTBi ðYi Þ ¼ > > Yi Ui t > > > > T Ui > > > i : 0
DLTBi ðYi Þ ¼
DSTBi ðYi Þ ¼
8 > > > > <
0
Yi Li > U > i Li > > : 1 8 > > > > <
s
1
Yi Ui > Li Ui > > > : 0
9 > > > > > > > if Li Yi ðxÞ Ti > > = if Yi ðxÞ < Li
> > > if Ti Yi ðxÞ Ui > > > > > > ; if Yi ðxÞ > Ui if Yi ðxÞ < Li if Li Yi ðxÞ Ui if Yi ðxÞ > Ui
s
if Yi ðxÞ < Li if Li Yi ðxÞ Ui if Yi ðxÞ > Ui
(5)
9 > > > > = > > > > ;
(6)
9 > > > > = > > > > ;
(7)
where Li, Ui and Ti are the lower, upper, and target values of response functions respectively, that are desired for response Yi (x), with Li < Ti < Ui. The exponents s and t determining how important it is to hit the target value. For s ¼ t ¼ 1, the desirability function increases linearly towards Ti; for s < 1, t < 1, the function is convex, and for s > 1, t > 1, the function is concave. Increasing exponents of desirability function, make the condition more difficult to reach an optimized solution. Optimization based on desirability concept consists of the following three steps: 1) Implement flow simulation based on DOE and determine all response functions by RSM. 2) Define individual desirability functions for each response. 3) Maximize the overall desirability D with respect to the controllable factors. 4. Results and discussion
Fig. 2. Predicted versus actual experiments for A) recovery factor, B) volumetric sweep efficiency and C) cumulative water production.
on the idea that performance of a product is generally characterized by many response variables. In many situations, these responses or quality characteristics are controlled by a set of independent factors. The desirability method finds operating conditions that provide simultaneously the most desirable response values. For each response Yi (x), a desirability function di (Yi) assigns numbers between 0 and 1 to the possible values of Yi (x), with di (Yi) ¼ 0 representing a completely undesirable value of Yi and di (Yi) ¼ 1 representing a completely desirable or ideal response value. The individual desirability's are then combined to give overall desirability (D) using the weighted geometric mean:
Equations (1)e(3) generally consist of three groups of expressions, the main factors, the interactions between factors and the square power of some factors. The constant in the first of equations implies that when factors are in their middle values, the mean value of the response is equal to that constant. The terms with negative coefficient cause reduction of response that is useful in reducing cumulative water production, while terms with positive sign tend to increase the recovery factor and volumetric sweep efficiency. The main effect of each factor is shown in Fig. 3. These factors have different effects on the three response functions trend. As indicated in the main effect plot for recovery factor, reservoir permeability and tubing head pressure have the most effect on recovery variation. This variation is very sharp for reservoir permeability and tubing head pressure of less than 100 md and 1000 psi respectively. After reservoir permeability and tubing head pressure;
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Table 3 Properties and results of 12 new blind simulation tests. Run#
X1
X2
X3
X4
X5
X6
Actual RF
Predicted RF
Actual Ln (Ev)
Predicted Ln (Ev)
Actual Ln (Wp)
Predicted Ln (Wp)
1 2 3 4 5 6 7 8 9 10 11 12
0.76 0.03 0.67 0.49 0.73 0.61 0.05 0.73 0.66 0.22 0.61 0.05
0.47 0.97 0.38 0.99 0.78 0.49 0.22 0.57 0.27 0.49 0.40 0.96
0.84 0.67 0.17 0.87 0.53 0.48 0.47 0.91 0.42 0.53 0.43 0.98
0.45 0.00 0.61 0.12 0.22 0.28 0.13 0.87 0.88 0.94 0.62 0.77
0.44 0.74 0.60 0.18 0.14 0.55 0.94 0.88 0.54 0.21 0.06 0.81
0.04 0.23 0.93 0.84 0.78 0.37 0.05 0.02 0.67 0.80 0.11 0.44
63.45 75.08 75.21 85.47 89.47 82.77 74.09 78.33 71.04 70.75 67.34 75.19
