IMPROVING THE NUMERICAL ACCURACY OF HYDROTHERMAL RESERVOIR SIMULATIONS USING THE CIP SCHEME WITH THIRD-ORDER ACCURACY

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1 PROCEEDINGS, Thirty-Seventh Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, January 30 - February 1, 2012 SGP-TR-194 IMPROVING THE NUMERICAL ACCURACY OF HYDROTHERMAL RESERVOIR SIMULATIONS USING THE CIP SCHEME WITH THIRD-ORDER ACCURACY Mitsuo Matsumoto Energy & Mineral Resources Department, Idemitsu Kosan Co., Ltd , Toranomon, Minato-ku, Tokyo , Japan mitsuo.matsumoto@si.idemitsu.co.jp ABSTRACT We have developed a numerical algorithm using the CIP scheme, or its enhanced version, the RCIP scheme, for improving the numerical accuracy when simulating fluid flows with a high gradient of temperature or chemical concentration in hydrothermal reservoirs. The CIP and RCIP schemes achieve third-order accuracy, and they can effectively control the numerical diffusion that often causes serious numerical errors when using the conventional first-order upstream-difference scheme. The developed algorithm consists of the following two procedures: 1. Time integration of the equations governing the conservation of mass and enthalpy; 2. Determining the thermodynamic functions and their derivatives by solving the systems of nonlinear and linear equations. This paper presents practical numerical techniques by describing the numerical algorithms using not only the CIP or RCIP scheme but also the conventional first-order upstream-difference scheme, for reference purposes. Using these algorithms, several numerical simulations of one-dimensional cold sweep are demonstrated. From the numerical solutions, we conclude that: 1. the algorithm using the CIP or RCIP scheme can effectively control the numerical diffusion; 2. the algorithm using the RCIP scheme yields more realistic solutions by controlling the overshoot that is caused when using the CIP scheme; 3. spatial steps that are four to eight times longer can be used without compromising numerical accuracy, by replacing the conventional firstorder upstream-difference scheme with the RCIP scheme. INTRODUCTION Subsurface fluid flows with a high gradient of temperature or chemical concentration often appear in a hydrothermal reservoir. One example is a cooling front propagated by the injection of cold water into the reservoir cold sweep. Accurate simulations of this effect are important for predicting the productivity of reservoirs during the operation of geothermal power plants. Another example is the flow of a tracer injected into the reservoir. Because the tracer did not exist originally, the injected tracer forms a sharp concentration peak in the reservoir. Simulating the tracer flow accurately and matching it with the observed changes in tracer concentration at the observation wells provide us with useful constraints when constructing a reliable numerical model of the reservoir. Although these high-gradient flows play an important role, attempts to simulate them often entail overcoming the drawback of insufficient numerical accuracy. The transportation of enthalpy or a chemical component due to advection is governed by the advection equation. This equation is known as one of the hyperbolic-type partial differential equations that are difficult to solve accurately using conventional numerical techniques such as the firstorder upstream-difference scheme. The major origin of the numerical error is the numerical diffusion that severely blunts the sharp gradient and yields failed numerical solutions. Improving the numerical accuracy is essential for simulating high-gradient flows. We have improved the numerical accuracy when simulating the high-gradient flow in hydrothermal reservoirs by using the constrained interpolation profile CIP scheme Takewaki et al., 1985 and its enhanced version, the rational function CIP RCIP scheme Xiao et al., Compared to the conventional first-order upstream-difference scheme,

