SPE The direct application of this work is to predict the IPR for a given system directly from rock-fluid properties and fluid properties.

Transcription

1 SPE New Perspectives on Vogel Type IPR Models for Gas Condensate and Solution- Gas Drive Systems R.A. Archer, University of Auckland, Y. Del Castillo, and T.A. Blasingame, Texas A&M University Copyright 003, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the 003 SPE Production Operations Symposium, Oklahoma City, OK, 3-5 March 003. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes 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 where and by whom the paper was presented. Write Librarian, SPE, P.O. Box , Richardson, TX , U.S.A., fax This work also includes a discussion of the Vogel IPR for solution-gas drive systems. The original work proposed by Vogel is based on an empirical correlation of numerical simulations for a solution-gas-drive system. Our work provides a critical validation and extension of the Vogel work by establishing a rigorous, yet simple formulation for flowrate-pressure performance in terms of effective permeabilities and pressure-dependent fluid properties. The direct application of this work is to predict the IPR for a given system directly from rock-fluid properties and fluid properties. This formulation provides a new mechanism that can be used to couple flowrate and pressure behavior for solutiongas-drive systems and it may be possible to extend the concept to gas condensate reservoir systems. Introduction Solution Gas-Drive Systems The first presentation of an inflow performance relationship (or IPR) concept was made by Rawlins and Schellhardt 1 in In Fig. 1 we provide a reproduction of this figure where Rawlins and Schellhardt used this plot to illustrate the effect of liquid loading on production performance. Abstract In this work we propose two new Vogel-type Inflow Performance Relations (or IPR) correlations for gas condensate reservoir systems. One correlation predicts gas production the other predicts condensate production. These correlations link reservoir rock and fluid properties (dewpoint, temperature, and endpoint relative permeabilities) to the flowrate-pressure performance for the system. The proposed IPR relationships for compositional reservoir systems are based on data from over 3000 compositional reservoir simulation runs with various fluid properties and relative permeability curves. The resulting IPR curves for gas condensate systems are quadratic in nature like the Vogel IPR trends (the Vogel profile generally presumed for the case of a solution gas-drive reservoir system). However in the gas condensate case the coefficients in the quadratic relationship vary significantly depending on the richness of the condensate and the relative permeability. A model to predict these coefficients was developed using an alternating conditional expectation approach (optimal non-parametric regression). Figure 1 Primitive IPR plot for a gas well experiencing liquid loading (circa 1935) (after Rawlins and Schellhardt (ref. 1)). In 1954 Gilbert introduced the concept of an "inflow performance relationship" (or IPR) for the purpose of optimizing producing rates and flowing bottomhole pressures. In Fig. we reproduce Gilbert's IPR diagram for orientation. In 1968 Vogel 3 established an empirical relationship based on reservoir simulation results for a solution gas-drive reservoir. Vogel used twenty-one reservoir data sets to generate a broad suite of cases. In this work Vogel noted that the shape of the pressure (p wf ) versus production (q o ) curves were very similar at various values of cumulative oil production. As such, Vogel defined the dimensionless inflow performance curve by dividing the pressures and flowrates at each point by the

2 R.A. Archer, Y. Del Castillo and T.A. Blasingame SPE intercepts on the respective x and y-axes (i.e., the average reservoir pressure on the y-axis and the maximum oil flowrate on the x-axis). Figure Primitive IPR plot for a gas well experiencing liquid loading (circa 1935) (after Gilbert (ref. )). In performing this work, Vogel elected to produce a "reference curve" that is an average of the various depletion cases for a given reservoir scenario. Vogel recognized that the liquid (oil), gas (dry gas), and solution gas drive cases have distinct behavior trends and was simply trying to produce a mechanism for predicting production performance with a relatively simple result. In Fig. 3a we present the Vogel plot illustrating the liquid (oil), gas (dry gas), and solution gas-drive cases. Figure 3b IPR behavior for solution-gas drive systems at various stages of depletion the "reference curve" is the correlation presented by Vogel (after Vogel (ref. 3)). For wells producing below the bubblepoint pressure, Fetkovich 4 derived the following generalized IPR equation using pseudosteady-state theory and a presumed linear relationship for the liquid (oil) or gas mobility functions (i.e., k o /(µ o B o ) or k g /(µ g B g )): qo qo,max n p wf = 1... () p An illustrative schematic plot of the Fetkovich concept for a linear mobility trend is shown in Fig. 4. Figure 3a IPR schematic plot for oil, gas, and solution gas drive systems (after Vogel (ref. 3)). The "Vogel correlation" for IPR behavior in a solution gasdrive system is given by: qo pwf pwf = (1) qo,max p p In Fig. 3b we reproduce the Vogel plot illustrating the IPR behavior for solution-gas drive systems at various stages of depletion the "reference curve" is the Vogel correlation. Figure 4 Mobility-pressure behavior for a solution-gas drive reservoir (after Fetkovich (ref. 4)). Fetkovich considered the oil phase for a solution gas-drive system but this concept has also been extended to the gas phase. The most important aspect of Fig. 4 is the presumed linearity of the mobility function for pressures below the bubblepoint. Richardson and Shaw 5 presented a generalized inflow performance relationship for solution gas reservoirs as a function of the parameter, ν. For the Vogel case ν=0., and Richardson