67.22 74.84 79.46 83.78 91.34 83.75 75.35 74.84 75.26 67.76 70.17 74.71
0.74 1.15 2.35 1.06 3.29 1.44 2.95 3.86 1.39 3.06 2.77 3.96
0.34 0.75 2.28 0.67 3.30 1.22 2.63 3.65 1.17 2.95 2.48 4.12
13.55 9.28 14.50 9.80 7.02 6.96 14.73 10.32 5.65 10.69 14.87 14.15
10.84 11.20 12.91 11.09 11.35 5.13 13.07 12.51 8.37 11.21 12.57 13.96
Table 4 The simulation result of 18 heterogeneous and 6 homogeneous models. Run
Kv/Kh
V
KG
(Kv/Kh)2
V2
(KG)2
Kv/Kh*V
Kv/Kh*KG
V*KG
Kv/Kh*V*KG
RF
Ev
Ln Wp
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.1 0.01 0.1 1 0.1 0.01 0.1 1 0.1 0.01 0.1 1 0.1 0.01 0.1 1 0.1 0.01 0.1 1 0.1 0.01 0.1 1
0 0.1 0.1 0.1 0 0.3 0.3 0.3 0 0.5 0.5 0.5 0 0.65 0.65 0.65 0 0.8 0.8 0.8 0 0.9 0.9 0.9
9.92 9.92 9.92 9.92 9.5 9.5 9.5 9.5 88.71 88.71 88.71 88.71 86.9 86.9 86.9 86.9 56.97 56.97 56.97 56.97 27 27 27 27
0.01 0.0001 0.01 1 0.01 0.0001 0.01 1 0.01 0.0001 0.01 1 0.01 0.0001 0.01 1 0.01 0.0001 0.01 1 0.01 0.0001 0.01 1
0 0.01 0.01 0.01 0 0.0841 0.0841 0.0841 0 0.2704 0.2704 0.2704 0 0.4225 0.4225 0.4225 0 0.64 0.64 0.64 0 0.7921 0.7921 0.7921
98.406 98.406 98.406 98.406 90.25 90.25 90.25 90.25 7869.5 7869.5 7869.5 7869.5 7551.6 7551.6 7551.6 7551.6 3245.6 3245.6 3245.6 3245.6 729 729 729 729
0 0.001 0.01 0.1 0 0.0029 0.029 0.29 0 0.0052 0.052 0.52 0 0.0065 0.065 0.65 0 0.008 0.08 0.8 0 0.0089 0.089 0.89
0.992 0.0992 0.992 9.92 0.95 0.095 0.95 9.5 8.871 0.8871 8.871 88.71 8.69 0.869 8.69 86.9 5.697 0.5697 5.697 56.97 2.7 0.27 2.7 27
0 0.992 0.992 0.992 0 2.755 2.755 2.755 0 46.1292 46.1292 46.1292 0 56.485 56.485 56.485 0 45.576 45.576 45.576 0 24.03 24.03 24.03
0 0.00992 0.0992 0.992 0 0.02755 0.2755 2.755 0 0.461292 4.61292 46.1292 0 0.56485 5.6485 56.485 0 0.45576 4.5576 45.576 0 0.2403 2.403 24.03
65.13 76.04 65.37 55.88 64.53 75.66 64.18 53.72 79.82 81.11 80.42 78.28 79.74 80.79 79.82 77.4 78.04 79.93 79.16 76.86 73.48 78.6 76.82 76.08
42.47 36.46 42.54 35.74 42.04 36.69 42.15 34.1 57.1 55.78 57.81 55.04 57.03 55.47 57.16 54.07 54.85 50.54 56.35 53.51 51.15 46.22 51.96 53.67
14.55 9.47 14.53 16.27 14.51 9.55 14.55 16.30 14.91 13.62 14.72 15.49 14.91 13.71 14.83 15.62 14.97 13.14 14.93 15.67 14.91 9.94 13.90 14.97
permeability anisotropy, production rate, aquifer size and perforated interval are the most effective factors respectively. Aquifer size has the highest effect on volumetric sweep efficiency and cumulative water production. For aquifer sizes greater than 10, volumetric sweep efficiency and cumulative water production increases sharply. As revealed in Fig. 4, the interaction plot for recovery factor; increasing tubing head pressure reduces recovery factor regardless of interaction with other factors. Production rate has different interaction effect with different factors. Increasing length of perforated interval has negative effect on recovery factor when reservoir permeability is low. Among all interaction terms, interaction between perforated interval and production rate is very important because of its economical effect on production and development of water drive gas reservoirs. Based on the perforated interval length, the recovery factor can be maximized or minimized with increasing production rate. Therefore, this important interaction effect will be explained more after optimization section. In order to determine the optimum condition to produce from water drive gas reservoirs in efficient manner, it is required to consider the relative importance of each response function simultaneously. This objective can be well developed by the application of the desirability concept. In the first step of simultaneous optimization of all response functions, the lower (Li) and upper (Ui) bounds of desirability equations (6) and (7) should be determined.
