1 B5 A NEW FAST FOURIER TRANSFORM ALGORITHM FOR FLUID FLOW SIMULATION LUDOVIC RICARD, MICAËLE LE RAVALEC-DUPIN, BENOÎT NOETINGER AND YVES GUÉGUEN Institut Français du Pétrole, 1& 4 avenue Bois Préau, 92852 Rueil malmaison, France Abstract Fluid flow simulators are usually based upon finite difference, finite volume or also finite element schemes. As these methods can be CPU-time consuming, we develop an alternative approach involving Fast Fourier Transforms to simulate steady-state, single-phase flow. In this paper, two algorithms are envisioned. They apply to heterogeneous porous media submitted to periodic boundary conditions. Basically, the algorithms are designed to solve sequentially the pressure equation in the frequency space. For the first algorithm, the convergence rate is proportional to the permeability contrast. It can be pretty slow for porous media with high permeability contrasts, such as turbidite systems. Then, a variation is introduced leading to the second algorithm. The convergence rate gets proportional to the square root of the permeability contrast, which results in a significant speed up. The performed numerical tests point out the efficiency and accuracy of the developed FFT-based flow solver. Introduction Geologic models of petroleum reservoirs are usually very detailed and discretized over grids with a huge number of grid blocks. As a result, they cannot be provided at once to fluid flow simulators: the required computational times would be prohibitive. The geologic model has to be upscaled, yielding a reservoir model with a reduced number of grid blocks. Except for special cases as stratified media, the numerical upscaling techniques are usually preferred. Among all of these methods, three main types are identified: local, extended local and global techniques. They all call for steady-state, single-phase flow simulations. If we do focus on the global approach, which is believed to be the most accurate, the pressure equation has to be solved for the whole fine geologic model. This step can be tackled only if a fast flow simulator is available. Many flow simulators, mainly based upon finite elements or finite volumes, have been developed. They apply to a wide variety of flow problems. A few years ago, Van Lent and itanidis (1989) introduced a novel FFT-based flow simulator. However, as they solved the pressure equation assuming that the variance of the log-permeability is very small compared to one, their technique was unsuitable for heterogeneous media. Parallely, a fast FFT-based technique was proposed to compute stresses and strains in composite media (Moulinec and Suquet; 1998; Eyre and Milton, 1999). As there is an obvious analogy between Hooke s law, which states that stress is proportional to strain, and Darcy s law, which states that flow is proportional to pressure gradient, we start from this stress-strain solver and extend it to the case of flow. Then, we perform a few numerical experiments in order to grasp the accuracy and the efficiency of the proposed techniques. Framework For steady-state, incompressible, single-phase flow, the continuity equation is: div ( U ) = ( 1 9th European Conference on the Mathematics of Oil Recovery Cannes, France, 3 August - 2 September 24
2 where U is the Darcy velocity. It is given by Darcy s law: U = P ( 2 is viscosity, P is pressure. As the porous medium is heterogeneous, permeability is considered as a random space function. For simplicity, it is assumed to be a scalar in each grid macro block. The pressure gradient P is split into two terms: its average P and a perturbation p. A basic scheme for porous media with moderate permeability contrasts The first proposed algorithm derives from the work of Moulinec and Suquet (1998). Auxiliary problem Before going further, we focus on an auxiliary problem, that is fluid flow in a porous homogeneous medium of a given permeability, noticed, submitted to a velocity field τ. We will see in the following sections that the value of is essential. The basic set of equations is given by: These equations are reformulated in the frequency space: Symbol ^ stands for Fourier transform and ξ is the wave vector. This equation reduces to: Γ is the Green operator associated to. General problem U = p + τ ( 3 div( U ) = ˆ U = i ξpˆ + ˆ τ ( 4 iξuˆ = ξ ξ ˆ ) τ ˆ pˆ = Γ with Γ = ξ ξ At this point, we come back to our basic problem, that is flow in a heterogeneous porous medium of permeability submitted to a pressure gradient P. We introduce a homogeneous reference permeability field so that: ( 5 U = P + P P = ( ) = τ P P P ( 6 The τ field is given by τ = P.