3 SPE 8058 New perspectives on Vogel Type IPR Models for Gas Condensate and Solution-Gas Drive Systems 3 and Shaw propose that the ν-parameter be optimized for a given case. Richardson and Shaw provide an interesting schematic plot to illustrate the behavior of the IPR function above and below the bubblepoint pressure (see Fig. 5 for a reproduction of this schematic plot). 6). The purpose of presenting these mobility-pressure profiles was to establish the "stability" of the mobility profile for a give depletion level which would confirm the use of an IPR model based on a "snapshot" of reservoir performance. An example mobility-pressure profile taken from the Wiggins, et. al. reference is shown in Fig. 6. Figure 5 Schematic IPR behavior for a solution-gas drive reservoir note the "linear" and "quadratic" characteristic regions (for p>p b and p<p b, respectively) (after Richardson and Shaw (ref. 5)). Particular to this work, Richardson and Shaw discuss procedures for testing wells where the objective is to use production data to solve for the ν-parameter directly for example, using a two or three-rate test. The "general" IPR model presented by Richardson and Shaw is given as: qo pwf pwf = 1 v (1 v)... (3) qo,max p p We will note in advance that we will use the formulation given by Eq. 3 to derive our new IPR models for gas and condensate behavior in gas condensate reservoir systems. Another attempt to quantify the IPR behavior for solution gasdrive systems was presented by Wiggins, et. al. 6 where this result is a polynomial series given in terms of q o /q o,max and p wf / p. The Wiggins, et. al. result is given by: 3 qo pwf pwf pwf = 1+ a1 + a + a qo,max p p p... (4) Where the a 1, a, a 3,... a n coefficients are determined based on the mobility function and its derivatives at the average reservoir pressure ( p ). This is a relatively impractical approach because of the derivatives involved but, we must recognize that the IPR behavior can be related to fundamental flow theories. As an aside, Wiggins, et al. (ref. 6) also present plots of oil mobility as functions of pressure (taken at various flowrates) as a means of calibrating their proposed IPR model (i.e., Eq. Figure 6 Oil mobility profiles as a function of pressure various flowrates ("Case ")) (after Wiggins, et al. (ref. 6)). We will comment that the "double-linear" trend illustrated in Fig. 6 is consistent with other work produced for solution gas drive systems, and is somewhat in dispute with the model proposed by Fetkovich (see Fig. 4). Introduction Gas Condensate Systems In this section we discuss methods to represent IPR behavior for gas condensate reservoir systems. Analogs and references to the solution gas-drive system are common for the case of a gas condensate system, but our goal is to isolate the important factors/elements that must be addressed in order to correlate IPR behavior for gas condensate reservoir systems. Inflow performance relationships have been considered for gas-condensate reservoir systems by several authors. The case of a gas condensate reservoir is more complex because of the evolution of the condensate bank as the reservoir is depleted. Fussell 7 addressed this issue using a 1-D radial compositional simulator. O'Dell and Miller 8 presented results which show that even a minor region of condensate blockage/banking can substantially reduce the deliverability of the well. Fevang and Whitson 9 presented a gas-rate flow equation for gas condensate systems, which employs a pseudopressure function (in terms conventional formation volume factors and gas-oil-ratios) as a mechanism to account for the effect of condensate banking. This result is given by: kh p krg k q R ro g = 1 dp [ r r s] + s 141. ln( e / w) 3/4 + p gbg ob wf µ µ o... (5)