2 In an idealized hydrothermal system composed of pure water and rocks, the mass of fluid per unit volume M is expressed as follows: 1 Figure 1: Solutions of the advection equation. The red, yellow, and blue lines denote the analytic solution AS, numerical solution obtained using the CIP scheme CIP, and that obtained using the firstorder upstream-difference scheme USD, respectively. The time and spatial step lengths are 0.01 and 0.025, respectively. these schemes can control the numerical diffusion effectively by achieving third-order accuracy Figure. 1. This paper presents practical techniques for applying the CIP and RCIP schemes to hydrothermal reservoir simulations. First, we describe the equations governing the subsurface fluid flow and physical properties of water. Second, we describe numerical algorithms using not only the CIP or RCIP scheme but also the conventional first-order upstreamdifference scheme for reference purposes. Finally, we demonstrate several numerical simulations of a onedimensional problem involving cold sweep, and we show how the CIP and RCIP schemes effectively improve the numerical accuracy. GOVERNING EQUATIONS Conservation of Mass and Enthalpy We adopt the mathematical model proposed by Faust and Mercer 1979 for describing the subsurface fluid flow. This model is also adopted in the reservoir simulator HYDROTHERM Kipp et al., 2008, which is a well-known multi-purpose simulator. In this model, we regard the rocks as porous media. We assume that fluid exists in the pores of rocks. The fluid flow is governed by Darcy s law. In this paper, we consider only one-dimensional problems in the Cartesian coordinate system, and for simplicity, we assume that the physical properties of rocks are constant. The numerical techniques described in this paper, however, can be immediately applied with minor modifications to multi-dimensional problems entailing variable physical properties of rocks. where ϕ and ρ f represent the porosity of rocks and density of fluid, respectively. The total enthalpy per unit volume H, including the enthalpy of fluid and rocks, is defined as follows: 2 where h, ρ r, c r, and T represent the specific enthalpy of fluid, density of rocks, specific heat capacity of rocks, and temperature, respectively. The first and second terms of the right-hand side of Eq. 2 represent the enthalpy of fluid and that of rocks, respectively. The principles of conservation of mass and enthalpy govern the changes in the mass of fluid M and total enthalpy H, respectively, as follows: 3 4 where t, x, u, λ, q M, and h src represent the time, coordinate, volumetric flux of fluid, thermal conductivity, mass flow rate of a source, and specific enthalpy of the fluid flowing from a source into the reservoir, respectively. The volumetric flux of fluid obeys Darcy s law. Physical Properties of Water For determining the physical properties of water, we have referred to the empirical equation of state IAPWS-95 IAPWS, 1996 and the empirical equation for calculating the viscosity IAPWS, 1997, developed by the International Association for the Properties of Water and Steam IAPWS. Although these empirical equations can simulate the physical properties of water with high accuracy, the computational load is too large to calculate them in numerical simulations. To speed up the calculation of the physical properties of water, we have developed the electronic steam table GANSEKI Matsumoto, GANSEKI provides a module that can be imported to an arbitrary numerical code written in Fortran 90/95, and it approximates the pressure, specific enthalpy, and viscosity of water on the basis of the abovementioned empirical equations. A comparison of the calculations using GANSEKI and the original empirical equations is shown in Figure 2. As shown

3 Figure 2: Changes in pressure, specific enthalpy, and viscosity of water depending on density and temperature calculated using the electronic steam table GANSEKI and the empirical equations developed by IAPWS. in Figure 2, the independent variables of GANSEKI are density and temperature. When developing GANSEKI, we divided the density-temperature space into a number of rectangular cells, whose lengths are 0.1 to 46.5 kg/m 3 and 0.8 to 20.0 deg. C. At the four corners of each cell, the values of the functions to be approximated and their derivatives are memorized in the database. Then, changes in the functions in each cell are approximated using bicubic interpolation. The relative error of this approximation is at most %. The total data size of GANSEKI is 7.8 MB. The calculation load of GANSEKI is less than one-tenth of that of the original empirical equations. ALGORITHM 1: FIRST-ORDER UPSTREAM- DIFFERENCE SCHEME In this section, we describe the numerical algorithm, using the conventional first-order upstreamdifference scheme for reference purposes. As shown in a later section, the numerical solutions using this algorithm coincide with those obtained using the well-known reservoir simulator HYDROTHERM. This algorithm consists of the following two procedures: 1. Integrating the discretized forms of Eqs. 3 and 4 explicitly with time, for yielding the mass of fluid M i n+1 and total enthalpy H i n+1 at the next time step, where i and n are the indexes of time and spatial steps, respectively; n+1 2. Substituting M and H for M i and H n+1 i, respectively, in Eqs. 1 and 2 and solving the system of nonlinear equations formed by these two equations numerically for determining the thermodynamic functions of water at the next time step. Time Integration In the first procedure, we adopt the time-splitting method and integrate the advection and nonadvection terms of Eqs. 3 and 4 separately as follows: For the advection terms: 5

4 6 18 For the non-advection terms: 19 where where Δt, Δx, k, and μ represent the time step length, spatial step length, permeability of rocks, and viscosity of fluid, respectively. The viscosity of fluid is a function of fluid density and temperature, and it can be computed using the electronic steam table GANSEKI. The non-advection terms Eqs. 7 to 9 are integrated with time using the finite difference method as follows: When integrating the advection terms Eqs. 5 and 6 with time, we adopt the first-order upstreamdifference scheme as follows: with The total enthalpy H i n+1 at the next time step can be computed by summing up as follows: 24 Thus, the time integration of Eqs. 3 and 4 can be reduced to that of the advection Eqs. 5 and 6 and non-advection Eqs. 7 to 9 equations. The source of the serious numerical error due to numerical diffusion is the time integration of Eqs. 5 and 6. As shown later, we intend to overcome this problem by improving the numerical accuracy when integrating these two equations by using the CIP or RCIP scheme. We have adopted explicit time integration to simplify the numerical algorithms, although the available time step lengths are shorter by several orders of magnitude, as compared to implicit time integration. We have preferred the simplicity of the numerical algorithms to shorter calculation time for the first stage of our attempt to apply the CIP and RCIP schemes. We intend to adopt implicit time integration to reduce the calculation time after verifying the adoptability of these schemes.

