Far East Journal of Applied Mathematics

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1 Far East Journal of Applied Mathematics Volume, Number, 29, Pages This paper is available online at 29 Pushpa Publishing House EVELOPMENT OF SOLUTION TO THE IFFUSIVITY EQUATION WITH PRESCRIBE-PRESSURE BOUNARY CONITION AN ITS APPLICATIONS TO RESERVOIRS EXPERIENCING STRONG WATER INFLUX ASEP KURNIA PERMAI Reservoir Engineering Research Group Faculty of Mining and Petroleum Engineering Bandung Institute of Technology Jalan Ganesa Bandung 432, Indonesia Abstract oublet and Blasingame [3] introduced a semi-analytical solution to the radial single-phase diffusivity equation with prescribed-flux boundary condition. However, the influx rate is practically more difficult to predict than the pressure and this difficulty has been restricting the application of the solution. Permadi et al. in [6-8, ], demonstrated this limitation in predicting the performance of a reservoir experiencing natural water influx or water injection. For that reason, a diffusivity equation with prescribed-pressure boundary condition may be more appropriate to model such a case. This paper introduces the latest development of the prescribed-pressure solution to such a non-linear partial differential equation and demonstrates its application to reservoirs with strong water influx. It is shown that the solution can appropriately model the reservoir behavior which is in fact very common in practice especially in Indonesian oil fields. 2 Mathematics Subject Classification: Kindly provide. Keywords and phrases: Kindly provide. Received August, 29

2 2 ASEP KURNIA PERMAI Nomenclature B = formation volume factor, RB/STB c t = total compressibility, psi - h = formation thickness, ft k = permeability, md p = pressure, psia p ext = external boundary pressure, psia p i = initial reservoir pressure, psia p wf = flowing bottomhole pressure, psia khδp p = = 4.2qBμ dimensionless pressure function kh p ext = [ p i p ext ] dimensionless external boundary pressure 4.2qBμ Δ p = pressure drop in the reservoir, psia q = production rate, STB/day q = 4.2qBμ kh( p ) = i p wf dimensionless rate function r = radial distance, r-direction r = r = dimensionless radius r w r e = reservoir drainage radius, ft r = r = dimensionless reservoir drainage radius e e r w r w = wellbore radius, ft t = time, days.6237kt t = = 2 φμ c t r w dimensionless time function μ = fluid viscosity, cp φ = porosity, fraction

3 EVELOPMENT OF SOLUTION TO THE IFFUSIVITY EQUATION 3. Introduction A semi-analytical solution to the radial single-phase flow equation in porous media (commonly called in the petroleum industry as the radial single-phase diffusivity equation) was developed by oublet and Blasingame []. The physical reservoir model of this solution was a single well centered in a bounded circular reservoir with a constant production rate at the inner boundary and prescribed flux at the outer boundary. Through the use of decline type curves, the authors demonstrated that the solution could be applied for analysis and interpretation of production data from reservoirs experiencing natural water influx or pressure support via water injection. They claimed that their solution, though derived and implemented as a single-phase solution, has been successfully validated with results from numerical simulation and field performance data. However, the use of a prescribed-flux boundary leads to some kind of difficulty in its application. The injection rate is practically more difficult to control than the injection pressure and the influx rate is usually more difficult to predict than the pressure. This major limitation of the so-called prescribed flux model has been shown by Permadi et al. in [6-8]. Also, its application to model a five-spot water injection that had been explored by Rawati and Permadi [] indicated that for certain cases the solution gave unsatisfactory results. Above all, the use of aquifer pressure rather than influx rate is more common and useful in petroleum reservoir engineering definition and calculations. The objective of this paper is to introduce the latest development of the semianalytical solution to the single-phase diffusivity equation when prescribed-pressure at the outer boundary is imposed. It is shown in this paper that the solution could be fairly straightforward to develop and is practically applicable using the methodology commonly used in the petroleum industry. 2. evelopment of Solution The system for which the solution was developed consists of a single well located at the center of a circular reservoir with prescribed-pressure condition at the outer boundary. Therefore, when the well is producing at a constant rate, its initialboundary value problem (IBVP) in dimensionless variables may be written as follows (see Permadi [6]):