4 4 R.A. Archer, Y. Del Castillo and T.A. Blasingame SPE Fevang and Whitson suggested that the pseudopressure integral could be evaluated by expanding the integral into three regions: (taken from ref. 9) Region 1: An inner, near-wellbore region where both gas and oil flow simultaneously (at different velocities). Region : A region of condensate "buildup" where only gas is flowing. Region 3: A region containing single-phase (original) reservoir gas. The producing GOR, PVT properties, and gas-oil relative permeabilities are required in order to calculate the pseudopressure function (Eq. 5). While Eq. 5 is straightforward i.e., an integral in terms of pressure and saturation-dependent functions, these functions can not be known accurately in advance which renders Eq. 5 as a possible analysis relation, but not a predictive (or modelling) relation. It is worthy to note that Fevang and Whitson (ref. 3) comment that critical oil saturation (S oc ) has no effect on gas condensate well deliverability. While it is not our intention to dispute this comment, we will note that our work has addressed the influence of the following parameters on gas condensate well deliverability: relative permeability, fluid composition, dewpoint pressure, reservoir temperature, S oc, and S gr. In this work, we provide a simple methodology to estimate IPR functions for wells in gas-condensate reservoir systems without the requirement of gas-oil ratio and saturation profiles in the reservoir as a function of pressure. The new IPR approach will be developed using a large database of compositional reservoir simulation cases. We use the form of the Richardson and Shaw model, 5 but the parameter ν will vary depending on the various properties of a given reservoir fluid (as noted by the group of parameters we have identified as influential in the case of gas condensate reservoir systems (see above)). Method Correlation of IPR Behavior for Gas Condensate Reservoir Systems In this study we use compositional reservoir simulation as the mechanism to perform numerical experiments in order to characterize the IPR behavior for gas and condensate liquid performance. This section outlines the construction of the numerical model in terms of gridding and the selection of relative permeability profiles and reservoir fluid properties. Model Construction A 1-D radial grid is used for the simulation model since our main purpose is to evaluate the productivity of a single well in a gas condensate reservoir system. Bottomhole pressures and production rates are of critical importance for IPR correlations and as such, we used fine grid cells near the wellbore to properly represent the well productivity losses due to the build up of condensate near the well, while maintaining coarser grids for the remainder of the reservoir. Malachowski et al. 10 concluded that fine grids should be used for gas condensate reservoir simulation or the predicted condensate saturation near the wellbore will be underestimated. As is customary, the sizing of the radial grid cell sizes follows a logarithmic distribution with the ratio between two consecutive cell sizes being The maximum cell size is 500 ft. The most convenient (and arguably the most appropriate) production constraint is that of a constant bottomhole pressure. A small initial time step is used to address the onset of condensate banking. A typical simulation is conducted for approximately 10 years in order to assess the effect of condensate banking on ultimate recovery. The grid cells have a uniform thickness of 30 ft and the reservoir model is considered homogeneous and isotropic with an absolute permeability of 5 md and a porosity of 0 percent. The simulation model was initialized from an initial pressure equal to the dewpoint pressure of the selected reservoir fluid which means that initially, the only fluid in the reservoir is gas, as the water saturation is set to zero. Relative Permeability Curves Seven (7) different sets of relative permeability curves are used in this study in order to assess the influence of relative permeability on the IPR character for gas condensate systems. Corey-type relative permeability curves as well as other relative permeability curves are use to address a spectrum of relative permeability behavior. The seven sets of curves are shown in Figs. 7 and 8. Figure 7 Oil relative permeability data sets for the condensate reservoir simulation study. The relative permeability curves have different endpoint saturations and different shapes. A set of "straight line" relative permeability curves are included as an "extreme" case (Data Set 5). Data Set 4 is designed to represent an "abrupt" change in oil relative permeability and is taken from another study of gas condensate reservoir systems.

5 SPE 8058 New perspectives on Vogel Type IPR Models for Gas Condensate and Solution-Gas Drive Systems 5 Figure 8 Gas relative permeability data sets for the condensate reservoir simulation study. It could be argued that we could have considered more cases of relative permeability behavior however, we believe that this set of relative permeability cases represents an appropriate suite of cases that would typically be encountered in practice. Reservoir Fluids In this work we considered six (6) reservoir fluid samples four (4) of these samples were synthetic fluids and the remaining two samples were taken from studies for the Cusiana and Cupiagua gas condensate fields in Colombia (South America). The composition of the synthetic fluids is shown in Table 1 and we note that these four fluids have varying degrees of richness. Table 1 Composition of synthetic reservoir fluid samples (Fluids 1-4) (variations of fluids presented by Roussenac (ref. 11)). Mole Fraction Fluid C C C Initial Molecular Weight of the Fluid Mixture Fluid M mix Tables A-1 and A- describe the Cusiana fluid sample and are provided in the Appendix of this paper. The sample composition for "Fluid 5" is detailed in Table A-1 and it is clear that there are too many components for practical reservoir simulation purposes. Data from a constant composition expansion experiment (CCE) were matched to tune a version of the Peng-Robinson equation of state (EOS) for this fluid. After tuning, the components were grouped (or lumped) using Whitson's method 1. The resulting groups are shown in Table A-3. In ref. 13, Jaramillo presents further detail regarding the fluid characterization for this Cusiana sample. The final reservoir fluid considered was based on a sample from the Cupiagua field ("Fluid 6"). This fluid composition is given in Table A-4. Similar to the fluid sample from Cusiana field, data from a constant composition expansion experiment were used to tune a Peng-Robinson EOS model. The pseudocomponents used to represent this fluid in the compositional simulation cases are given in Table A-5. Additional detail and discussion of the fluid characterization for this sample is given by Guerra. 14 Assumptions Used in the Simulation Model The following assumptions are used in the development and execution of the reservoir simulation models for this case: Interfacial tension effects and non-darcy flow effects are not considered. Capillary pressure effects are not considered. Near wellbore effects (saturation, pressure, and mobility) are accurately represented by a refined grid. Phase equilibrium is accurately calculated by the EOS. A reduced permeability zone (skin) is not considered. Gravitational segregation of the condensate is not considered. No compositional gradients are considered. Strategies for the Development of the IPR Database We first validated the simulation model and considered the following parameters/functions (or combination of parameters/ functions) in order to generate the IPR database: Reservoir Temperature: T = 30, 60, 300 Deg F Critical Oil Saturation: S oc = 0, 0.1, 0.3 Residual Gas Saturation: S gr = 0, 0.15, 0.5 Relative Permeability: 7 sets of k ro -k rg data Fluid Samples: 4 synthetic cases, field samples We note that temperature is the primary control variable for dewpoint pressure (i.e., a change in temperature yields a change in dewpoint pressure). For each case (i.e., a particular combination of the parameter/functions considered), 30 to 45 simulation scenarios were conducted using constant (but different) bottomhole pressures. Each simulation was begun at the dewpoint pressure of the fluid at the conditions selected with our purpose being to evaluate only the phase behavior of the condensate fluid in the two-phase region, where gas condensate and gas coexist together while encountering significant changes in composition. The maximum gas (G) and condensate (N) volumes were determined for each simulated case (with respect to the separator conditions) in order to evaluate the recovery of these

