Assuming that the well s productivity index is constant, the oil flow rate can be calculated as follows:

Transcription

1 SPE 404 New Inflow Performance Relationship for Solution-Gas Drive Oil Reservoirs Mohamed Elias, SPE, H. Ahmed El-Banbi, SPE, K.A. Fattah, and El-Sayed Ahmed Mohamed El-Tayeb, SPE, Cairo University Copyright 009, Society of Petroleum Engineers This paper was prepared for presentation at the 009 SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana, USA, 4 7 October 009. 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 have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract The Inflow Performance Relationship (IPR) describes the behavior of the well s flowing pressure and production rate, which is an important tool in understanding the reservoir/well behavior and quantifying the production rate. The IPR is often required for designing well completion, optimizing well production, nodal analysis calculations, and designing artificial lift. Different IPR correlations exist today in the petroleum industry with the most commonly used models are that of Vogel s and Fetkovitch s. In addition to few analytical correlations, that usually suffers from limited applicability. In this work, a new model to predict the IPR curve was developed, using a new correlation that accurately describes the behavior the oil mobility as a function of the average reservoir pressure. This new correlation was obtained using 47 actual field cases in addition to several simulated tests. After the development of the new model, its validity was tested by comparing its accuracy with that of the most common IPR models such as Vogel, Fetkovitch, Wiggins, and Sukarno models. Twelve field cases were used for this comparison. The results of this comparison showed that: the new developed model gave the best accuracy with an average absolute error of 6.6 %, while the other common models are ranked, according to their accuracy in the following order to be Fetkovich, Sukarno, Vogel, and Wiggins, with average absolute errors of 7 %,. %, 3.7 %, and 5.7 respectively. The new developed IPR model is simple in application, covers wide range of reservoir parameters, and requires only one test point. Therefore, it provides a considerable advantage compared to the multipoint test method of Fetkovich. Moreover, due to its acuuracy and simplicity, the new IPR provides a considerable advantage compared to the widely used method of Vogel. Introduction For slightly compressible fluids, the productivity index is given by: kh k J = ro r + e r w S o B... () ln( / ).75 μ o Therefore, the variables that affecting the productivity index and in turn the inflow performance are essentially those that are pressure dependent parameters (µ o, B o, and k ro ). Fig. schematically illustrates the behavior of those variables as a function of reservoir pressure. Above the bubble-point pressure, kro equals unity and the term (k ro /µ o B o ) is almost constant. As the pressure declines below p b, the gas is released from solution which can cause a large decrease in both k ro and (k ro /µ o B o ). Assuming that the well s productivity index is constant, the oil flow rate can be calculated as follows: qo ( ) = J p r p wf. () Eq. suggests that the inflow into a well is directly proportional to the pressure drawdown. Evinger and Muskat (949) observed that when the pressure drops below the bubble-point pressure, the inflow performance curves deviates from that of the simple straight-line relationship as shown in Fig., therefore, the above relationship is not valid for two-phase flow or in case of solution gas drive reservoires.

2 SPE 404 Fig.- Effect of p r on B o, µ o, and k ro (After Ahmed, T. ) Many IPR correlations addressed the curvature in Fig. of the inflow performance curves in case of solution gas drive oil reservoirs in which the bubble point pressure is the initial reservoir pressure. Based on the literature survey, the most known IPR correlations can be subdivided into empirically derived and analytically derived correlations. Some of the most known empirical derived correlations are Vogel 3 (968), Fetkovich 4 (973), Kilns and Majcher 5 (99), Wiggins 6 (993), and Sukarno et al. 7 (995). Some of the most known analytical derived correlations are Wiggins et al. 8, 9 (99, 99), and Del Castillo, Yanil et al. 0, (003). The Empirical derived Correlations Fig.- The inflow performance curve below the bubble- point pressure (After Ahmed, T. ) Vogel's Method In 968, Vogel 3 used a computer program based on Weller s assumptions in 966 for solution gas drive reservoirs to predict inflow performance curves. Vogel used twenty-one reservoir data sets to develop the following IPR: q p o wf = 0. q o,max p r p wf 0.8 p r... (3) Vogel s correlation gave a good match with the actual well inflow performance at early stages of production but deviates at later stages of the reservoir life. Therefore, this will affect the prediction of inflow performance curves in case of solution gas drive reservoirs, because at later stages of production the amount of the free gas that comes out of the oil will be greater than the amount at the early stages of production. Fetkovich's Method In 973, Fetkovich 4 developed an IPR based on the data from multi-rate tests forty different oil wells from six fields. In addition, the general treatment of the inflow performance for the solution gas drive provided by Raghavan 3 under pseudo-steady state conditions was used. The following relation gives the oil flow rate as introduced by Raghavan 3 : p r k ro ( S o ) qo = J dp p μ wf o B... (5) o Where: S o is the oil saturation, and J the modified productivity index for this case, and is defined by:

3 SPE J = 4. kh ln( r e / r w ).75+ S... (6) The form given by Eq.5 is not useful in a practical sense- the integral must be reduced to a simple function of pressure. The approach suggested by Fetkovich 4 in the pursuit of Eq.5, is that he assumed the oil mobility function is a "simple" function of pressure. For example, Fetkovich 4 proposed the following relationship between the oil mobility function and p r : k ro ( S o ) = x μ o B o p r.... (7) Where, x is simply constant. Fetkovich 4 concluded that Eq.7 is too simplistic to model all of the changes of the pressure and saturation-dependent properties during depletion. Finally, the "Fetkovich form" of the IPR equation is given as the "backpressure" modification form, which is written as: n p q o wf = q o p. (8),max r This model requires a multi-rate test to determine the value of the backpressure exponent (n). As indicated, the main parameter that has the great effect on the Fetkovich's model is the oil mobility as a function of the average reservoir pressure, which assumed to be a linear relationship as illustrated in Fig.3. Fig.3- Mobility-pressure behavior for a solution gas drive reservoir (After Fetkovich 4 ) Klins and Majcher's Method In 99 based on Vogel s work, Klins and Majcher 5 developed the following IPR correlation that takes into account the change in bubble-point pressure and reservoir pressure due to the depletion in solution gas drive reservoirs. N q p p o wf wf = (8) q,max p r p o r S = 0 Where, N is defined by the following equation: p r N = * + 00 p p b b ( ).. (9) Wiggins's Method (993) In 993, Wiggins 6 developed a generalized empirical three phase IPR similar to Vogel 3 correlation to overcome the problem of applying his developed analytical model in 99. This correlation is: q p p o wf wf = (0) q o,max p r p r

4 4 SPE 404 Sukarno and Wisnogroho's Method In 995, Sukarno and Wisnogroho 7 developed the following IPR equation based on simulation results that attempts to account for the flow-efficiency variation caused by rate-dependent skin: q o pwf pwf = pwf FE qo, max p r = 0 p r pr S 3... () Where: 3 p 3 wf p wf p wf FE = a o + a + a + a... () p r p r p r a 3 i = b oi + bi S + b i S + b3i S.. (3) In Eq.3, S is the skin factor, and a o, a, a, a 3, b oi, b i, b i, and b 3i are the fitting coefficients that are shown in Table.. Table - Constants for Sukarno and Wisnogroho b oi b i b i b 3i a o a a a The Analytical Derived Correlations Wiggins's Method (99) In 99, Wiggins, Russell, and Jennings 8,9 studied the three-phase inflow performance for oil wells producing oil, water, and gas in a homogeneous, bounded reservoir. They started from the basic principle of mass balance with the pseudo-steady state solution to develop the following analytically IPR correlation: 3 4 q p p 3 p 4 p o C wf C wf C wf C = wf.. (4) q o,max D p r D p r D p r D p r Where, C, C, C 3 C n, and D coefficients are determined based on the oil mobility function and its derivatives taken at the average reservoir pressure (p r ) as given by: \ \\ \\\ k k k k C = ro + ro + ro + ro. (5) μobo = μ = 6 0 obo μ 0 obo = 0 μobo pd pd pd pd= 0 \ \\ \\\ k k k C = ro + ro + ro. (6) μo B o = μ 4 p 0 o B o p = 0 μo B o D D p D = 0 \ \ \ \ \ C = kro + kro 3... (7) 6 μo Bo 6 = 0 μo Bo pd pd = 0 \ \ \ k C = ro 4 (8) 4 μ o B o p D = 0 \ \\ \\\ D= kro + kro + kro + kro.. (9) μo B o 6 4 = 0 μo B o = 0 μ p p o B o p = 0 μo B o D D D p D = 0 Wiggins, et al. 8, 9 found that the main reservoir parameters that play a major role in the inflow performance curve are relative permeability and fluid properties (i.e., the oil mobility function). The major problem in applying the Wiggins's analytical IPR is its requirement for the mobility derivatives as a function of average reservoir pressure, which is very difficult in practice. Therefore, in 993 Wiggins 6 developed the empirical IPR equations that previously presented (i.e., Eq.0) from this analytical IPR model by assuming a third degree polynomial relationship between the oil mobility function and the average reservoir pressure. Wiggins, et al. 8, 9 also presented plots of the oil mobility as a function of the average reservoir pressure taken at various flow rates to establish the "stability" of the oil mobility profile for a given depletion level. An example to the oil mobility-pressure profile that is presented by Wiggins, et al. 8 is shown in Fig.4.

5 SPE Fig.4- The oil mobility profiles as a function of pressure- various flow rates (Case, after Wiggins, et al. 8 ) Del Castillo, Yanil's Method In 003, another theoretical attempt to relate the IPR behavior with the fundamental flow theories is developed by Del Castillo, Yanil et al. 0,. In this model, a second-degree polynomial IPR is obtained with a variable coefficient (v), or the oil IPR parameter that may in fact be a strong function of pressure and saturation. The starting point for this development is the pseudo-pressure formulation for the oil phase, which is given as: μ p k p = o B o ro po ( p) dp k ro p p μ base o B (0) o n In that work, Del Castillo, Yanil et al. 6, 7 presumed that the oil mobility function has a linear relationship with the average reservoir pressure as given below: k ro f p r e d p p r o B = ( ) = +. () μ o r Where e, and d are constants established from the presumed behavior of the oil mobility profile. Fig.3 refers to the physical interpretation of Eq.. Substituting with Eq. in Eq.0 and manipulating, the following equation could be presented: q p p o wf ( ) wf = v v. () q o,max p r p r Specifically, the ν-parameter is given as: v= (3) d + p r e Wiggins, et al. 8, 9 and Del Castillo and Yanil et al. 0, relationships can only be applied indirectly or inferred, by estimating the oil mobility as a function of the average reservoir pressure to construct the IPR curve. Future Inflow Performance Relationship In the absence of saturation data as a function of reservoir pressure, three simple methods can be used to predict future inflow performance curves. First Method (Vogel's Method) This method was provided by Vogel 3 and provides a rough approximation of the future maximum oil flow rate (q o, max ) f at the specified future average reservoir pressure (p r ) f from the following equation: p ) ) ) ) r f p r f ) ) qo, max = q + f o,max (4) p p r p p r p

