A block preconditioner for two-phase flow in porous media by mixed hybrid finite elements

2.1 Mathematical model

2.2 Numerical model

Given a porous domain $\mathrm{\Omega }$ partitioned into a set of nonoverlapping hexahedra, let ${ℰ}^h$ and ${ℱ}^h$ denote the collection of the grid elements and faces of size $N_e$ and $N_f$, respectively. With reference to the characteristic hexahedral element of the grid depicted in Figure 1, the pressure unknowns are located on the element centroid, $p^E$, and on each face barycenter, ${\pi }_p$. The water saturation, $S_w^E$, instead, is computed only on the element centroid. Let also $z^E$ and ${\pi }_z$ be the depth (positive downward) of the element centroid and faces, respectively. In the following, we describe the discretization of the PDEs in Equations (1b), (5a), and (5b).

Equation (1b) is integrated by using the ${\mathbb{P}}_0$ approximation space for pressures ($p^h$ and ${\pi }_p^h$) and water saturation ($S_w^h$), while the MHFE method in the lowest-order Raviart–Thomas ${ℝ𝕋}_0$ space⁴⁹ is used to discretize the velocity (${\mathbf{v}}_{\alpha }^h,\alpha =o,w)$. In a 3-D setting, the latter space consists of local piecewise trilinear vector functions, ${\mathit{\eta }}_i^E(x,y,z)$, defined for each face belonging to element E. These basis functions satisfy the two properties recalled in Nardean et al..⁵⁰ The unknowns $p^E$ and $S_w^E$ express the average value of the relevant physical quantities within the element, whereas ${\pi }_p$ plays a similar role for the faces. The face pressures in the MHFE method act the part of Lagrange multipliers, whose introduction is aimed at enforcing the continuity of the normal component of the fluxes across the inter-element faces.

4.1 Test 0

The objective of this preliminary test is to analyze the structure of ${\tilde{S}}_1$ and ${\tilde{S}}_2$ and its expected evolution during a typical unsteady simulation. In a waterflooding process, in fact, the number of nonzeros of $𝒥$ evolves as a result of the water propagation in the reservoir. It is indeed the change in the water saturation that causes an increase in the density of $𝒥$, with a limited role played by pressure. Assuming that the reservoir is initially fully saturated of oil and water is present at its irreducible saturation, the most significant stencil enlargement is noted in blocks $J_{sp}$, $J_{ss}$, and $J_{s\pi }$ (see also Appendix B.1), which are associated with the elemental water mass conservation equations. Knowing how the Schur complement structure evolves because of these variations is important to drive the selection of the local preconditioner.

We consider the planar reservoir in Figure 2A discretized with 300 cells, with homogeneous material properties and gravity taken into account. The simulation runs for 220 days and, at the end, half domain is water saturated. Figure 3A,B shows the pattern of ${\tilde{S}}_1$ at the beginning and end of the simulation, respectively. From an initially diagonal structure, the stencil of ${\tilde{S}}_1$ is progressively enriched by a limited number of fill entries, which remain pretty clustered around the main diagonal. Such entries generate a banded nearly-lower triangular structure resembling the one typically expected in purely advective problems. This inspection suggests that a simple diagonal scaling or an incomplete factorization with no fill-in are good candidates for the application of ${\tilde{S}}_1^{-1}$, the former especially during the first stages of a transient simulation. Moreover, this guarantees also that the use of the ${\tilde{S}}_1$ diagonal in the computation of ${\tilde{S}}_2$ is expected to provide a good approximation of $S_2$.