6 6 R.A. Archer, Y. Del Castillo and T.A. Blasingame SPE fluids. For all cases the pressure of the separator was set to a pressure of 14.7 psia and a temperature to 60 deg F. Condensate (liquid) and gas production rates were recorded at eight selected "stages" of cumulative condensate (liquid) and cumulative gas production (i.e., gas or condensate "depletion ratios" and in turn, these data were tabulated as a function of bottomhole flowing pressure. The deliverability curves for condensate (liquid) and dry gas were constructed as follows: p wf versus q o for a given cumulative oil production (N p ) p wf versus q g for a given cumulative gas production (G p ) IPR Calibration (regression to obtain IPR trends) Using the results from the database described above, we conducted a simultaneous regression of the data for all of the depletion stages for a particular case in order to estimate the average reservoir pressure and the maximum condensate or gas rate for each depletion stage. In this work we always consider 8 (eight) depletion stages per case (see Table ). Table Fixed values of condensate and gas cumulative production used to generate a particular IPR trend (fraction or N p /N and G p /G will vary according to initial reservoir conditions). Depletion N p G p Stage (STB) (MSCF) 1 5.0x x x x x x x x x x x10 6.5x x x x x10 7 Our regression work also solves for the ν-parameter for a particular case (for clarity we note that ν o and ν g are solved independently for a given case). In our regression formulation we define the following variables: ( qθ ) i, j y i, j = (θ is the oil or gas phase)... (6) ( qθ, max ) j ( p wf ) i xi, j = (p wf is constant for a given simulation)... (7) ( p) j We cast the optimization problem into the following double summation form, where the SOLVER algorithm in MS Excel 15 is used to minimize the "residual function," J θ for a particular phase: ( n) 8 p wf = = J = θ yi, j 1 ν θ xi, j (1 νθ ) x.(8) i, j j 1 i 1 We note that the 8 (eight) term in Eq. 8 is the number of depletion stages considered for a given condensate or gas scenario we have chosen to always use 8 depletion stages, regardless of the scenario (condensate or dry gas). The (n) pwf variable is total number of simulations per case (i.e., the number of individual p wf runs per case) generally this number is between 30 and 45, depending on the reservoir pressure at a given time. We comment that we have cast this problem into an "absolute error" form, as opposed to the typical "least squares" formulation. This is relevant the SOLVER algorithm is completely general, and we believe that this form yields better results than the least squares formulation for this problem. Results IPR Trends and Correlations for Gas Condensate Systems Base IPR Trends We developed base IPR trends for each gas and gas condensate dataset (within a particular case) using the regression procedure described in the previous section (there are a total of 6 cases). After the simulation of each case, the rate and pressure results are plotted as "dimensional" IPR curves (i.e., plots of the flowrate of condensate or gas versus the flowing bottomhole pressure). Once the regression procedure has been applied, we then present the "base" IPR curves in a "non-dimensional" (or dimensionless) format in the same manner as the Vogel IPR function. Examples of "dimensional" and "non-dimensional" IPR trends for gas and gas condensate (Case 16) are shown in Figs We consider Case 16 to be a "typical" gas condensate case (see Table A-6 for a complete inventory of all cases developed in this work). In Fig. 9 (the dimensional gas IPR plot for Case 16) we note an excellent correlation of the rate and pressure functions (including the extrapolated trends). Figure 9 Example of a "dimensional" IPR trend (gas) (Case