6 6 SPE 404 Second Method (Fetkovich's Method) A simple approximation for estimating future (q o, max ) f at (p r ) f was proposed by Fetkovich 4. The relationship has the following mathematical form: 3.0 p ),max ),max ) r f q o = q o.. (5) f p p ) r p Third Method (Wiggins's Method) Wiggins 6 extended the application of Eq.4 to predict future performance by providing the following relationship: p ) ) ) ) r f p r f q o,max = q o,max (6) f p p ) p ) p r p r Summary of Literature Survey As indicated, the empirical correlations suffer from the limitation of their application range as they depend largely on the data used in their generation, and its lack of accuracy. In addition, they are not explicitly function of reservoir rock and fluid data, which are different from one reservoir to another. On the other hand, the analytical correlations suffer from their difficulty to be applied due to its requirement to the oil mobility profiles and its derivatives in addition to the assumptions used in their development. As discussed, the main parameter that affects the productivity index and in turn the inflow performance curves is the oil mobility function (k ro /µ o B o ) and its relation to the average reservoir pressure. Therefore, the relationship between the oil mobility function and the average reservoir pressure should be accurately determined. In addition, the most common equation that represents a basic start point for the development of any IPR equation, in case of solution gas drive reservoirs is Eq.0,which mainly a fuction of the oil mobility (k ro /µ o B o ). Most of the perviously empirical derived IPR equations did not take into their consideration the whole effect of the oil mobility function, this in turn largely reduce the accuracy, power, and utility of these equations. Even though the models that took into their consideration the effect of this function, such as the models of Fetkovich 4 and Wiggins 6, assumed the relationships between this function and p r, as the linear form and the third polynomial form for Fetkovich and Wiggins, respectively. In fact, these linear and polynomial forms do not accurately describe the general behavior of the oil mobility function with the average reservoir pressure with an accurate manner. On the other hand, some of analytical derived IPR equations did not considered the effect of this function, except the models of Wiggins, et al. 8, 9 Del Castillo, Yanil et al. 0,. Wiggins s model is so complicated because it requires the oil mobility represented in its derivates as a function of the average reservoir pressure, this is greatly difficult in application. Del Castillo, Yanil s model is not accurate, this is because Del Castillo, Yanil assumed a liner relationship between the oil mobility function and p r, which in turn reduce the accuracy of this model. Another parameter should be considered in the selecting of the IPR method, is the aspect of conducting the flow tests. It is evident that test costs have to be taken into consideration. Finally, the range of applicability will also influence the selecting of the IPR methods to predict the well performance. Accordingly based on the literature survey in this work, it is necessary to: Develop a new, more general, simple, and consistent method to correlate inflow performance trends for solution gas drive oil reservoirs. This new method takes into consideration the behavior of the oil mobility function with the average reservoir pressure without the direct knowledge of this behavior. Determine the applicability and accuracy of the proposed new model by applying it on different field cases with a comparison with some of the most known and used IPR equations, considering a wide range of fluid, rock, and reservoir characteristics. Test some of the available IPR methods on field data. Address the prediction of future performance from current test information. The New Developed IPR Model In this work, a single well 3D radial reservoir model using MORE 4 reservoir simulator was built. The reservoir simulation was used to investigate the shape and in turn the relationship between the oil mobility function and the average reservoir pressure. Then, a new IPR equation was derived based on the resulted oil mobility-pressure profile; this new IPR is mainly a function of the relationship between the oil mobility and the average reservoir pressure. Then, forty-seven field cases (published cases) were used to develop an empirical relationship between the oil mobility and average reservoir pressure. Thus, obtaining a new IPR model that is explicitly function of the oil mobility that is highly affect the IPR model.