The structure of ${\tilde{S}}_2$ is more complicated. We can split this matrix into three contributions (see Equation (15)): $B_1=J_{\pi \pi }-J_{\pi p}{J}_{pp}^{-1}{J}_{p\pi }$, $B_2=J_{\pi s}-J_{\pi p}{J}_{pp}^{-1}{J}_{ps}$, and $B_3=J_{s\pi }-J_{sp}{J}_{pp}^{-1}{J}_{p\pi }$, with ${\tilde{S}}_2=B_1-B_2{\tilde{S}}_1^{-1}{B}_3$. The pattern of $B_1$ (Figure 4A) does not change over time and that of $B_2$ slightly increases (Figure 4B,C). By distinction, $B_3$ reflects the waterflooding evolution. It is almost empty at the beginning, with nonzeros only in the rows linked to the injection wells (Figure 4D), then it is progressively filled as the cells are flooded (Figure 4E). The evolution of block $B_3$ modifies the weight of the product $P=B_2{\tilde{S}}_1^{-1}{B}_3$ over $B_1$. At the beginning of the simulation, $P\approx 0$ and ${\tilde{S}}_2\approx B_1$. Then, the ${\tilde{S}}_2$ stencil enlarges as $B_3$ is filled and P becomes more influential. Looking at Figure 4F,G, it is also expected that both a Jacobi and a zero fill-in factorization should not be effective to approximate the application of ${\tilde{S}}_2^{-1}$. In this case, an incomplete factorization with fill-in, for instance, would be preferable.

Finally, we compare the eigenspectrum of ${\tilde{S}}_1$ and ${\tilde{S}}_2$ with that of the exact Schur complements $S_1$ and $S_2$ at the end of the simulation (Figure 5). The eigenvalue distribution largely overlaps for both Schur complements, proving that the inexact computations are expected to be good approximations with no detrimental effect to the original conditioning. Notice also that the eigenvalues of ${\tilde{S}}_1$ are real and rather clustered away from zero.

4.2 Test 1

The simulated time interval spans 600 days of production covered in 196 time steps, whose size is dynamically adjusted during the simulation following the approach in Abushaikha et al..²¹ The maximum time step size is four days with the Courant–Friedrichs–Lewy (CFL) number^{54, 55} up to 8.56. The main model results are displayed in Figure 6 in terms of pressure, water saturation and fluid velocity. Moreover, Figure 7 shows the water front advancement from the injector toward the producer captured at some representative times along the vertical section connecting the wells. The shape of the profiles recalls that of the classical Buckley–Leverett benchmark,⁵⁶ since the simulated physical process is the same and the model setting is similar.

Figure 8A shows the number of iterations to converge of the first Jacobian system per time step. It can be seen that the performance stabilizes after the maximum time step size is achieved, which, in this case, occurs at the 57th step ($t=46.1$ d). After this point, the average performance appears to be a meaningful indicator. The profiles are obtained with different approximations for ${\tilde{J}}_{pp}^{-1}$, ${\tilde{S}}_1^{-1}$, and ${\tilde{S}}_2^{-1}$, as reported in Tables 2 and 3.

As to the block preconditioner, we first analyze the effect of the inexact application of $J_{pp}^{-1}$, $S_1^{-1}$, and $S_2^{-1}$ on the overall performance. This investigation allows to understand where most of the efforts are needed to optimize the solver effectiveness, by identifying the blocks requiring a higher-quality local approximation. We used either diagonal scaling, or incomplete factorization, or exact application of the inverse in different combinations. In Table 2, we mainly considered the first two terms, that is, $J_{pp}^{-1}$ and $S_1^{-1}$, whereas, in Table 3, the appropriate setting for the second block is evaluated again and we focused on the approximation of $S_2^{-1}$. The analysis is carried out considering both the global and local (average) performance of the preconditioned linear solver.