7 SPE 8058 New perspectives on Vogel Type IPR Models for Gas Condensate and Solution-Gas Drive Systems 7 Figure 10 Example of a "non-dimensional" IPR trend (gas) (Case 16). In Fig. 10 we present the "non-dimensional" (or dimensionless) gas IPR trend for Case 16 we note good correlation of the "dimensionless" trends, and would comment there is a cluster of points in the low pressure region (probably the least accurate portion of data). We are satisfied that this case is effectively represented by the "Vogel" IPR model, and would suggest that this behavior is "typical" for a gas condensate system. correlation of the rate and pressure data with the proposed "Vogel" IPR model. In fact, the correlation for this case is so strong that we could consider the Vogel IPR model to be the general model for condensate (liquid) production performance. Obviously other cases must validate such a suggestion, but we do note that virtually all cases in this work exhibit a similar performance (for the condensate phase) as illustrated in Fig. 11. In Fig. 1 we present the "non-dimensional" (or dimensionless) condensate IPR trend for Case 16, as with the dimensional trend (Fig. 11), the correlation shown in Fig. 1 is extraordinary. One could argue that such correlations might be expected due to the flexible nature of the general Vogel model (Eq. 3) however, we must recognize that the data in Fig. 1 represents the simultaneous regression of 8 depletion stages (from individual simulation cases). Such performance should be considered both relevant and unique and should further validate the used of the general Vogel model for representing the IPR behavior of gas condensate behavior (gas and liquid phases). Figure 11 Example of a "dimensional" IPR trend (condensate) (Case 16). The "dimensional" condensate IPR trend for Case 16 is presented in Fig. 11 we immediately note an excellent Figure 1 Example of a "non-dimensional" IPR trend (condensate) (Case 16). As noted above, the IPR base data (i.e., the ν γ and ν ο parameters as well as the fluid and rock fluid properties) are summarized in Table A-6 for all 6 cases considered in this work. Global Correlations IPR Behavior for Gas Condensate Systems Orientation: In the previous section we demonstrated that gas condensate behavior can be correlated using the general Vogel model (Eq. 3). This is a significant contribution however, it is our intention to provide a general or "global" correlation of IPR behavior for gas and condensate production performance. A quick check of Table A-6 confirms the intuitive concept that

8 8 R.A. Archer, Y. Del Castillo and T.A. Blasingame SPE the parameters (ν γ and ν ο ) are not the same for a particular case. The flow mechanisms and controlling factors are different for the gas and condensate phases and must be accounted for separately. In other words, we must develop separate "global" correlations for the gas and condensate cases. Initial Efforts Univariate Plots: The starting point for a global correlation is to attempt to assess the influence of individual parameters as well as the influence of groups of parameters. Our approach is systematic in that our first effort is to consider the behavior of the ν γ and ν ο parameters with respect to the oil-in-place/gas-in-place ratio (N/G) and the initial molecular weight of the fluid mixture (M mix,i ) (Figs ). Figure 14 Correlation of the ν o parameter as a function of the oil-in-place/gas-in-place ratio (N/G) (all data). Another "logical" comparison is that of the ν γ and ν ο parameters with the initial molecular weight of the fluid mixture (M mix,i ) this comparison is shown in Fig. 15 for the gas cases and in Fig. 16 for the gas condensate cases. Figure 13 Correlation of the ν g parameter as a function of the oil-in-place/gas-in-place ratio (N/G) (all data). The correlation attempts shown in Figs. 13 and 14 (i.e., ν γ and ν ο versus the oil-in-place/gas-in-place ratio (N/G)) clearly show a "clustered" trend of data however, we would no consider this to be particularly useful since ν γ and ν ο vary for each fluid sample. This indicates that more variables will be required. The oil-in-place/gas-in-place ratio (N/G) would not be an ideal variable for correlation as we wish to present a generalized correlation using (N/G) would be "problematic" in that we would have to have knowledge of this ratio, which in practice we typically do not. Our choice of using (N/G) is more of a validation we wanted to establish whether or not the (N/G) parameter exerts dominance on the ν γ and ν ο parameters. Figs. 13 and 14 suggest a strong, but not dominant influence. Figure 15 Correlation of the ν g parameter as a function of the initial molecular weight of the fluid mixture (M mix,i ) (all data). In Figs. 15 and 16 we (again) note clustering of data however, we also note that there are multiple values of the parameters (ν γ and ν ο ) for a given value of M mix,i. This suggests that the univariate correlation of the IPR parameters ν γ