Values of Li and Ui for individual response function are calculated by considering the probability distribution functions for the independent variables and performing Monte Carlo simulation. In this regard, based on previous studies and available information (Peake et al., 2005), we assigned log-normal distribution for the three factors: reservoir horizontal permeability, permeability anisotropy, aquifer size and triangular distribution for gas production rate, perforated thickness, and tubing head pressure. Table 5 shows the values of Li and Ui for all three response functions. We optimized all three functions simultaneously by considering three values for exponent of desirability function (s in equations (6) and (7)) and four values of relative importance (wi in equation (4)) for recovery factor. Optimization results are given in Tables 6 through 8 for s ¼ 0.1, 1 and 10 respectively. In the referred tables, when relative importance of recovery factor (wRF) is equal to 100%, it means the recovery factor will be optimized and the other two responses are not important. In this research, the relative importance of volumetric sweep efficiency and cumulative water production are assumed equal. For instance, in Table 6, when relative importance of recovery factor is 75%, the relative importance for each of the volumetric sweep efficiency and cumulative water production will be 12.5%. It is required to assign relative importance for individual responses to optimize all three functions simultaneously based on overall desirability (D). As indicated in Tables 6 through 8, overall
Fig. 3. Main effects plot for A) recovery factor, B) volumetric sweep efficiency and C) cumulative water production. A main effect is present when different levels of a factor affect the response differently. The steeper the slope of the line, the greater the magnitude of the main effect.
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Fig. 4. Interaction plot for recovery factor. An interaction is present when the effect of one factor depends on the level of the other factor. Parallel lines in an interaction plot indicate no interaction. The greater the difference in slope between the lines, the higher the degree of interaction.
factors, optimum criteria are different. Regardless of the desirability function weight and relative importance of functions, this study shows that reducing the gas production rate from water drive gas reservoirs can optimize the recovery factor, volumetric sweep efficiency and cumulative water production. Variation of recovery factor with normalized aquifer size and reservoir horizontal permeability for different values of Kv/Kh is given in Fig. 5. While reservoir horizontal permeability is less than 70.8 md, recovery factor decreases by increasing aquifer size because of early breakthrough time and increasing water production. However, recovery factor increases by increasing aquifer size (<14%) for horizontal permeability greater than 70.8 md due to decreased rate of reservoir pressure decline. This interaction between reservoir and aquifer allows further gas to be produced prior to abandonment at the bottom hole pressure limit. Although this behaviour is not
Table 5 Values of Li and Ui for recovery factor, volumetric sweep efficiency and cumulative water production. Response function
Li
Ui
Optimization type
RF (%) Ln (Ev) Ln (Wp)
46 0.23 0.6
93 4.17 19
Maximum Maximum Minimum
desirability decreases with increasing exponent of desirability function. This means that by increasing value of s, it will be more difficult to reach optimum condition. However; increased accuracy of the simultaneous optimization solution compensates this reduction of overall desirability. The results of the simultaneous optimization of response functions (Tables 6 through 8) show that by combining various levels of
Table 6 Optimization with s ¼ 0.1 for desirability function of the STB and LTB type. Run#
wRF
X1
X2
X3
X4
X5
X6
RF
Ln (Ev)
Ln (Wp)
D
1 2 3 4
100% 75% 50% 25%
1 1 1 1
1 1 0.374 0.758
1 0.515 0.333 0.394
1 1 1 1
0.737 0.717 0.859 0.778
1 0.758 1 0.616
98.96 93 93 90.47
e 3.453 3.132 3.112
e 9.416 8.231 7.694
1 0.99 0.98 0.97
Table 7 Optimization with s ¼ 1 for desirability function of the STB and LTB type. Run#
wRF
X1
X2
X3
X4
X5
X6
RF
Ln (Ev)
Ln (Wp)
D
1 2 3 4
100% 75% 50% 25%
1 1 1 1
1 0.657 0.256 0.758
1 0.273 0.377 0.394
1 1 1 1
0.737 0.939 0.778 0.778
1 0.964 1 0.616
98.96 93 93 90.47
e 2.972 3.228 3.112
e 7.709 8.573 7.694
1 0.91 0.82 0.74
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267
Table 8 Optimization with s ¼ 10 for desirability function of the STB and LTB type. Run#
wRF
X1
X2
X3
X4
X5
X6
RF
Ln (Ev)
Ln (Wp)
D
1 2 3 4
100% 75% 50% 25%
1 0.939 1 1
1 0.313 0.879 0.758
1 0.293 0.414 0.394
1 1 1 1
0.737 0.798 0.697 0.778
1 1 0.819 0.616
98.96 93 92.98 90.47
e 3.027 3.228 3.112
e 8.051 8.573 7.694
1 0.359 0.132 0.05
Fig. 5. Contour plot for recovery factor versus normalized values of aquifer size (Vaq) and reservoir horizontal permeability (Kres) for A) Kv/Kh ¼ 0.01, B) Kv/Kh ¼ 0.1 and C) Kv/Kh ¼ 1.