Starting from Table 2, the outcome of run 1 tells that the diagonal approximation for $J_{pp}^{-1}$ is appropriate for this block. Replacing the application ${\tilde{S}}_1^{-1}$ with an ILU(0), as in run 2, leads to a marginal increase in the total number of linear iterations (2.7%). By distinction, a diagonal approximation of ${\tilde{S}}_1^{-1}$ causes a 120.6% increase in the global iteration count. Therefore, a more detailed comparison is needed by taking into account the computational effort to compute and apply the preconditioner. In any case, the outcome of this analysis suggests that the overall solver performance mainly relies on the quality of the approximation for ${\tilde{S}}_2^{-1}$, as expected from the preliminary analysis carried out on Test 0. This is investigated in Table 3. Here, two incomplete factorization variants, that is, ILU(0) and threshold-based ILU($\tau$), have been tested for ${\tilde{S}}_2^{-1}$. The tests have been carried out for both the diagonal scaling and ILU(0) factorization of ${\tilde{S}}_1^{-1}$. The results show that, although ILU($\tau$) can significantly reduce the cumulative linear iteration count, ${\overline{it}}_l$, and solving time, ${\bar{t}}_s$, this strategy typically does not pay off because of the higher set-up cost, ${\bar{t}}_p$. We also observe that the difference between diagonal scaling and ILU(0) factorization for ${\tilde{S}}_1^{-1}$ is not as significant in terms of total time. Nevertheless, the latter appears to be more effective, hence it is chosen as default option in the next tests. Furthermore, the effect on the preconditioner density, $\mu$ and $\bar{\mu }$, is almost negligible.

The analysis in Section 4.1 highlighted that ${\tilde{S}}_2$ tends to become denser as the simulation proceeds. This result is confirmed by the outcome of Table 3, which shows that the preconditioner density is likely to increase slowly but steadily during the simulation when ILU(0) is used for ${\tilde{S}}_2$, whereas the opposite trend is noted with ILU($\tau$). Specifically, compare the density values in the columns $\bar{\mu }$ and $\mu$ (runs 1,2 vs. 3–8). In any case, filtration techniques can be applied to sparsify ${\tilde{S}}_2$ and stabilize the memory occupation, if necessary. Of course, the application of such techniques can affect the set-up cost.

The performance of the block preconditioner is also benchmarked against that of global incomplete LU factorizations with no or variable fill-in. The ILU decomposition is built upon the Jacobian matrix preliminary reordered by means of the reverse Cuthill–McKee algorithm.²⁶ The outcome is provided in Table 4. While with ILU(0) GMRES does not converge within 1000 iterations in most of the linearized systems, threshold-based ILU delivers better results but at a significant higher computational time with respect to our block preconditioner, where the greatest share is due to the set-up cost (compare, in this regard, columns $t_p$, $t_s$, ${\bar{t}}_p$, and ${\bar{t}}_s$ in Tables 3 and 4). Notice also that reducing the threshold value, $\tau$, below 0.01 results in a degradation of the preconditioned solver performance, most likely caused by the introduction of detrimental near-zero entries in the triangular factors.

4.3 Test 2

In this test, gravity is taken into account to evaluate its impact on the solver performance, whereas the production scenario and the other model settings are the same as in Test 1. The simulated time interval spans 250 days, with a maximum time step size of 0.5 days. This value is lower with respect to the previous test and it was set to keep the number of nonlinear iterations below 6–7. The simulation takes 526 time steps to conclude, with the maximum size achieved at the 37th step ($t=5.79$ d) and the CFL number up to 2.26. Figure 9A shows the water saturation field at the end of the simulation on the whole domain, while Figures 9B–D report the solution along a vertical section through the wells. The injected water propagates through the reservoir toward the producer, and simultaneously tends to accumulate at the bottom due to the action of gravity.

Table 5 reports the main results of the preconditioner performance analysis, which can be compared with those in Table 3. Based on the cumulative number of nonlinear iterations and the smaller time interval simulated than in Test 1, we can say that the problem is more challenging for the outer nonlinear solver, but the effect of gravity on the performance of the preconditioned linear solver appears to be negligible. The number of linear iterations, $i{t}_l$, in fact, along with the density, $\mu$, are similar to those in Test 1. This result is consistent with the observations in Section 4.1. The extra nonzero block appearing with gravity, $J_{\pi s}$, is required only in the computation of the product P for ${\tilde{S}}_2$ (specifically in $B_2$). We observed also that the importance of the P term is not constant during a simulation, but evolves depending on the water distribution within the reservoir. Even in this application, the threshold-based incomplete factorization, though succeeding in deflating the iteration count significantly (runs 2–4), does not outperform ILU(0) in terms of CPU time. A similar behavior of the preconditioner density, which grows during the simulation when an ILU(0) approximation is used, can be noted as well.