9 SPE 8058 New perspectives on Vogel Type IPR Models for Gas Condensate and Solution-Gas Drive Systems 9 and ν ο with M mix,i will not yield unique correlations. We recognize this issue and move towards an attempt to generate multivariate correlations of the IPR parameters. For example, we will segregate the most influential variables (T, p d, S oc, S gr, M mix,i, and the groups C1, C, C7+) then correlate ν γ and ν ο simultaneously with all variables or with optimal groups of variables. We establish the optimal grouping(s) of variables using non-parametric regression. 16 Figure 17 Correlation of the ν g parameter as a function of the ν o parameter (Cartesian format plot). Figure 16 Correlation of the ν o parameter as a function of the initial molecular weight of the fluid mixture (M mix,i ) (all data). Before proceeding to the task of multivariate regression we compare the ν γ and ν ο parameters in Figs. 17 and 18 to establish if these parameters are strongly correlated. Intuition suggests that if the ν γ and ν ο parameters are correlated, such a correlation may be coincidental as the ν γ and ν ο parameters were estimated in separate regressions. Regardless, correlation of the ν γ and ν ο parameters would be useful, and upon inspection of Fig. 17 (the Cartesian format plot of ν γ versus ν ο ) we find that an apparent correlation exists (ν γ ν ο ). While this correlation is fairly weak we conclude that the correlation may be of sufficient accuracy to predict one variable (e.g., ν γ ) from the other rather than to generate two separate correlations for ν γ and ν ο. In Fig. 18 we present the log-log format plot of ν γ versus ν ο. In this plot we note a similar "apparent" correlation of these parameters and we provide an approximate relation to orient the reader to this apparent correlation. We do not advocate using either the linear or power law relations presented on Figs. 17 and 18 (respectively), but we do suggest that these apparent correlations by used to orient future correlations. Figure 18 Correlation of the ν g parameter as a function of the ν o parameter (log-log format plot). Non-Parametric Multivariate Correlation of ν γ and ν ο Table A-6 was used as input for a non-parametric algorithm that can be applied to univariate or multivariate datasets. 16 The specific version of the algorithm used in this work is called GRACE (see ref. 16) where the GRACE algorithm is designed to perform a non-parametric regression. GRACE can also be instructed to provide a "parametic" regression by fitting generalized functions (in this case quadratic polynomials) to the non-parametric transform functions generated internally in the GRACE algorithm.

10 10 R.A. Archer, Y. Del Castillo and T.A. Blasingame SPE From ref. 16 we have the following description of the functionality of the GRACE algorithm. "The GRACE algorithm is based on the concept of developing non-parametric transformations of the dependent and independent variables. Moreover, the transformations are constructed pointwise based only on the data. The final result is a maximum correlation between the dependent and multiple independent variables with a minimum error." In cases where the only difference between runs was a differing set of relative permeability curves, a single representative simulation run was retained. This selection procedure reduced the total number of cases for analysis to 54. Several combinations of independent variables (e.g., T, p d, S oc, S gr, M mix,i, and the groups C1, C, C7+.) were used in the GRACE regression of the IPR parameter data. After considerable testing of variables (and combinations of variables) we developed what we consider to be the most representative model for ν o. This model includes C1, M mix,i, p d, S gr, S oc, and T as independent variables. Using the GRACE algorithm, the average error associated with the prediction of ν o from this model is 9.3 percent (this is a non-parametric correlation it is the minimum error possible). shown in Fig. 19. We note that the performance of this form of the correlation degrades at the upper end of the ν o scale. The equations defining this model are: C1Tr = C C MTr = M + 0.1M pdew, Tr = 7.81E 08 p dew pdew Sgr, Tr = S gr Sgr Soc, Tr TTr SumTr = 7.03 S oc.8 Soc = 8.9E 06T T = C1Tr + MTr + pdew, Tr + Sgr, Tr + Soc, Tr + TTr and finally, νo is estimated using : ν o = Sum Tr SumTr (6) Figure 19 Correlation of the ν o parameter using a parameterized correlation of the results of the nonparametric regression. In order to generate a correlation that can be of practical use, we must use the point-wise transform functions constructed by the GRACE algorithm and fit these transform (data) functions with functional forms (such as polynomials). Using quadratic polynomials to represent the individual transform functions, the average error increased (as would be expected) to 1.40 percent. The results from this "parametric" correlation is Figure 0 Correlation of the ν g parameter using a parameterized correlation of the results of the nonparametric regression. The same set of independent variables were also used in the model of the IPR parameter for gas. The GRACE model achieved an average error of 3.87 percent. When quadratic functions were used to fit the transform functions the average error increased to 4.51 percent. Unlike the model for the liquid case the "parametric" model from GRACE matches the data for all ranges of the ν g parameter. A comparison of the predicted and actual ν g data is given in Fig. 0. As an overall comment we will note that the shape of the computed IPR trends is actually quite insensitive to the actual values of ν o and ν g. For example, 5 percent variations in these values causes a relatively small change to the shape of the corresponding IPR trends. We view the models proposed