Fig. 6. Contour plot for recovery factor versus normalized values of gas production rate (Qg) and perforated thickness (Hp) for A) Kv/Kh ¼ 0.01, B) Kv/Kh ¼ 0.1 and C) Kv/Kh ¼ 1.
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predicted by material balance equation, it has been observed in Columbus Basin gas fields and also by simulation study for the Mahogany 20 Gas Sand (McMullan and Bassiouni, 2000; Hallam and Maria, 2003; Ali-Nandalal and Gunter, 2003). Contour plot of recovery factor versus normalized production rate and perforated interval for different values of Kv/Kh is shown in Fig. 6. In this figure, gas recovery factor increases with increasing production rate for perforated interval values greater than 60% of total pay zone thickness. In other words, when perforated interval is less than 60% of the total gas column thickness, recovery decreases with high values of production rate due to decreased gasewater ratio. Therefore, the well will be shut in because of the economical limit of gasewater ratio. Considering relative importance of volumetric sweep efficiency and the cumulative water production, recovery factors greater than 80% can be obtained for perforation length of less than 44% and production rate between 60 and 70 MMSCF/D. In other words, although accelerating production decreases ultimate recovery factor by 3.2e8.4%, it may be economically favourable over much shorter timescales. Results of the optimization reveal that for tubing head pressures between 500 psi to 1000 psi, all three responses will be optimized.
Kh reservoir horizontal permeability KG geometric mean of reservoir horizontal permeability Kv/Kh permeability anisotropy Li lower bound for the response function MMSCF/D million standard cubic feet per day Qg gas production rate RF gas recovery factor S&t weight of the desirability function THP tubing head pressure Ti target value for the response function Ui upper bound for the response function Vaq aquifer size V DykstraeParsons coefficient wi relative importance of response function Wp cumulative water production X1 coded variable for reservoir horizontal permeability X2 coded variable for permeability anisotropy X3 coded variable for aquifer size X4 coded variable for gas production rate X5 coded variable for perforated thickness X6 coded variable for tubing head pressure Yi (X) value of response function at X
5. Conclusions References In the present study, optimization of production rate from water drive gas reservoirs based on the concept of desirability has been investigated. Part of reservoir and aquifer with properties described in Section 2 and Table 1 was simulated and the effect of six factors on the recovery factor, volumetric sweep efficiency and cumulative water production was established by the experimental design techniques and response surface methodology. It is very important to note that the results and conclusions brought as an insight of this study are just valid under the specific model and assumed properties and its ranges. In addition to review the most important mechanisms and effective parameters, we aimed to introduce the useful methodology for analysis of the water drive gas reservoirs. The optimum conditions calculated using desirability functions and the main conclusions are as follow: 1. Accelerating production decreases ultimate recovery factor because of poor volumetric sweep efficiency and early water breakthrough. 2. Considering economic strategies, although accelerating production decreases ultimate recovery factor by 3.2e8.4%, it may be economically favourable over much shorter timescales. 3. Increasing production rate from 60 to 120 MMSCF/D increases gas recovery factor at least by 8% for perforated intervals greater than 60% of the gas column thickness. 4. Recovery factors greater than 80% can be obtained with production rates between 60 and 70 MMSCFD for perforated intervals less than 44% of gas column thickness. 5. Recovery factor increases by increasing aquifer size (<14%) for reservoir permeability greater than 70.8 md because of slow rate of reservoir pressure decline and increased amount of produced gas prior to abandonment at the bottom hole pressure limit. 6. Decreasing tubing head pressure below 1000 psi optimizes the production from water drive gas reservoirs. Nomenclature Di (Yi (X))desirability value for Yi (X) Ev volumetric sweep efficiency Hp perforated thickness k number of response functions
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