7 SPE Mobility-Reservoir Pressure Relationship Production rate and pressure results from six simulation cases were used to develop the inflow performance curves. Table presents the ranges of reservoir, rock, and fluid parameters used in the six simulation cases. The saturation and pressure information was also used to develop the mobility function profiles. The general simulation assumptions that were used in building the reservoir model can be summarized as follows: 3D radial flow into the well bore The reservoir initially at the bubble point pressure Vertical well at the center of the formation The well is completed through the whole formation thickness. Homogeneous, bounded reservoir Isothermal conditions exist No initial O.W.C. exist Capillary pressure is neglected Interfacial tension effects and non-darcy flow effects are not considered. Results of the Simulator Fig.5 shows the behavior of the mobility as a function of the pressure at different values of the flow rate during the two-phase flow for simulation case No. (Case-S). The other five cases are shown in Appendix A, and the shapes of Fig.A. through Fig.A.5 curves are typical of trend in addition to Fig.5. Therefore, based on the six simulation cases, a reciprocal relationship between the oil mobility function and the average reservoir pressure was assumed and gives an acceptable and good match with the calculated simulator data as shown in Fig.6. Table - Ranges of reservoir parameters - simulation cases Rock/Fluid Property Range Units Average reservoir pressure, p r psia Bubble point pressure, p b psia Reservoir temperature, T F Oil specific gravity relative to water, γ o dimensionless Gas specific gravity relative to air, γ g dimensionless Water specific gravity, γ w.0.5 dimensionless Water viscosity, μ w cp Initial solution gas oil ratio, R soi Mcf/STB Initial oil formation volume factor, B oi. -.5 bbl/stb Initial oil viscosity, μ oi cp Z-factor dimensionless k (S or ) fraction k ( S wc ) 0..0 fraction k (-S wc -S or ) fraction Reservoir radius, r e ft Formation thickness, h ft Absolute permeability, k md The Oil Mobility Function STB/day 5000 STB/day 0000 STB/day STB/day STB/day STB/day The Avergae Reservoir Pressure, Psia Fig.5- Mobility-pressure behavior for solution gas drive oil reservoir-case S

8 8 SPE Standard Deviation, Relative Error, S = r = The Oil Mobility The Avearge Reservoir Pressure, psia Fig.6- Reciprocal model- Case S Derivation of the new IPR Equation The starting point for the derivation is the definition of the oil-phase pseudopressure for a single well in a solution gas drive reservoir and the pseudo-steady state flow equation for the oil-phase. In this work, a new form for the oil mobility function at different values of the average reservoir pressure (i.e., the reciprocal relationship) is obtained from the simulation study that performed on MORE simulator using the six simulation cases. This reciprocal relationship was used as: k ro o B =... (7) μ o a p r + b p r Where, a and b are constants established from the presumed behavior of the mobility profile. Substituting Eq.7 in Eq.0 and manipulating (the details are provided in Appendix B), the following new IPR equation will be introduced: q ln( α p + ) o wf =.. (8) q o, max ln( α p r + ) Where: α is the oil IPR parameter for the new IPR model. The productivity index can be numerically calculated from the following developed equation: kh PIα = +... (9) ln( r e / r w ) 0.75 S α p r + Eq.8 is the proposed new IPR equation. As recognized, the α-parameter is not "constant," therefore, forty-seven field cases (published cases) were used to develop the following two empirical relationships between the variable coefficient α which represent the oil mobility function and average reservoir pressure: When the average reservoir pressure-range is less than or equal to 600 psia the following relationship was developed: α =.. (30) a. p r + b Where: a= -0.98, and b= When the average reservoir pressure-range is greater than or equal to 600 psia the following relationship was developed: α = c + d. p e. p f. p 3 g. p 4 h. p 5 r + r + r + r + r. (3)

9 SPE Table 3 shows the constants of Eq.3. Table 3- Constants of Eq.3 Constant Value c = d = e = f = g = 4.98E E E E-7 h=.85e- Eq.30 and Eq.3 were developed based on the following developed chart between the α-parameter and the average reservoir pressure (i.e. Fig.7) that was generated from the forty-seven field cases. Table 4 introduces the ranges of data used in the development of these two equations. Finally, Eq.8, Eq.30, and Eq.3 represent the new developed IPR model. α = The Variable Coeffiecint, α Methodology to use the New IPR Model The Average Reservoir Pressure, psia Fig.7- The variable coefficient α and the average reservoir pressure chart Step. If the average reservoir pressure is less than or equal to 600 psia, therefore, the oil IPR parameter (α) is calculated using Eq.30 as follows: α = 0.98 p r If the average reservoir pressure is greater than or equal to 600 psia, therefore, the oil IPR parameter (α) is calculated using Eq.3 as follows: Ε 06 p r.4ε 09 p r Ε 3 p r 6.48Ε 7 p r +.85Ε pr Step. Calculate q o, max using Eq.8 and any given test point: ln( α p wf ( test) + ) q o,max = q o ( test) /[ ] STB/day ln( α p r + ) Step3. Assume several values for p wf and calculate the corresponding q o using Eq.8: ln( α p wf + ) q o = q o,max [ ] STB/day ln( α p r + ) Step4. For future IPR, calculate α f using the future value of p r ) f using Eq.30 or Eq.3 according to the value of p r ) f: Step5. Solve for q o, max, at future conditions using Fetkovich s equation (i.e., Eq.5) as follows:

10 0 SPE 404 qo, max ) = ) f q o, max p r p p r ) ) 3.0 f STB/day p Step6. Generate the future inflow performance-curve by applying Eq.8 as follows: ln( ) max ) α f p wf + q o = q o, f [ ] STB/day ln( α f p r ) f + ) Table 4- Ranges of data used in the development α-p r relationship Properties Data range. Fluid properties data: Gas specific gravity API gravity of oil 0-50 Water specific gravity Initial oil formation volume factor (bbl/stb) Initial oil viscosity ( cp) Initial solution gas oil ratio,( Mscf/STB) Bubble point pressure (psia) Up to Rock properties data: Porosity Absolute permeability (md) Irreducible water saturation Residual oil saturation ( W/O) Residual oil saturation (G/O) Critical gas saturation Total compressibility ( psi - ) 0.33 x x 0-6 Oil relative 0.0 and 0. S gc Reservoir and well dimension: Average reservoir pressure (psia) Up to 7000 Drainage area (acres) 0-80 Formation thickness (feet) 0-8 Reservoir radius ( feet) Well bore radius (feet) Reservoir temperature (deg. F) Validation of the New IPR Model To verify and validate the new developed IPR model, information from twelve field cases were collected and analyzed to get the present inflow performance, and two-field cases were collected and used to predict the future IPR curve. The ranges of fluid properties, rock properties, and reservoir data of these cases are included in Table 4. Each field case uses actual field data which representing different producing conditions. In order to test the accuracy and reliability of the new developed IPR model, which is single point method, it will be compared to some of the other two-phase IPR methods currently available in the industry. These methods are those of Vogel 3 (single point method), Fetkovich 4 (multi-point method), Wiggins 6 (single point method), and Sukarno 7 (single point method) for the present inflow performance. And, Vogel 3, Fetkovich 4, Wiggins 6 for the future inflow performance Elias, Mohamed 5, presents the complete details of the comparison analysis while the cases analyzed for present and future performances are summarized in Table 5 and Table 6, respectively. Field Case No.: Carry City Well Frederic Gallice and Michael L. Wiggins 6 presented multirate-test data for a well producing from the Hunton Lime in the Carry City Field, Oklahoma. The test was conducted in approximately weeks during the well, which was producing at random rates, rather than in an increasing or decreasing rate sequence. The average reservoir pressure was 600 psia, with an estimated bubble-point pressure of 530 psia and an assumed skin value of zero. The multi-rate test of this well is summarized in Table 7. Table 8 presents the predictions of the well s performance for the test information at a flowing bottomhole pressure of 94 psia, which representing a 5 % of the pressure drawdown. As can be observed, the maximum well deliverability varies from 550 to 465 STB/D. The largest flow rate was calculated with Wiggins s IPR, while the smallest rate was obtained using Fetkovich model. Fig.8 shows the resultant IPR curves for the different methods of calculations such as Vogel, Fetkovich, Wiggins, and Sukarno in comparison with the actual field data and the new developed IPR model. It is clear from this figure that the method of the new

11 SPE 404 developed IPR model is succeed to estimate the actual well performance. In addition, it can be clearly concluded from this figure that the methods of the new developed IPR model and Fetkovich s model are nearly estimate the maximum oil flow rate for this well more accurately than the other models, and as indicated, the other methods overestimate the actual performance. Table 5- Validation field cases analyzed for the present performance Case Case Name Case Type p r, psia Carry City Well 9 Vertical Well 600 Case XB Vertical Well Well 4 6 Vertical Well, Layered Reservoir Well 8, West Texas 30 Vertical Well, Low Pressure Well, Gulf of Suez, Egypt 3 Vertical Well 00 6 Well M Vertical Well 30 7 Well M00 3 Vertical Well Well 3-Field C 8 Vertical Well Well E, Keokuk Field 8 Vertical Well 70 0 Well A, Keokuk Field 30 Vertical Well 734 Well TMT-7 30 Vertical Well, Low Pressure 868 Well A 30 Vertical Well 785 Table 6- Validation field cases analyzed for the future performance Case Test Chronology Case Name Reference p r, psia Present Well M Future Well M Present Well A, Keokuk Field-934 Future Well A, Keokuk Field Table 7- Test data- Field Case No. Test Data p wf, psia q o, STB/D

12 SPE 404 Table 8- Prediction of the performance of Case No. at 5 % of the pressure drawdown Field Data The new IPR method Vogel method Fetkovich method Wiggins method Sukarno method p wf, psia q o, STB/D q o, STB/D q o, STB/D q o, STB/D q o, STB/D q o, STB/D The average absolute errors percent between the actual flow-rate data and the calculated rate for the five IPR methods that used in this study are shown in Fig.9 for the comparison. It is clear from this figure that the new developed IPR model has the lowest average absolute error percent that is 6.47 %, while the average absolute error percent for Fetkovich s method is 8.56 %. The other singlepoint methods have average absolute errors percent ranging from 0. to 3.3 % for Sukarno and Wiggins, respectively. In summary, the new model provided the best estimates of well performance for this case s entire range of interest. The multipoint method of Fetkovich tends to do a better job of predicting well performance than the other three single-point methods. Overall, the single-point methods of Vogel, Wiggins, and Sukarno provided similar great average differences in this case. As indicated in this work, the more important relationship to evaluate well performance is the relationship between the oil mobility function and the average reservoir pressure, and this was clearly demonstrated from the value of the average absolute error percent that resulted from using the new developed IPR model. Bottom Hole Flowing Pressure, psia Test Point Sukarno Wiggins Fetkovich Vogel The new IPR Field Flow Rate, STB/day Fig.8- The predicted inflow curves by the different used methods in comparison to the actual field data for Case No.

