AUTOMATING UP-SCALING OF RELATIVE PERMEABILITY CURVES FROM JBN METHOD ESCALAMIENTO AUTOMÁTICO DE CURVAS DE PERMEABILDIAD RELATIVA DEL MÉTODOS JBN

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1 AUTOMATING UP-SCALING OF RELATIVE PERMEABILITY CURVES FROM JBN METHOD ESCALAMIENTO AUTOMÁTICO DE CURVAS DE PERMEABILDIAD RELATIVA DEL MÉTODOS JBN JUAN D. VALLEJO 1, JUAN M. MEJÍA 2, JUAN VALENCIA 3 Petroleum Engineering, Research Group of Flow and Transport Dynamics in Porous Media, Universidad Nacional de Colombia, jdvallejor@unal.edu.co Ph.D., Associated Professor, Research Group of Flow and Transport Dynamics in Porous Media, Universidad Nacional de Colombia, sede Medellín, jmmejiaca@unal.edu.co M.Sc., Research Group of Flow and Transport Dynamics in Porous Media, Universidad Nacional de Colombia, sede Medellín, jdvalencl@unal.edu.co ABSTRACT Relative permeability curves are a key data when any analysis is performed in a hydrocarbon reservoir. Many methods have been proposed to measure rel perm curves. One of the most complete rel perm measurement methods is the steady state method. However, it is time-consuming and consequently not often used in the industry. One alternative is the unsteady JBN method, based on the fractional flow theory. The JBN method is very simple and efficient, being one of the most used rel perm measurement methods in the oil and gas industry. Although the JBN methods is widely used, one of its major limitations is that measurements are only related to the core exit face. Therefore, the rel perm curves from JBN method are usually not representative of the core. Therefore, significant errors can be introduced in reservoir studies based on rel perm measurements from JBN tests. We present a method for calculating relative permeability curves using an inverse problem approach: we adapted a reservoir simulation tool in order to find the rel perm curves matching both, the measured oil recovery and pressure drop data with the JBN test. The AdRelPerm tool allows for different rel perm models (Corey, LET) and a general shape based on B- Splines interpolation. The results show the efficacy of the method and its potential to estimate a closer permeability curves of a core and minimize the uncertainty reservoir studies. KEYWORDS: History matching, optimization, relative permeability, simulation 1. INTRODUCTION Relative permeability curves largely determine the flow of fluids through porous media and thus they are a key variable in any analysis performed in reservoir engineering. Many methods have been developed for their calculation, which range from core tests to mathematical models and computer simulations. Conventionally, relative permeability curves are obtained from core tests. Steady state method was the first in been introduced. In this one both the wetting and not wetting phase are injected simultaneously at constant rate and constant pressure drop to make sure that a steady state is reached and Darcy equation can be used to calculate the relative permeability [1]. Although this is considered a direct method to calculate the relative permeability curves it has a great defect, it is time consuming because the steady state could take days to be stablished [2]. Because of the above, the unsteady state method emerged. In this one phase is injected into a core saturated with other phase producing a displacement. Data of pressure drop across the core and produced volumes of the phases are recollected to calculate the relative permeability curves performing an adjustment of the measured data with the graphic method JBN [3]. This method is the most used in the oil industry due to its simplicity and efficiency but it comes with some limitations. It is based on the

2 Buckley-Leverett theory so it does not consider the capillary pressure and big mistakes can be committed because that effect is high in the exit face of the core and especially in low permeability cores. Other fact is that the relative permeability curves are representative only in the exit face of the core, so errors are frequently present in high heterogeneous media [4] [5]. To remedy all the limitations described above an inverse problem process is used along with the data from the displacement test to measure the relative permeability curves. With the inverse problem, the parameters of a model are calculated from measured data. In the case of relative permeability curves, it is sought to estimate the parameter of a model that describe the behavior of the curves. 2. METHODOLOGY The inverse problem is based on the minimization of an objective function that compares the measured data from the displacement test and the results of a flow simulation that replicate the test. There are four items to consider in this process: 1. Flow simulator 2. Model that represent the behavior of the relative permeability curves 3. Objective function 4. Optimization algorithm A computational program was developed to automate this process. It was called AdRelPerm Tool Flow Simulator The used flow simulator was developed inside the Research Group of Flow and Transport Dynamics on Porous Media from Universidad Nacional de Colombia. This is an extended black oil simulator capable to work with cartesian and radial meshes developed using Fortran program language Relative Permeability curves models AdRelPerm tool allows to work with three different models: corey, LET and b-splaine. They vary in complexity being corey the simplest one and b-spline the most robust Corey This is the most used model to represent relative permeability curves due to its simplicity. It assumes that the curves represent an exponential shape. w = w(s or ) ( S w S wc ) n w o = o(s wc ) (1 S w S wc ) n o (1) (2) Here n w and n o are the corey exponents of the phases water and oil respectively. Those two coefficients are the ones that the AdRelPerm tool looks for.