5 Determining the Thermodynamic Functions In the second procedure, we determine the thermodynamic functions, such as pressure P i n+1 and temperature T i n+1, by using the mass of fluid M i n+1 and total enthalpy H i n+1 determined in the previous procedure. We assume that the fluid density ρ f,i n+1 and temperature T i n+1 are the independent unknowns and that the other variables are functions depending on these unknowns. Since the porosity of rocks ϕ is assumed to be constant, we can immediately obtain the fluid density ρ f,i n+1 from Eq. 1. Thus, we can reduce Eqs. 1 and 2 to a single nonlinear equation whose unknown is solely temperature T i n+1, as follows: where where we regard the enthalpy of fluid h as the function of the already obtained fluid density ρ f,i n+1 and temperature T i n+1. We can compute this function using the electronic steam table GANSEKI. After solving Eq. 25 using the Newton-Raphson method, pressure P i n+1 is determined using GANSEKI again. Finally, the enthalpy of fluid H f,i n+1 and that of rocks H r,i n+1 defined in Eqs. 11 and 12 are computed using the determined thermodynamic functions. ALGORITHM 2: CIP AND RCIP SCHEMES Time Integration Next, we describe the numerical algorithm using the CIP or RCIP scheme by modifying the algorithm presented in the previous section. In the first procedure, we consider not only Eqs. 3 and 4 but also the derivatives of Eqs. 3 and 4 as follows: where By using the CIP scheme, the pair of advection equations 5 and 30 can be integrated with time as follows: where By using the time-splitting method, the time integration of Eqs. 26 and 27 can be reduced to that of the following advection and non-advection equations, 45

6 46 47 equations 32 to 34 are integrated using the finite difference method as follows: The pair of advection equations 6 and 31 can be integrated in the same manner. We also consider the RCIP scheme that can control over- and undershoot effectively. To pdopt the RCIP scheme, we replace Eqs. 43, 44, 46, and 47 as follows: 60 with Note that the finite differences of Eqs. 21 to 23 in space are used for calculating α, β f, and β r in Eqs. 58 to 60. The derivative of the total enthalpy n+1 Ψ i at the next time step can be obtained by summing up as follows: Determining the Thermodynamic Functions and Their Derivatives In the second procedure, we consider not only Eqs. 1 and 2 but also the derivatives of Eqs. 1 and 2 as follows: where if Otherwise,. The non-advection equations 7 to 9 are integrated as shown in Eqs. 21 to 23. The non-advection 63 The derivatives of the enthalpy of fluid are determined using the electronic steam table GANSEKI. Since all the thermodynamic functions are determined by solving Eq. 25, we can solve the system of linear equations formed by Eqs. 62 and n+1 63 for δ i and τ n+1 i. Finally, the derivatives Ψ n+1 f,i and Ψ n+1 r,i defined in Eqs. 35 and 36 are computed using the determined thermodynamic functions and their derivatives.