13 SPE Average Errors % The new IPR 6.47 Vogel 6 Fetkovich 8.56 Wiggins 3.3 Sukarno 0. 0 Models Fig.5.- The average absolute errors percent at 5 % drawdown for Case No. Field Cases Summary for the Present Inflow Performance The additional cases and their analysis are presented in detail in Ref. 0. Table 9 presents a summary of the average absolute errors percent that was obtained for each method in each one of the twelve case studies that were examined. As indicated, the method of the new developed IPR model always provided the most reliable estimates of the actual well data analyzed. It has the lowest value of the total average absolute error percent, which is 6.6 % in comparison with that of Fetkovich's method, which has a reasonable average absolute error percent of 7 % but is still higher than the method of the new developed IPR model. The other methods always provided less accurate values for the pressure-rate estimates of the actual well data that used in this analysis. The method of the new developed IPR model tends to do a better job of predicting well performance than the other methods, and this it may be due to assume an accurate relationship between the oil mobility function and the average reservoir pressure (i.e., the Reciprocal Relationship). Overall, the single-point methods of Vogel, Wiggins, and Sukarno provided great average absolute errors percent in the cases examined-. to 5.7 %. However, the following comment should be introduced based on the above table Case No.4, the Vogel's model provided the best estimates of well performance for this case. The single point method of Wiggins tends to do a good job of predicting well performance in this case. Finally, it can be concluded from this case that the new IPR model has some limitations in case of low-pressure reservoirs, which have a reservoir pressure less than 000 psia. This is because there were no sufficient data below 000 psia in case of development the new IPR model, therefore it is recommended in this case to use Vogel s model. Field Case Summary for the Future Performance The analyses of the two future field cases are presented in detail in Ref. 5. Table 0 presents a summary of the average absolute errors percent that was obtained for each method in each one of the two future case studies that were examined. As indicated, the method of the new developed IPR model always provided the most reliable estimates of the actual well data analyzed. It has the lowest value of the total average absolute error percent, which is 5 %. The other methods always provided less accurate values for the pressure-rate estimates of the actual future well data that used in this analysis. Overall, the other methods of Fetkovich, Vogel, and Wiggins provided great average absolute errors percent in the cases examined range from 5.5 % and 3.5 % for Fetkovich and Vogel, respectively. However, general comment can be presented based on Table 0 and the future cases analyzed in this work, the average absolute error percent is almost as great for the Fetkovich's method as compared to the method of the new developed IPR model % compared to 5 %. As indicated, the main reasons for that are: ) Fetkovich considered the relationship between the mobility function and the average reservoir pressure is linear relationship (see Fig.3), but the new IPR model considered this relation is reciprocal relationship (see Fig.5 and Fig.6) which cover the entire range of interest more accurately. ) The backpressure equation parameter (n) of Fetkovich IPR equation (Eq.8) does not take into consideration the change in the average reservoir pressure. 3) The new model IPR parameter (α) of Eq.8 takes into consideration the change in the average reservoir pressure (see Eq.30 and Eq.3).

14 4 SPE 404 Table 9- Summary of the average absolute error percent-the twelve cases Total Average Errors % Case The new IPR Vogel Fetkovich Wiggins Sukarno Average % Table 0- Summary of the average absolute errors percent-future field cases Total Average Errors % Future Field The new Vogel Fetkovich Wiggins Case IPR Average % Conclusions In this work, we reviewed the most commonly used IPR models, also, we developed new IPR model. The new IPR was compared to the most commonly used models using field data ( field cases). Based on this work, we can conclude the following:. A general correlation for α- parameter that represent the oil mobility as a function of p r was developed by using 47 field cases. A new method to construct and predict the IPR curve for solution gas drive reservoirs was developed by using this general correlation of α- parameter. The validity of the new IPR model was tested through its application on field cases in comparison with the behavior of the most common methods that are used in the industry. The results of this validation showed that the new IPR model ranked the first model that succeeded to predict the behavior of the IPR curve for the examined field cases, while the other models of Fetkovich, Sukarno, Vogel, and Wiggins ranked the second, the third, the forth, and the fifth, respectively 3. The new IPR model requires one test point and is as accurate or more than Fetkovich s model which requires three test points 4. The new developed IPR outperformed all available IPR models except at low pressures (Less than 000 psia). At these low press Vogel s correlation was found to be the most accurate model.

15 SPE Nomenclature A Drainage area of well, sq ft a Reciprocal model constant, dimensionless a 0,a,a,a 3 Constants for Sukarno and Wisnogroho, dimensionless API stock tank oil liquid gravity in, O API b Reciprocal model constant, dimensionless b 0,b,b,b 3 Constants for Sukarno and Wisnogroho, dimensionless B o B g C A C,C,C 3,C 4, D Oil formation volume factor, bbl/stb Gas formation volume factor, bbl/scf Shape constant or factor, dimensionless Wiggins's constants, dimensionless d, e Del Castillo, Yanil's constants, dimensionless h Formation thickness, ft J Productivity index of the reservoir (PI), STB/psi n deliverability exponent for Fetkovich, dimensionless N Oil IPR parameter for Klins's equation, dimensionless p b Bubble Point Pressure, psia p D Dimensionless pressure p e Pressure at the outer boundary, psia PI α Productivity index from the new IPR model, STB/psi p r Average reservoir pressure, psia p wf Bottom hole flowing pressure, psia q o Oil flow rate, STB/D q o, max Maximum oil flow rate, STB/D r e Drainage Radius, ft R s Solution gas-oil ratio, scf/stb r w Well Radius, ft S Radial flow skin factor, dimensionless S o Oil saturation, fraction T Reservoir temperature, ο F K ro Mobility ratio at zero dimensionless pressure μoβo p D =0 \ K ro μoβo p D =0 \\ K ro μoβo p D =0 \\\ K ro μoβo p D =0 Mobility ratio first derivative at zero dimensionless pressure Mobility ratio second derivative at zero dimensionless pressure Mobility ratio third derivative at zero dimensionless pressure α Oil IPR parameter for the new IPR model, dimensionless γ Euler's constant ( ) γ g γ o μ o Δp Gas gravity, fraction Oil gravity, fraction Oil Viscosity, cp Pressure drawdown, psi