11 SPE 8058 New perspectives on Vogel Type IPR Models for Gas Condensate and Solution-Gas Drive Systems 11 as having definite qualitative value but we would caution against the use of these models in cases where the values of the independent variables are beyond the ranges of the variables used to develop the proposed models. C1Tr = 4.3 C C1.07 MTr = M M 7.10 pdew, Tr = 1.06e 08 p dew pdew Sgr, Tr = 0.9 Sgr 0.6 Sgr Soc, Tr TTr SumTr = 6.9 Soc.7 Soc = 1.76e 06 T T = C1Tr + MTr + pdew, Tr + Sgr, Tr + Soc, Tr + TTr and finally, ν g is estimated using : ν g = Sum Tr SumTr (7) Validation In order to test the performance of our proposed IPR model we compared the performance of the proposed model to two validation cases. The first case is a synthetic case based on rock, rock-fluid, and fluid properties similar to those cases we constructed to generate the correlations. This approach was used because we were unable to find a suitable field data case in the published literature for which to compare our proposed IPR model. Both gas and condensate performance was compared for this synthetic case. We did find a single literature case with which we could compare our proposed IPR model unfortunately, only the gas performance data were available for this case (Xiong, et al. 17 ). For the synthetic case, sets of gas and condensate IPR curves were developed and correlated to obtain the appropriate ν o and ν g values (along with the q g,max, q o,max, and p avg variables). For the synthetic oil case the ν o was determined to be 0., while the GRACE model (using quadratic transform functions) predicted 0.8. For the synthetic gas the ν g estimate was 0.54 while the GRACE model predicted These values are arguably a subset of the database used to generate the correlations we should therefore accept the model estimates as reasonable. In order to estimate ν g for the Xiong, et al. 17 data it was necessary to fit Eq. 3 to this dataset. Upon regression we obtained an estimate of 0.11 for ν g. In comparison, the GRACE model predicted a ν g estimate of We conclude that these (admittedly limited) cases validate the concept and encourage utilization of the proposed IPR model for gas condensate systems. Summary and Conclusions 1. Vogel IPR Correlation Model: The generalized form of the Vogel IPR model (Eq. 3) can be used for the performance prediction of oil, gas, and condensate systems. The validity of this model is well-established empirically, and there are also semi-analytical validations of this model for solution gas-drive systems.. IPR Behavior for Gas Condensate Systems: This work has validated the concept of correlating the IPR behavior for gas condensate systems for both the gas and condensate phases. This "correlation" of behavior was based on the use of Eq. 3 as the base model. We were able to generate representative "dimensional" and "non-dimensional" IPR trends for every case considered in this work (6 cases total). 3. IPR Correlations for Gas Condensate Systems: In this work we used Eq. 3 as the IPR model and correlated the resulting ν γ and ν ο parameters using non-parametric regression to identify the most influential parameters in particular: T, p d, S oc, S gr, M mix,i, and the groups C1, C, C7+. Using the results of the non-parametric correlations, we were able to generate correlations (based on fitting functional approximations to the transform functions generated by the non-parametric regression algorithm) for both the gas and condensate phases. These correlation, while somewhat cumbersome, are considered to be both representative and accurate. Nomenclature B g = Gas formation volume factor, scf/rcf B o = Condensate formation volume factor, RB/STB h = Formation thickness, ft k = Formation permeability, md k rg = Relative permeability to gas, fraction k ro = Relative permeability to condensate, fraction J θ = Residual function for optimization, fraction q g = Gas flowrate, MSCF/D q g,max = Maximum gas flowrate, MSCF/D q o = Condensate (oil) flowrate, STB/D q o,max = Maximum condensate (oil) flowrate, STB/D G = Original gas-in-place, MSCF N = Original condensate (oil)-in-place, STB p bar = Average reservoir pressure, psia (text and figures) p = Average reservoir pressure, psia (text and figures) p d = Dewpoint pressure, psia p wf = Flowing bottomhole pressure, psia r e = Reservoir drainage radius, ft r w = Wellbore radius, ft R s = Solution gas-oil ratio, scf/stb S o = Condensate saturation, fraction S oc = Critical condensate saturation, fraction S gr = Residual gas saturation, fraction T = Reservoir temperature, Deg F s = Radial flow skin factor, dimensionless µ g = Gas viscosity, cp µ o = Condensate viscosity, cp ν g = Gas IPR parameter (for Eq. 3)