3 macro As stated above, the pressure gradient is the sum of its average P Thanks to the auxiliary problem, we obtain the following solution: P Pˆ which can be rewritten in the real space as: This expression can be reformulated as: First algorithm ( ξ ) = Γˆ ˆ τ macro = Pˆ ˆ macro ξ and a perturbation p. P = Γ τ + P. ( 8 m macro m 1 m 1 P = P + Γ U + Γ P. ( 9 Equation 5 is the key of the proposed algorithm. FFT stands for fast Fourier transform. ( 7 Initialization: macro P = P and U = P macro using FFT, compute ˆP. ( m) ( m) Iterate m+1: Pˆ and U are known. ( m) 1) using FFT, compute Uˆ ; ( m 2) set ) ( m) ( m Pˆ 1 ( ξ) ˆ ( ξ) Uˆ ˆ + = Γ + Γ ( ξ). Pˆ ) ( ξ) when ξ ( m and ) macro Pˆ + 1 = P ; 3) using the inverse FFT, compute P m+1 ; ( m+1 ) ( m+1) 4) compute U = P ; m+ 1 U ( m+ 1) 5) compute the effective permeability eff = ( m+ 1) P ( m+ ) ( m) 6) set m = m+1 and go back to step 1 until 1 eff eff ε. Because of the use of FFTs, computations are very fast. However, they are restricted to periodic boundary conditions and regular reservoir grids. Convergence The convergence of equation 6 has been analysed in a number of papers (e.g., Eyre and Milton, 1999). In the worst case, the series optimally converges when: ( ) + max. min = ( 1 2 ( m) ( m 1) m macro The rate of convergence is given by P P = γ P where: max( ) + 1 γ = min( ). max ( 11 1 min( ) 9th European Conference on the Mathematics of Oil Recovery Cannes, France, 3 August - 2 September 24
4 As γ is lower than 1, the developed algorithm converges whatever the starting guess. However, it can be very slow for media exhibiting high permeability contrasts. A basic scheme for porous media with higher permeability contrasts In order to speed up the first proposed algorithm, we consider another expression of τ (Eyre and Milton, 1999): τ 2 macro ( I γz) = P ( 12 with γ = 2 Γ I and z = rewritten as a sequence: +. Again, following Neumann expansion, this expression can be 2 ( m+ 1) macro ( m) τ = P + τ γ z. ( 13 The heart of the second FFT-based flow solver is described below. Initialization: τ = 2 P macro ( m) Iterate m+1: τ is known. ( m+1) ( m) 1) compute Q = zτ ; 2) using FFT, compute ˆ m+1 Q ; ( m+ 1 3) compute ˆ τ ) ( ξ ) = ˆ γ zˆ, ξ ( m+ 1 and ) ( ˆ τ =τˆ ) ; ( m+1) 4) using the inverse FFT, compute τ ; 5) set m = m+1 and go back to step 1 until τ 6) When done, derive P and U from τ. ( m + 1) ( m ) τ ε. As above, the rate of convergence depends strongly on. Eyre and Milton (1999) states that the proposed iterative scheme converges optimally when = min( ) max( ) and that the ( m) ( m 1) m macro convergence rate is given by P P = γ P with: max( ) + 1 min( ) γ =. ( 14 max( ) 1 min( ) As γ is lower than 1, the developed algorithm converged whatever the starting guess. Numerical tests A few numerical experiences are performed to evidence the potential of the two suggested FFTbased algorithms to solve a steady-state, incompressible, single-phase flow problem. We consider a lognormal permeability field (Figure 1). The mean and the variance of the logarithm of the permeability values are 3 and 1, respectively. The spatial distribution of these values is characterized by an isotropic spherical variogram with a correlation length of 2 grid blocks. The permeability field is discretized over a 5x5 grid, the size of each grid block being 4x4
5 meters. The studied flow problem is defined by Equations 1) and 2) with a fluid viscosity of 1 cp. Boundary conditions are assumed to be periodic: the model is repeated infinitely in each direction and a pressure gradient of,3 kpa/m is applied in the X-direction. This flow problem is solved using (1) a common centered finite-difference solver, (2) the first FFT-based algorithm, and (3) the second FFT-based algorithm. Pressures and Darcy velocities are compared in Figure 2, Figure 3 and Figure 4. We check that the computed pressures and velocities look the same, whatever the solver. However, there are tiny differences which can be explained, at least partly, by the distinct natures of the solvers. First, Darcy velocities are not computed at the same locations (Figure 5). Finite-difference solvers provide velocities at the interfaces between adjacent grid blocks, while the described FFT-based algorithms return the velocities at the centers of the grid blocks. Second, to estimate Darcy velocities, the finite-difference solvers call for the preliminary estimation of interblock transmissibilities. In this paper, the term transmissibility refers to the petroleum use (Aziz and Settari, 1979) which differs from the hydrogeological one (Marsily, 1986). In other words, transmissibility between two adjacent grid blocks is a function of their permeabilities. To compute transmissibilities from permeabilities, most finite-difference simulators use the harmonic average rule, which is theoretically justified for one-dimensional flow solely. Other weighting formulas have been proposed, such as the arithmetic mean, the geometric mean, or the more general power average mean, but their justifications remain empirical (Toronyi and Farouq Ali, 1974; Pettersen, 1983). On the contrary, the proposed FFTbased solvers deal with permeabilities at once: they do not introduce any kind of approximated transmissibility. Table 1 shows a more detailed comparison between the first and the second FFT-based algorithms. Again, the studied permeability field is the one depicted in Figure 1. The flow problem is solved for decreasing ε values. We recall that ε controls the stopping criteria of the iterative schemes. The second algorithm is clearly more efficient than the first one. For a given ε, it converges with a smaller number of iterations. In addition, the estimated effective permeability stabilizes for higher ε. For the studied case, the effective permeability is estimated to be 22.23 md. To reach this value, the first FFT-based algorithm needs 19 iterations, while the second one requires 21 iterations. Table 1. Comparison between the two proposed FFT-based algorithms. ε is a constant used to stop the iterative procedures. eff is the effective permeability computed at the last iteration. ε 1 st FFT-based algorithm 2 nd FFT-based algorithm Number of iterations eff (md) Number of iterations eff (md) 1-2 22 22.37 21 22.23 1-3 46 22.27 31 22.23 1-4 6 22.26 41 22.23 1-5 1 22.26 51 22.23 1-6 19 22.23 61 22.23 9th European Conference on the Mathematics of Oil Recovery Cannes, France, 3 August - 2 September 24
6 Figure 1. Studied 5x5 permeability field. Figure 2. Pressures computed using a finite-difference solver (DF), the first FFT-based algorithm (FFT1) and the second FFT-based algorithm (FFT2). Figure 3. Darcy velocities along the X axis computed using a finite-difference solver (DF), the first FFT-based algorithm (FFT1) and the second FFT-based algorithm (FFT2). Figure 4. Darcy velocities along the Y axis computed using a finite-difference solver (DF), the first FFT-based algorithm (FFT1) and the second FFT-based algorithm (FFT2).