4 4 ASEP KURNIA PERMAI r r r p r = p. t (2.) Initial condition: Uniform pressure distribution p ( r, t ), r r, t =. (2.2) = e Inner boundary condition: Constant rate production p r =, r =, >. t r (2.3) Outer boundary condition: Prescribed-pressure p ( r, t ) p ( t ), r = r, t >. (2.4) = ext e Analogous models to those used by oublet and Blasingame were applied for modeling the outer boundary pressure. In this case, we may use models of: Constant pressure condition (constant at initial pressure) p ( ) =. (2.5) ext t Step pressure condition (impulse change of pressure) pext ( t ) = p U ( t t ). (2.6) ext, start Ramp pressure condition (smooth change of pressure) pext ( t ) = p [ exp( t t )]. (2.7) ext, start Note that the negative signs in Equation (2.6) and Equation (2.7) are required because the flow is across the boundary, toward the well. The dimensionless external boundary pressure, p ext ( t ), can be transformed to Laplace domain in the following way: ut pext ( u) = L [ pext ( t )] = pext ( t ) e dt. (2.8) The solution to this so-called prescribed-pressure model had been introduced by Permadi [6]. However, details of the development of the solution had not been presented in the literature until the publication by Permadi and amargalih [9] in 2. The most recent status of this development is denoted by the work of de Jong and Permadi [2]. The following is the summary of the derivation of the solution.

5 EVELOPMENT OF SOLUTION TO THE IFFUSIVITY EQUATION General solution Applying the Laplace transformation to Equation (2.), see van Everdingen and Hurst [3], results in an ordinary differential equation as follows: r r r p r = up p ( r, t = ). (2.9) The second term on the right hand side of Equation (2.9) vanishes in satisfying the initial condition. This leaves the equation that is exactly in the same form as the following Modified Bessel Equation (see Appendix B of Lee et al. [5]): d 2 dx which has the general solution of y 2 dy + ηy = x dx (2.) y = AI ( ηx) + BK ( η ). (2.) x Therefore, the general solution to the above IBVP is given by p ( r u) = AI ( ur ) + BK ( ur ). (2.2) 2.2. Particular solution, To obtain the particular solution, we will find coefficients A and B of Equation (2.2) subject to the specific inner and outer boundary conditions. Recall recurrence relationships for the first derivatives of the Modified Bessel Functions as follows: From Rice and o [] on page 37, we have d dx p ηz ( ) ( ) p+ ηx + Z ηx, Z = J, Y, K x p [ Z p ( ηx)] = (2.3) p ηz ( η ) + ( η ) = p+ x Z x, Z x p or from Appendix B of Lee et al. [5] we also have d dx d dx [ f ( x) ] = f ( x) I[ f ( )], (2.4a) I x [ f ( x) ] = f ( x) K[ f ( )]. (2.4b) K x

6 6 ASEP KURNIA PERMAI Using these relations, after multiplying through by r we have from Equation (2.2) r dp dr = A ur I ( ur ) B ur K ( ur ). (2.5) Now, applying Laplace transform to the inner boundary condition gives p r r r = =. u (2.6) Satisfying this condition, we substitute Equation (2.6) to Equation (2.5) to obtain p r r r = = A ui( u ) B uk( u ) =. u (2.7) Now, from Equation (2.4) and Equation (2.8), the outer boundary condition is given by p ( re, u) = pext ( u). (2.8) Satisfying this condition, we substitute Equation (2.8) to Equation (2.2) to obtain ( r u) = AI ( ur ) + BK ( ur ) p ( ). (2.9) p e, e e = ext u We solve Equation (2.7) and Equation (2.9) simultaneously for A and B and we obtain A = K ( u ) p ( u) ext K( u ) I( ure ) + K( ure ) I( u ) + u I( ure ) K( u ) ui ( u )[ K ( u ) I ( ur ) + K ( ur ) I ( u )] e e ui ( ), (2.2) u u B = I ( u ) p ( u) ext K( u ) I( ure ) + K( ure ) I( u ) I( ur ) + e u u[ K ( u ) I ( ur ) + K ( ur ) I ( u )]. (2.2) e e