7 NUMERICAL SIMULATIONS Conditions We present several numerical simulations using the abovementioned numerical algorithms. In these simulations, we considered a one-dimensional problem of cold sweep in a hydrothermal reservoir, as shown in Figure 3. We simulated the change in pressure and temperature in a one-dimensional space whose length was 2.0 km. We assumed that the physical properties of rocks are constant, as shown in Table 1. The initial pressure and temperature were uniformly 20 MPa and 250 deg. C, respectively. Cold water with a specific enthalpy of 550 kj/kg the specific enthalpy of saturated water at 131 deg. C was injected at a source located 800 m away from the left boundary. The flow rate of the source was kg/s/m 2. The pressure and temperature at the two boundaries remained constant at the initial values. Numerical Solutions and Discussions Figure 4 shows the numerical solutions of the abovementioned problem obtained using the algorithms with the first-order upstream-difference, CIP, and RCIP schemes. For reference purposes, each of these solutions overlays the numerical solution obtained using the well-known reservoir simulator HYDROTHERM under the same conditions. The common time and spatial step lengths were years and 100 m, respectively. Since the injection of cold water at the source, the cooling fronts at which the temperature begins to decrease from the initial value propagate to both sides of the one-dimensional space. Most of the space has been cooled down over 30 years. The numerical solution obtained using the algorithm with the first-order upstream-difference scheme is compared with that obtained using HYDROTHERM Figure 4a. Both solutions precisely coincide with each other. This ensures the validity of the numerical solutions using the first-order upstream-difference scheme. Figures 4b and 4c show the numerical solutions obtained using the algorithm with the CIP and RCIP schemes, respectively, with third-order accuracy. Both solutions simulate the temperature profiles at the cooling fronts more sharply than the solution with the first-order upstream-difference scheme Figure 4a by controlling the numerical diffusion effectively. The RCIP scheme yields a more realistic solution than the CIP scheme by controlling the overshoot at the cooling fronts. We have confirmed the validity of the numerical solution obtained using the RCIP scheme by studying the convergence of the numerical solutions. Figures 5 and 6 show the simulated temperature profiles using the RCIP scheme with a spatial step length of 100 m and the first-order upstream-difference scheme with Figure 3: One-dimensional problem for the simulations. Table 1: Physical properties of rocks for the simulations. Property Value Porosity 0.10 Density kg/m 3 Permeability m 2 Thermal conductivity Specific heat capacity Table 2: 1.0 W/m/K J/kg/K Pairs of time and spatial step lengths for the simulations. Time step length [years] Spatial step length [m] spatial step lengths of 12.5 to 100 m. Refer to Table 2 for the time step lengths used for various spatial step lengths. We find that the spatial and temporal profiles of temperature simulated using the first-order upstream-difference scheme converge with that simulated by using the RCIP scheme as the time and spatial step lengths decrease. From this convergence, we can conclude that the numerical solution obtained using the RCIP scheme is more accurate than that obtained using the first-order upstream-difference scheme with the common spatial step length. Figures 5 and 6 show that the RCIP scheme can yield a simulation with accuracy that is comparable with that of the first-order upstream-difference scheme, even if we adopt spatial steps that are four to eight times longer. This advantage becomes the critical factor, especially when performing three-dimensional simulations, because the number of grid points can be reduced by a factor of several tens to hundreds by adopting the RCIP scheme instead of the first-order upstream-difference scheme. The lack of numerical accuracy possibly results in seriously incorrect predictions of the change in the reservoir temperature. As shown in Figure 5, the location of the cooling front differs by a few hundred

8 meters in the profiles simulated using the RCIP and first-order upstream-difference schemes with the same spatial step length of 100 m. The temperature difference between the two profiles is up to around 20 deg. C. The time when the temperature begins to decrease i.e., when the cooling front passes also differs by around ten years, as shown in Figure 6. Such differences can be a critical factor in predicting the change in the productivity of hydrothermal reservoirs and in studying the feasibility of operating a geothermal power plant based on the numerical reservoir model. CONCLUSIONS From the numerical solutions presented in this paper, Figure 4: Changes in pressure and temperature simulated using the first-order upstream-difference a, CIP b, and RCIP c schemes. The numerical solution under the same condition obtained using the reservoir simulation HYDROTHERM is also shown for reference purposes. The time and spatial step lengths are years and 100 m, respectively.

9 Figure 5: Temperature profiles 15 years after the initial state obtained using the RCIP scheme RCIP with a spatial step length of 100 m and the first-order upstream-difference scheme USD with spatial step lengths of 12.5 to 100 m. we can conclude that: 1. the developed algorithm using the CIP and RCIP schemes can effectively control numerical diffusion; 2. the algorithm using the RCIP scheme yields more realistic solutions by controlling the overshoot that appears using the CIP scheme; 3. we can use spatial steps that are four to eight times longer without compromising numerical accuracy, by replacing the conventional firstorder upstream-difference scheme with the RCIP scheme. REFERENCES Faust, C. R., and Mercer, J. W Geothermal Reservoir Simulation 1 Mathematical Models for Liquid- and Vapor-Dominated Hydrothermal Systems, Water Resources Res., 15, IAPWS 1996 Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance of General and Scientific Use, 18p. IAPWS 1997 Revised Release on the IAPS Formulation 1985 for the Viscosity of Ordinary Substance, 15p. Kipp, K. L., Jr., Hsieh, P. A., and Charlton, S. R Guide to the Revised Ground-Water Flow and Heat Transport Simulator: Figure 6: Changes in temperature 1,500 m away from the left boundary obtained using the RCIP scheme RCIP with a spatial step length of 100 m and the first-order upstream-difference scheme USD with spatial step lengths of 12.5 to 100 m. HYDROTHERM Version 3, USGS Techniques and Methods, 6-A25, 160p. Matsumoto, M Numerical Simulation of Two-phase Hydrothermal System Using a Simple Explicit Scheme and Parallel Computing Development and Application of the Electronic Steam Table GANSEKI, J. Geotherm. Res. Soc. Japan, 33, in Japanese with an English abstract. Takewaki, H., Nishiguchi, A., and Yabe, T The Cubic-Interpolated Pseudo-Particle CIP Method for Solving Hyperbolic-Type Equations, Journal of Computational Physics, 61, Xiao, F., Yabe, T., and Ito, T Constructing Oscillation Preventing Scheme for Advection Equation by Rational Function, Comput. Phys. Commn., 93, 1-12.