12 1 R.A. Archer, Y. Del Castillo and T.A. Blasingame SPE ν o = Condensate IPR parameter (for Eq. 3) References 1. Rawlins, E.L. and Schellhardt, M.A.: Backpressure Data on Natural Gas Wells and Their Application to Production Practices, Monograph Series, USBM (1935) 7.. Gilbert, W.E.: "Flowing and Gas-Lift Well Performance," Drill. and Prod. Prac., API (1954) Vogel, J.V.: "Inflow Performance Relationship for Solution-Gas Drive Wells," paper SPE 1476 presented at the 1968 SPE Annual Fall Meeting, Dallas, October Fetkovich, M.J.: "The Isochronal Testing of Oil Wells," paper SPE 459 presented at the 1973 SPE Annual Fall Meeting, Las Vegas, Nevada, 30 September-3 October. 5. Richardson, J.M. and Shaw A.H: "Two-Rate IPR Testinga Practical Production Tool," JCPT, (March-April 198) Wiggins, M.L., Russell, J.E., Jennings, J.W.: "Analytical Development of Vogel-Type Inflow Performance Relationships," SPE Journal (December 1996) Fussell, D.D.: "Single-Well Performance Predictions for Gas Condensate Reservoirs," JPT (July 1973) O'Dell, H.G. and Miller, R.N.: "Successfully Cycling a Low Permeability, High-Yield Gas Condensate Reservoirs," JPT (January 1967) Fevang, O and Whitson, C.H.: "Modelling Gas-Condensate Well Deliverability," paper SPE presented at the 1995 SPE Annual Technical Conference and Exhibition, Dallas, -5 October. 10. Malachowski, M.A., Yanosik, M.A., and Saldana, M.A.: "Simulation of Well Productivity Losses Due to Near Well Condensate Accumulation in Field Scale Simulations," paper SPE presented at the 1995 SPE Annual Technical Conference and Exhibition, Dallas, Texas, -5 October. 11. Roussennac, B.: Gas Condensate Well Analysis, M.S. Thesis, Stanford University, June Whitson, C.H., and Brule, M.R.: Phase Behavior, Monograph Series, SPE, Richardson, Texas (000) 0, Jaramillo, J.M.: Vertical Composition Gradient Effects on Original Hydrocarbon in Place Volumes and Liquid Recovery for Volatile Oil and Gas Condensate Reservoirs, M.S. Thesis Texas A&M University. December 000, College Station. 14. Guerra, A.: Analysis of the Dynamics of Saturation and Pressure Close to the Wellbore for Condensate Reservoirs as a Tool to Optimize Liquid Production, M.S. Thesis Texas A&M University. May 001, College Station. 15. MS EXCEL (Office 10), Microsoft Corporation (00). 16. Xue, G., Datta-Gupta, A., Valko, P., and Blasingame, T.A.: "Optimal Transformations for Multiple Regression: Application to Permeability Estimation from Well Logs," SPEFE (June 1997), Xiong, Y., Sun, Le., Sun Li., and Li S.: "A New Method for Predicting the Law of Unsteady Flow Through Porous Medium on Gas Condensate Well," paper SPE presented at the 1996 SPE Program Conference, Calgary, Canada, 8 April- 1 May. Acknowledgements The first author would like to acknowledge that this work was performed at Texas A&M University.

13 SPE 8058 New perspectives on Vogel Type IPR Models for Gas Condensate and Solution-Gas Drive Systems 13 Appendix Table A-1 Molar compositions for the Cusiana reservoir fluid sample (Fluid 5) (from ref. 13). Mole fraction Molecular weight Components ž i M i M i ž i N C CO C C IC NC IC NC Benzene C Toluene C C C C C C C C C C C C C C C C C C C C C C C C M mix Table A- Separator test data (54 deg F) for the Cusiana reservoir fluid sample (Fluid 5) (from ref. 13). Gas Specific Gravity Pressure Temperature GOR (air = 1.0) (psig) ( o F) (scf/stb) γ g Table A-3 Pseudocomponents for the Cusiana reservoir fluid sample (Fluid 5) (from ref. 13). Critical Pseudo- Mole Molecular Pressure Com- Fraction Weight p c ponent Components ž i M i (psig) CO GRP1 N-C GRP C-C GRP3 IC4 to C GRP4 Toluene to C GRP5 C11 to C GRP6 C17 to C Critical Critical Tem- Critical Com- Pseudo- perature Volume pressibility Com- T c v c Factor ponent Components (deg F) (ft 3 /lb-mole) z i CO GRP1 N-C GRP C-C GRP3 IC4 to C GRP4 Toluene to C GRP5 C11 to C GRP6 C17 to C

14 14 R.A. Archer, Y. Del Castillo and T.A. Blasingame SPE Table A-4 Molar compositions for the Cupiagua reservoir fluid sample (Fluid 6) (from ref. 14). Mole fraction Molecular weight Components ž i M i M i ž i N C CO C C IC NC IC NC Benzene C Toluene C C C C C C C C C C C C C C C C C C C C C C C C M mix Table A-5 Pseudocomponents for the Cupiagua reservoir fluid sample (Fluid 6) (from ref. 14). Critical Pseudo- Mole Molecular Pressure Com- Fraction Weight p c ponent Components ž i M i (psia) GRP1 N - C GRP CO - C GRP3 C3 to NC GRP4 IC5 to Toluene GRP5 C7 to C GRP6 C11 to C GRP7 C18 to C Critical Critical Tem- Critical Com- Pseudo- perature Volume pressibility Com- T c v c Factor ponent Components (deg F) (ft 3 /lb-mole) z i GRP1 N - C GRP CO - C GRP3 C3 to NC GRP4 IC5 to Toluene GRP5 C7 to C GRP6 C11 to C GRP7 C18 to C

15 Table A-6 IPR Correlation Results from Compositional Reservoir Simulation Gas Condensate Reservoir Systems. N/G Mole Fraction M mix k r Fluid S oc S gr p dew T (STB/ C1 C1-C3 C4-C6 C7 + ρ init (lb m / v o v g Case set set (frac) (frac) (psi) (deg F) MSCF) (frac) (frac) (frac) (frac) (lb m /ft 3 ) lb-mole) (dim-less) (dim-less)