4.4 Test 3

The homogeneous isotropic permeability field of Test 1 and 2 is here replaced by a highly heterogeneous and anisotropic distribution retrieved from the SPE10 data set (see Figure 10A,B). Notice the wide range of values, spanning approximately ten orders of magnitude, and the abrupt discontinuities, especially in the vertical permeability (Figure 10B), which make this test numerically challenging. The porosity distribution, displayed in Figure 10C, follows that of the SPE10 model as well. Given the similar performance of the linear solver with and without gravity and the greater computational burden for the outer nonlinear solver observed in Test 2, in this application we decided to neglect the gravitational effects. The simulation runs for 250 days with a maximum time step size of 1.5 days, achieved at the 42nd step ($t=16.95$ d), and a total number of 198 steps. The CFL value is up to 30.27. At the end of the simulation, large portions of the oil in the reservoir have been displaced by water and breakthrough occurs at the producer. Figure 11 provides some model insights in terms of water saturation at $t=80$ d and $t=250$ d.

Table 6 summarizes the results obtained for different approximations of ${\tilde{S}}_2^{-1}$. For two of them, the number of iterations to solve the first linearized systems vs. time step is displayed in Figure 8B. Although the permeability distribution is highly heterogeneous and anisotropic, the number of iterations approximately stabilizes after the maximum time step size has been achieved like in Test 1 (see also Figure 8A). By comparing the iteration counts, ${\overline{it}}_l$ and $i{t}_l$, of run 1 with the counterpart in Table 3, we can appreciate the effect of the ill-conditioning caused by the extremely wide range of permeability and, with a minor role, porosity values. We carried out a sensitivity analysis on the threshold value for ILU($\tau$) (runs 2-4), which appears now to pay off with respect to the zero-fill incomplete factorization (run 1). Despite the higher set-up cost, the cumulative total solving time, ${\bar{t}}_t$, reduces as the threshold $\tau$ is decreased, while the density growth is firmly under control. Moreover, the cumulative number of linear iterations is reduced by 2.6 to 6.5 times. The sensitivity analysis shows that a threshold value for the drop tolerance in the range between 0.001 and 0.005 appears to be a good trade-off between fast convergence and the need of restraining the memory occupation.

4.5 Test 4

Modeling the fluid flow with a high approximation accuracy in non-Cartesian grids using full tensor permeability properties is an important requirement for modern simulators. This is also a challenge for most discretization schemes, which is effectively addressed by the MHFE method.²¹ In the last test case, a dome reservoir is taken into consideration with this challenging setting. Porosity and permeability are again taken from the SPE10 data set with the horizontal/vertical anisotropy ratio up to 1.E5. The full permeability tensor is obtained from the original diagonal tensor by rotating the local element axes so as to follow the curvature of the dome formation, similarly as in Nardean et al..⁵⁰ The main effect of non-Cartesian grid and full tensor permeability on the model can be seen in the stencil of block $J_{\pi \pi }$, which is enlarged with respect to the previous setting. This is because the local matrices ${(B^E)}^{-1}$, which are block diagonal with a diagonal permeability tensor and a Cartesian grid, become full (see Equation (8)). Therefore, while in the former setting there are up to three nonzero entries in each row of $J_{\pi \pi }$, such limit increases to eleven in the latter. On the contrary, the stencil of the other blocks in the Jacobian is not affected by the change in the grid or permeability tensor type. Figure 12 shows some model insight in terms of water saturation distribution, while the computational performance of the linear solver is summarized in Table 7. As in Test 3, the threshold-based incomplete factorization performs better than the no-fill variant in terms of CPU time. The sensitivity analysis on the threshold $\tau$ shows that a value between 0.0005 and 0.001 can be a good compromise in terms of memory footprint and computational efficiency, which is a little lower than in Test 3. Notice also that the density of the preconditioner in this test is lower than in Test 3 when the same value for $\tau$ is used in the incomplete factorization.