16 6 SPE 404 References. Ahmed, T., and D. McKinney, P.: Advanced Reservoir Engineering, Page 34-36, Gulf Professional Publishing - Elsevier Inc., Oxford, UK (005).. Evinger, H.H. and Muskat, M.: Calculation of Theoretical Productivity Factors, Trans., AIME (94) Vogel J.V.: Inflow Performance Relationships for Solution Gas Drive Wells, JPT, Jan 968 (SPE 476). 4. Fetkovich: The Isochronal Testing of Oil Wells, Paper SPE 459 presented at the 973 SPE Annual Meeting, Las Vegas, NV, Sep. 30-Oct Klins, M.A., and Majher, M.W.: Inflow Performance Relationships for Damaged or Improved Wells Producing Under Solution-Gas Drive, (Paper SPE 985) JPT, Dec. 99, p Wiggins, M.L., Generalized Inflow Performance Relationship for Three-Phase flow Paper SPE 5458, Production Operations Symposium, Oklahoma City, OK, March -3 (993). 7. Sukarno, P. and Wisnogroho, A.: Generalized Two-Phase IPR Curve Equation under Influence of Non-linear Flow Efficiency, Proc., Soc of Indonesian Petroleum Engineers Production Optimization Intl. Symposium, Bandung, Indonesia (995) Wiggins, M.L.: Inflow Performance of Oil Wells Producing Water, PhD dissertation, Texas A&M University, TX (99). 9. Wiggins, M.L, Russel, J.L., and Jennings, J.W.: Analytical Inflow Performance Relationships for Three Phase Flow, Paper SPE 4055, presented at 99 Western Regional Meeting, California, April, Del Castillo, Y.: New Inflow Performance Relationships for Gas Condensate Reservoirs, M.S. Thesis Texas A& M University, May 003, College Station.. Archer, R.A., Del Castillo, Y., and Blasingame, T.A.: "New Perspectives on Vogel Type IPR Models for Gas Condensate and Solution-Gas Drive Systems," Paper SPE presented at the 003 SPE Production Operations Symposium, Oklahoma, OK, 3 5 March.. Weller, W. T.: Reservoir performance during two-phase flow, Journal Pet. Tech. (966) Raghavan, R., Well Test Analysis, Page 53-54, Prentice Hall Petroleum Engineering Series, Englewood Cliffs, New Jersey (993). 4. MORE Manual, Version 6.3, ROXAR (006). 5. Elias, Mohamed: New Inflow Performance Relationship for Solution Gas Drive Reservoirs, MS thesis, U. of Cairo, Giza, Egypt (009). 6. Gallice, Frederic and Wiggins, M. L.:"A Comparison of Two-Phase Inflow Performance Relationships" Paper SPE 88445, the 999 SPE Mid-Continent Operations Symposium, Oklahoma City, Oklahoma, March 8 3 (999). 7. Ribó, M. O.: Análisis de pruebas de presión en pozos de Cerro Prieto, Proceedings symposium in the field of geothermal energy, Agreement CFEDOE (Comisión Federal de Electricidad de México-Department of Energy of USA), San Diego California, USA, 3-9 (989). 8. Wiggins, M.L., Wang, H.-S.: A Two-Phase IPR for Horizontal Oil Wells Paper SPE 9430, Production Operations Symposium, Oklahoma City, OK, April 7-9 (005). 9. Abdel Salam, M.: Inflow Performance Relationship Correlation for Solution Gas Drive Reservoirs Using Non- Parametric Regression Technique, MS thesis, U. of Cairo, Giza, Egypt (008). 0. Kabir, C. S. and Kiog, G. R.:" Estimation of AOFP and Average Reservoir Pressure from Transient Flow-After-Flow Test Data: A Reservoir Management Practice" Paper SPE 9583, Rocky Mountain Regional/ Low-Permeability Reservoirs Symposium, Denver, CO, March 0- (995).. Haider, M.L.: Productivity Index, Paper SPE 937, Fort Worth Meeting (936).. Gallice, F.: A Comparison of Two-Phase Inflow Performance Relationships, MS thesis, U. of Oklahoma, Norman, Oklahoma (997).