7 Pi U Pi+1 i+1/2 + a) i Pi Ui, b) +i + i+1 Pi+1 Ui+1 + i+1 Figure 5. Discretization scheme for finite-difference computations (a) and for FFT computations (b). A similar comparison study is performed for a 5x5 categorical permeability field. We actually consider a permeability field with three facies of permeability 5 md, 3 md and 5 md (Figure 6). Thus, the permeability contrast equals 25. Again, we estimate pressures and Darcy velocities from a finite-difference scheme and from the second FFT-base algorithm. The first proposed algorithm was disregarded, because it could not deal with such a high permeability contrast. Results are reported in Figure 7. At first glance, we observe a good agreement between the finite-difference simulation and the FFT-based one. However, the finite-difference results, especially the Darcy velocities, seem to be smoother. This behavior is probably due to the use of transmissibilities. Indeed, transmissibilities are defined as a mean (in this case, we use the harmonic mean) of the permeabilities of adjacent grid blocks. Figure 6. 5x5 facies permeability field. 9th European Conference on the Mathematics of Oil Recovery Cannes, France, 3 August - 2 September 24
8 Figure 7. Comparison of pressures and Darcy velocities computed from a finite-difference scheme and the second FFT-based algorithm for the facies permeability described in Figure 6. Last, we focus on a permeability field consisting of one million of grid blocks. Its geostatistical properties are similar to those of the continuous permeability field depicted in Figure 1. Setting ε to 1-2 which seems to be enough to get relevant responses (Table 1), the second FFT-based algorithm converges with 11 iterations which is equivalent to 18 seconds of CPU time on a standard PC. For comparison purposes, we solve the same fluid flow problem using 3DSL, the Streamsim s simulator (Batycky et al., 1997) considering only the pressure solver: we disregard the streamline computations. 3DSL is presented as the industry s most advanced streamlinebased reservoir simulator. It involves finite-volume and multigrid techniques. With this solver, the required CPU time to get velocities and pressures is 2 minutes and 3 seconds. Thus, the second FFT-based algorithm contributes to a CPU-time decrease by a factor of about 7. Conclusions The following main conclusions can be drawn from this study: - Two distinct FFT-based algorithms are proposed to solve steady-state, single phase flow in porous heterogeneous media submitted to periodic boundary conditions. - These algorithms, which are presented as alternatives to standard finite volume or element solvers, turn out to be fast and accurate. - They exhibit an additional interesting feature, that is they do not call for any approximated transmissibility. Computations integrate permeabilities at once. - At this point, we intend to take advantage of these fast flow solvers to implement a global upscaling technique. References [1] Aziz,., and Settari, A., 1979, Petroleum reservoir simulation, London, Applied Science. [2] Batycky, R.P., Blunt, M.J., and Thiele, M.R., 1997, A 3D field-scale streamline based reservoir simulator, SPERE, 4, November. [3] Eyre, D.J., and Milton, G.W., 1999, A fast numerical scheme for computing the response of composites using grid refinement, Eur. Phys. J. AP, 6, 41-47. [4] Marsily, G. de, 1986, Quantitative hydrology, groundwater hydrology for engineers, Diego, Academic Press. [5] Moulinec, H., and Suquet, P., 1998, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Methods Appl. Mech. Engrg., 157, 69-94. [6] Pettersen, O., 1983, Finite-difference representation of permeabilities in heterogeneous models, SPE ATCE, SPE 1234. [7] Toronyi, R.M., and Farouq Ali, S.M., 1974, Determining interblock transmissibility in reservoir simulators, Journal of Petroleum Technology, 26(1), 77-78. [8] Van Lent T.J. and itanidis P.., 1989, A numerical spectral approach for the derivation of piezometric head covariance function. Water Resour. Res., 25(11):2287-2298.