7 EVELOPMENT OF SOLUTION TO THE IFFUSIVITY EQUATION 7 Now, substitute Equation (2.2) and Equation (2.2) into Equation (2.2) to obtain the final form of the particular solution of p( r, u) = I( ure ) K( ur ) K( ure ) I( ur ) 3 2 u [ I( ure ) K( u ) + K( ure ) I( u )] I ( ) ( ) ( ) ( ) ( ) ur K u K ur I u p u + + ext [ I ( ur ) K ( u ) + K ( ur ) I ( u )]. (2.22) At the well, i.e., at r =, the solution is given as e e p(, u) = I( ure ) K( u ) K( ure ) I( u ) 3 2 u [ I( ure ) K( u ) + K( ure ) I( u )] I ( ) ( ) ( ) ( ) ( ) u K u + K u I u + pext u [ I ( ur ) K ( u ) + K ( ur ) I ( u )]. (2.23) e 3. Computational Method At this point, we observe that the first part of the right hand side of Equation (2.23) is exactly the constant pressure outer boundary solution that had been presented by van Everdingen and Hurst [3]. Thus, we can write this solution generally as: p ( r =, u) = Constant pressure outer boundary solution + Prescribed-pressure model solution. (2.24) Accordingly, if we have p ( ) =, then the prescribed-pressure model ext t solution portion in Equation (2.23) is eliminated and we have the solution of constant pressure at the outer boundary. This could be a verification of our solution. In the case of constant pressure production, for simplicity in our computational inversion of the Laplace space solution, we assume that the relation given by van Everdingen and Hurst [3] is valid (the complete derivation of this identity can be found in Permadi [6]). We can then convert the constant rate dimensionless pressure solution to the constant pressure dimensionless rate solution in the Laplace domain as follows: q ( u) = 2 u p ( u). (2.25) e

8 8 ASEP KURNIA PERMAI Hence, rather than deal with the tedious algebraic equations in developing the constant pressure dimensionless rate solution, we simply use Equation (2.25). Furthermore, in our computational work, we do not need to substitute directly Equation (2.23) into Equation (2.25) since it requires us to store a much larger expression but we simply program the above sequence into the Laplace transform inversion procedure. In this case, we can use Gaver-Stehfest numerical inversion algorithms (see Stehfest [2]) in which the modified Bessel functions are evaluated using the algorithms provided by Cody and Stoltz []. 4. Application of Solution ecline Type Curves Applying the computational method described in the preceding section, a published work by Permadi and amargalih [9] and an unpublished research report by de Jong and Permadi [2] provided an extensive development of decline type curves that are commonly called Fetkovich-style type curves [4]. For the constant rate production case, these type curves used the following dimensionless pressure and dimensionless time variables as the plotting functions: p p = d, (2.26) ln r e 2 t t = d. (2.27) 2 r ln 2 2 e re For the constant pressure production case, the same dimensionless time function was used while the dimensionless rate function was given by q ln d = q re. (2.28) 2 As mentioned, the dimensionless rate function for constant pressure production case was obtained by converting p ( u) to q ( u) through the use of van Everdingen and Hurst identity, Equation (2.25). To validate these decline type curves, a finite difference numerical simulator was used to model water influx from the aquifer into the reservoir. 5. Results and iscussion Figure shows the validation of the generated decline type curves based on

9 EVELOPMENT OF SOLUTION TO THE IFFUSIVITY EQUATION 9 prescribed-pressure model solution, Equation (2.23), for the case of r = 8, t d, start = and p ext, =.75. For this validation, the finite difference simulation model was a one-dimensional, logarithmically-gridded circular reservoir system with a production well located at its center. The model assumed negligible capillary and gravity effects, uniform initial pressure, homogeneous and isotropic, producing interval is equal to the total formation thickness, no damages in the well, and no wellbore storage effects. The pressure-volume-temperature (PVT) data and oil-water relative permeabilities were set similar to those used by oublet and Blasingame [3], which is a water-wet, black-oil case. e Figure. Type curves validation using finite difference simulator for r = 8, t d, start =, p ext, =.75. Figure 2 displays an example of our decline type curves. It was generated when r e = 4. and we applied a step-pressure condition at the outer boundary, Equation (2.6). Figure 3 shows another set of our decline type curves for the same r e but we applied a ramp-pressure condition, Equation (2.7). The correlating parameters for these two type curves are the timing ( t start ) and strength of aquifer, i.e., pressure at the outer boundary ( p ext ). We can see obviously that the prescribed-pressure model solution depicts two major periods. First, the constant pressure-like period that happens immediately after the pressure declines due to production activity. This is represented by a horizontally straight-line curve. Second, the secondary depletion period that happens after the first part ends at a certain time. This is represented by a curved declining-stem. The constant pressure-like part happens where the q d at e