17 SPE Appendix A The Oil Mobility Function STB/day STB/day 0000 STB/day 000 STB/day The Average Reservoir Pressure, psia Fig.A.- Mobility-pressure behavior for solution gas drive oil reservoir-case S The O il M obility Function STB/day 0000 STB/day 000 STB/day The Average Reservoir Pressure, psia Fig.A.- Mobility-pressure behavior for solution gas drive oil reservoir-case S3 The Oil Mobility Function STB/day STB/day 0000 STB/day The Average Reservoir Pressure, psia Fig.A.3- Mobility-pressure behavior for solution gas drive oil reservoir-case S4

18 8 SPE 404 The Oil Mobility Function STB/day STB/day 0000 STB/day The Average Reservoir Pressure, psia Fig.A.4- Mobility-pressure behavior for solution gas drive oil reservoir-case S5 The Oil Mobility Function STB/day 0000 STB/day 3000 STB/day The Average Reservoir Pressure, psia Fig.A.5- Mobility-pressure behavior for solution gas drive oil reservoir-case S6 Appendix B In this Appendix, the derivation of the new IPR equation is based on the pseudo-steady state flow equation for a single well in a solution gas drive reservoir systems (pseudopressure formulation). In addition the relation between the mobility of the oil phase and the average reservoir pressure (i.e., Reciprocal relationship Mo =.0 / (a. p r + b) is used. The definition of the oil-phase pseudopressure for a single well in a solution gas drive reservoir is given as: : μ p k p = o B o ro po ( p) dp k ro p p μ base o B... (B.) o n The pseudo-steady state flow equation for the oil-phase in a solution gas drive reservoir is given by: pp ( p r ) = p po ( p wf )+ q o b o ss.. (B.) Where: μ r 3 b = 4. o B o (ln( e ss ) + S).. (B.3) k ro h r w 4 p n

19 SPE In this work, a new form for the oil mobility function at different values of the average reservoir pressure (i.e., the reciprocal relationship) is obtained from the result of the simulation study that performed on MORE simulator using the six simulation cases as follows: k ro o B =... (B.4) μ o a p r + b p r Where, a and b are constants established from the presumed behavior of the mobility profile. Solving Eq.B. for the oil rate, q o, the following equation for the oil flow rate can be presented: q o= p po ( p r ) p p ( p wf ) (B.5) b ss o Solving Eq.B.5 for the maximum oil rate, q o, max (i.e., at p wf = zero, or p po (p wf ) = 0): q = p ( ) ( = 0),max p p r p po p wf (B.6) o b ss o Dividing Eq.B.5 by Eq.B.6 gives the "IPR" form (i.e., q o /q o, max ) in terms of the pseudopressure functions, which yields: q p ( p ) p ( p ) o po r po wf =... (B.7) q o, max p po ( p r ) p po ( p wf = 0) At this point, it should be noted that, it is not the goal to proceed with the development of an IPR model in terms of the pseudopressure functions, p po (p)-rather, the goal is to develop a simplified IPR model using Eq.B.4 and Eq.B.7 as the base relations. Substituting Eq.B.4 into Eq.B., this yields: p o B o o B o p p ( p) = μ μ dp o k [ ] = + ) ro p a p+ b k ro a p n base p n [ ln( a. p b ] p pbase Or, μ ( ) o B o p p p = [ln( a p b) ln( a p base b)] o + + k.. (B.8) ro a p n Substituting Eq.B.8 into Eq.B.7, this gives: q [ln( a p r + b) ln( a p base + b)] [ln( a p wf + b) ln( a p base + b)] o = q o, max [ln( a p r + b) ln( a p base + b)] [ln( b) ln( a p base + b)] Or, q ln( a p r + b) ln( a p wf + b) o =.. (B.9) q o, max ln( a p r + b) ln( b) Rearranging Eq.B.9 gives the following form: q ln( ) ln( a p b) o a p r + b wf + =. (B.0) q o, max ln( a p r + b) ln( b) ln( a p r + b) ln( b) Or, q ln( ) ln( a p b) o a p r + b wf + =.. (B.) q o,max a p b a p b ln( r + ) ln( r + ) b b

20 0 SPE 404 Dividing the right term through Eq.B. by the term b gives the following form: a a q ln( p + ) ln( p + ) o = b r b wf (B.) q o, max a a ln( p r + ) ln( p r + ) b b Defining α = a/b and substituting this definition into Eq.B., this yields the following IPR form: q ln( α p + ) o wf = (B.3) q o, max ln( α p r + ) Where: α is the oil IPR parameter for the new IPR model. It is suggested that Eq.B.3 serves as an equation of the proposed new IPR model. The productivity index can be numerically calculated by recognizing that J must be defined in terms of pseudo-steady state flow conditions. For slightly compressible fluids, the oil flow rate is given by: k o h p r q o = μo B o r e r w ( ) p wf ( ln( / ) S)... (B.4) The productivity index is defined as the ratio of the total liquid flow rate to the pressure drawdown. For a water-free oil production, the productivity index was developed from Darcy s law for steady state radial flow of a single incompressible fluid and is given by: q q J = o = o.. (B.5) p r p wf Δp The above equation is combined with Eq.B.4 to give: k o h J = μo B o r e r w ( ln( / ) S ) (B.6) The oil relative permeability concept can be conveniently introduced into Eq.B.6 to give: kh k J = ro r + e r w S o B... (B.7) ln( / ) 0.75 μ o Eq.B.4 can be rewritten as a function of the oil IPR parameter (α) as given below: k ro = o B (B.8) μ o α p r + p r Substituting with Eq.B.8 to Eq.4.7, the following expression for the productivity index can be presented: kh PIα = + (B.9) ln( r e / r w ) 0.75 S α p r + Eq.4.9 determines the productivity index of the well.