3 LET This model was proposed for Lomeland, Ebeltoft, & Thomas [6]. It can represents the s shape that the relative permeability curves can adopt due to the low changes in their value at low and high saturation values. S wn Lw w = w max. (3) S Lw wn +E w (1 S wn ) T w (1 S wn ) L o o = o max. (4) (1 S wn ) L o+e o S To wn S wn = S w S wc (5) There are three parameter to adjust by curve, L w, E w, T w for the water phase and L o, E o, T o for the oil phase Cubic B-Spline The cubic B-spline interpolation can represent curves with a free form. The only imposed condition to the curves is that they must be always decreasing. 1 3 K rl = 1 [ t 3 t² t 1] [ P i P ] [ i ] (6) 3 0 P i P i+2 Here t is a parameter that range from 0 to 1 and depends on saturation. P i are the nodes with which the method interpolates the curve. Those nodes are the adjusting parameters Objective Function The objective function implemented in AdRelPerm tool consider the pressure drop across the core and the cumulative oil recovery of the displacement test. n OF = w(np simulatedi Np measuredi ) 2 + (1 w)(deltap simulatedi Deltap measuredi ) 2 i=1 (7) There W is a weight factor that indicates which data is more important to adjust Optimization Algorithm AdRelPerm tool uses the particle swarm optimization algorithm (PSO) proposed for Eberhart & Kennedy [7] to minimize the objective function. It is based in a number of particles (set of parameter to adjust) which follow a leader particle.

4 3. RESULTS OF VALIDATION To validate the provided results by the program the next methodology was performed: - Create a simulation of a displacement test using the commercial software CMG. The simulation was deployed with two different relative permeability curves, one based on the corey model and the other using the LET model, these are considered as the real curves from their respective test. - From the two simulations, the data necessary to run the AdRelPerm tool is extracted. - The relative permeability curves obtained from the tool are compared with the original ones. The basic information needed to perform the simulations is listed in Table 1. The simulated core is homogenous with constant petrophysic properties. For the two sets of data the AdRelPerm tool was run twice. In the first run the curves end points are known, and in the second run those are unknown. Also in the second test, the curve obtained using the JBN methodology was calculated and compared. Table 1. General information about the core, the displacement test and the relative permeability curves Core Properties Length 58.1 cm Diameter 2.54 cm Absolute Permeability md Porosity Displacement Test Conditions Injection Rate 0.3 cc/min Backpressure 2000 psi Relative Permeability Curves Sor c o (c) 1.0 w (Sor) The results from the first test are show in Figure 1. In both cases the tool was capable to find the right relative permeability curves. Figure 1. (a) Comparison of real and simulated curves knowing the end points using corey model, (b) Comparison of real and simulated curves when the end points are unknown using corey model w real o real w sim o sim w real o real w sim o sim

5 Figure 2. (a) Comparison of real and simulated curves knowing the end points using LET model, (b) Comparison of real and simulated curves when the end points are unknown using LET model w real o real w sim o sim w JBN o JBN w real o real w sim o sim w JBN o JBN Figure 2. shows the results when the LET based relative permeability curves were used. The match between the curves is not as perfect as in the case with corey model but the results are good enough. The maximum discrepancy is presented in the case that that end points are unknown, it evidences one problem with the freedom degrees in the optimization process. Increasing the number of adjusting parameters allows the optimization algorithm to find different solutions, regardless the solution is acceptable compared with the JBN curve obtained. 4. CONCLUSIONS - An automatic computational tool was created to adjust relative permeability curves from JBN test - The results show the accuracy of the tool predicting the relative permeability curves from homogenous cores. - The accuracy decreases when the amount of freedom degrees of the model of the relative permeability curves increase. - The calculated relative permeability curves with the AdRelPerm tool match better than curves from JBN method showing the effectiveness of the program. REFERENCES [1] S. Qadeer, W. Brigham and L. Castanier, "Techniques to Handle Limitations in Dynamic Relative Permeability Measurements," [2] A. Kantzas, Fundamental of Fluid Flow in Porous Media, [3] E. Johnson, D. Bosller and V. Naumann, "Calculation of relative permeability from displacement experiments," Society of Petroleum Engineers, [4] T. Tao and A. Watson, "Accuracy of JBN Estimates of Relative Permeability: Paart 1 - Error Analysis," Society of petroleum engineers, [5] G. Saviolli, M. Bidner and C. Grattoni, "The Influence of Capillary Pressure when Determining Relative Permeability from Unstady-State Corefloods," SPE, [6] F. Lomeland, E. Ebeltoft and W. Thomas, "A New Versatile Relative Permeability Correlation," [7] R. Eberhart and J. Kennedy, "Particle swarm optimization," IEEE Proccedings Neural Networs, 1995.

SPE Copyright 2001, Society of Petroleum Engineers Inc.

SPE Copyright 2001, Society of Petroleum Engineers Inc. SPE 69394 Scaling Up of Laboratory Relative Permeability Curves. An Advantageous Approach Based on Realistic Average Water Saturations M. A. Crotti, SPE, Inlab S.A. and R. H. Cobeñas, SPE, Chevron San