10 ASEP KURNIA PERMAI the well is constant resulted from the full support of the influx. The start of the second period will actually depend on the start of pressure at the reservoir boundary to decline ( t d, start ). Therefore, in the second period, the driving energy of the influx gives no more full support to drain the fluid in the reservoir. Referring to the unsteady-state theory of van Everdingen and Hurst [3], the secondary depletion represents the declining pressure at the reservoir-aquifer boundary that has been felt at the outer boundary of the aquifer. At this point, the flux into the reservoir has reached over its maximum value. Figure 2. Type curves for r = 4. using step pressure condition. e Figure 3. Type curves for r = 4. using ramp pressure condition. e

11 EVELOPMENT OF SOLUTION TO THE IFFUSIVITY EQUATION When the boundary-dominated flow occurs, i.e., when the outer boundary effect has been felt, the reservoir behavior starts to deviate as the pressure at the reservoirboundary starts to decline. When the pressure drops drastically as represented by step-pressure condition, a spike in the q d curve occurs as shown in Figure 2. When the pressure declines gradually as represented by ramp-pressure condition, a smooth change in the q d curve occurs as shown in Figure 3. If the pressure after the decline is kept constant (this is called prescribed limiting pressure) as in the case examined by de Jong and Permadi [2], then the reservoir experiences secondary depletion and accordingly the steady state condition fails to exist. At this condition, the ratio between the influx rate at the reservoir-aquifer boundary and the production rate at the well is no longer unity. The influx at the reservoir-aquifer boundary could be kept at a certain value as represented by prescribed-flux condition if its pressure declines continuously to provide the required pressure drop. 6. Conclusions. A solution to the diffusivity equation with prescribed-pressure conditions at the outer boundary has been developed. 2. The solution has been validated with results from finite-difference numerical simulation. 3. The solution leads to the development of new sets of decline type curves that could be used as a diagnostic tool for analysis and interpretation of production data from reservoirs experiencing natural water influx or water injection. References [] W. J. Cody and L. Stoltz, Transportable FORTRAN Algorithms for Modified Bessel Functions, Mathematics and Computer Science ivision, Argonne National Laboratory, Argonne, IL 988. [2] M. de Jong and A. K. Permadi, evelopment of decline type curves for reservoirs with strong water influx using prescribed limiting pressure of reservoir-aquifer boundary condition [Translated], Internal Research Report, epartment of Petroleum Engineering, Bandung Institute of Technology, Bandung, Indonesia, 27. [3] L. E. oublet and T. A. Blasingame, ecline curve analysis using type curve: water influx/waterflood cases, Paper SPE 3774 presented at the 995 Annual Technical Conference and Exhibition, allas, TX, October

12 2 ASEP KURNIA PERMAI [4] M. J. Fetkovich, ecline curve analysis using type curves, J. Petroleum Technology, Society Petroleum Engineers, 98. pages. [5] W. J. Lee, J. B. Rollins and J. P. Spivey, Pressure Transient Testing, Society of Petroleum Engineers Textbook Series, Vol. 9, Richardson, TX 23. [6] A. K. Permadi, Modeling simultaneous oil and water flow with single-phase analytical solutions, Ph.. issertation, Texas A & M University at College Station, College Station, TX, 997. [7] A. K. Permadi, Modeling two-phase oil-water flow using single-phase semi-analytical solutions: sensitivity analysis, Journal of Mineral Technology, Bandung Institute of Technology, Bandung, Indonesia, (5) (998). pages. [8] A. K. Permadi,.. Mamora and W. J. Lee, Modeling simultaneous oil and water flow in reservoirs with water influx or water injection using single-phase semianalytical solutions, paper SPE presented at the 998 SPE Asia Pacific Conference on Integrated Modeling for Asset Management, Kuala Lumpur, Malaysia, March. [9] A. K. Permadi and Y. amargalih, ecline type curves for reservoirs with waterflood or water influx using prescribed-pressure models at the reservoir outer boundary, Journal of Mineral Technology, Bandung Institute of Technology, Bandung, Indonesia 2(8) (2). Page no. [] H. Rawati and A. K. Permadi, The application of a radial single-phase semi-analytical solution on a five-spot patterned water injection case [Translated], Internal Research Report, epartment of Petroleum Engineering, Bandung Institute of Technology, Bandung, Indonesia, 26. [] R. G. Rice and.. o, Applied Mathematics and Modeling for Chemical Engineers, John Wiley & Sons, Inc., New York, 995. [2] H. Stehfest, Numerical inversion of Laplace transforms, Communications of the ACM 3() (97), [3] A. F. van Everdingen and W. Hurst, The application of the Laplace transformation to flow problems in reservoirs, Trans. AIME 86 (949),

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