Bradley T. Mallison

Compact Multiscale Finite Volume Method for Heterogeneous Anisotropic Elliptic Equations

The multiscale finite volume (MSFV) method is introduced for the efficient solution of elliptic problems with rough coefficients in the absence of scale separation. The coarse operator of the MSFV method is presented as a multipoint flux approximation (MPFA) with numerical evaluation of the transmissibilities. The monotonicity region of the original MSFV coarse operator has been determined for the homogeneous anisotropic case. For grid-aligned anisotropy the monotonicity of the coarse operator is very limited. A compact coarse operator for the MSFV method is presented that reduces to a 7-point stencil with optimal monotonicity properties in the homogeneous case. For heterogeneous cases the compact coarse operator improves the monotonicity of the MSFV method, especially for anisotropic problems. The compact operator also leads to a coarse linear system much closer to an M-matrix. Gradients in the direction of strong coupling vanish in highly anisotropic elliptic problems with homogeneous Neumann boundary d...

An Alternative to Streamlines for Flow Diagnostics on Structured and Unstructured Grids

Streamline-based methods can be used as effective post-processing tools for assessing flow patterns and well allocation factors in reservoir simulation. This type of diagnostic information can be useful for a number of applications, including visualization, model ranking, upscaling validation, and optimization of well placement or injection allocation. In this paper, we investigate finite-volume methods as an alternative to streamlines for obtaining flow diagnostic information. Given a computed flux field, we solve the stationary transport equations for tracer and time of flight by use of either single-point upstream (SPU) weighting or a truly multidimensional upstream (MDU) weighting scheme. We use tracer solutions to partition the reservoir into volumes associated with injector/producer pairs and to calculate fluxes (well allocation factors) associated with each volume. The heterogeneity of the reservoir is assessed with time of flight to construct flowcapacity/storage-capacity (F-vs.-U) diagrams that can be used to estimate sweep efficiency. We compare the results of our approach with streamline-based calculations for several numerical examples, and we demonstrate that finite-volume methods are a viable alternative. The primary advantages of finite-volume methods are the applicability to unstructured grids and the ease of implementation for general-purpose simulation formulations. The main disadvantage is numerical diffusion, but we show that a MDU weighting scheme is able to reduce these errors.

Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem

We present two new cell-centered nonlinear finite-volume methods for the heterogeneous, anisotropic diffusion problem. The schemes split the interfacial flux into harmonic and transversal components. Specifically, linear combinations of the transversal vector and the co-normal are used that lead to significant improvements in terms of the mesh-locking effects. The harmonic component of the flux is represented using a conventional monotone two-point flux approximation; the component along the parameterized direction is treated nonlinearly to satisfy either positivity of the solution as in 29, or the discrete maximum principle as in 9. In order to make the method purely cell-centered, we derive a homogenization function that allows for seamless interpolation in the presence of heterogeneity following a strategy similar to 46. The performance of the new schemes is compared with existing multi-point flux approximation methods 3,5. The robustness of the scheme with respect to the mesh-locking problem is demonstrated using several challenging test cases. A method for consistent interpolation in heterogeneous media is proposed.A finite-volume flux is split into sum of two-point flux approximation and transversal part.Methods preserving positivity and satisfying the discrete maximum principle are constructed.Parametrization of the split and unsplit flux is proposed to address locking issue.A strategy for the choice of the parameter is proposed.

Multidimensional upstream weighting for multiphase transport on general grids

The governing equations for multiphase flow in porous media have a mixed character, with both nearly elliptic and nearly hyperbolic variables. The flux for each phase can be decomposed into two parts: (1) a geometry- and rock-dependent term that resembles a single-phase flux; and (2) a mobility term representing fluid properties and rock–fluid interactions. The first term is commonly discretized by two- or multipoint flux approximations (TPFA and MPFA, respectively). The mobility is usually treated with single-point upstream weighting (SPU), also known as dimensional or donor cell upstream weighting. It is well known that when simulating processes with adverse mobility ratios, SPU suffers from grid orientation effects. An important example of this, which will be considered in this work, is the displacement of a heavy oil by water. For these adverse mobility ratio flows, the governing equations are unstable at the modeling scale, rendering a challenging numerical problem. These challenges must be addressed in order to avoid systematic biasing of simulation results. In this work, we present a framework for multidimensional upstream weighting for multiphase flow with buoyancy on general two-dimensional grids. The methodology is based on a dual grid, and the resulting transport methods are provably monotone. The multidimensional transport methods are coupled with MPFA methods to solve the pressure equation. Both explicit and fully implicit approaches are considered for time integration of the transport equations. The results show considerable reduction of grid orientation effects compared to SPU, and the explicit multidimensional approach allows larger time steps. For the implicit method, the total number of non-linear iterations is also reduced when multidimensional upstream weighting is used.

Monotone Multi-dimensional Upstream Weighting on General Grids

The governing equations for multi-phase flow in porous media often have a mixed elliptic and (nearly) hyperbolic character. The total flux for each phase consists of two parts: a geometry and rock dependent term that resembles a single-phase flux and a mobility term representing fluid properties and rock-fluid interactions. The geometric term is commonly discretized by two or multi point flux approximations (TPFA and MPFA, respectively). The mobility is usually treated with single point upstream weighting (SPU), also known as dimensional or donor cell upstream weighting. It is well known that when simulating processes with adverse mobility ratios, e.g. gas injection, SPU yields grid orientation effects. For these physical processes, the governing equations are unstable on the scale at which they are discretized, rendering a challenging numerical problem. These challenges must be addressed in order to avoid systematic biasing of simulation results and to improve the overall performance prediction of enhanced oil recovery processes. In this work, we present a framework for multi-dimensional upstream weighting for multi-phase flow with gravity on general two-dimensional grids. The methodology is based on a dual grid, and the resulting transport methods are provably monotone. The multi-dimensional transport methods are coupled with MPFA methods to solve the pressure equation. Both explicit and fully implicit approaches are considered for treatment of the transport equations. The results show considerable reduction of grid orientation effects compared to SPU, and the explicit multi-dimensional approach allows larger time steps. For the implicit method, the total number of non-linear iterations is also reduced when multi-dimensional upstream weighting is used.

Hybrid Upwinding for Two-phase Flow in Heterogeneous Porous Media with Buoyancy and Capillarity

Numerical simulation of subsurface flow requires an efficient solution strategy for the partial differential equations governing coupled multiphase flow and transport in porous media. A common characteristic of geological porous media is their highly heterogeneous structure. Heterogeneity is a challenge for numerical simulation, as variations by orders of magnitude in the permeability and porosity fields create a wide range of time scales in the transport problem. Therefore, in the solution strategy, the temporal discretization of choice is often the unconditionally stable fully implicit method. However, the nonlinear systems arising from this discretization – often solved with Newton’s method with global damping –, are highly nonlinear and difficult to solve. Thus, the overall computational cost is strongly dependent on the nonlinear convergence rate, and enhancing this nonlinear convergence property is key to speed up flow simulation. We improve the robustness and nonlinear convergence with an efficient fully implicit, finite-volume scheme for immiscible two-phase flow in the presence of viscous, buoyancy, and capillary forces. Each term in the numerical flux is treated separately based on physical considerations to obtain a differentiable and robust discretization. Following the Implicit Hybrid Upwinding strategy (Lee et al., Advances in Water Resources, 2015), the viscous term is upwinded based on the sign of the total velocity, whereas the directionality of the gravity part is determined by the density differences. In this paper, the emphasis is on the discretization of the capillary flux, which is composed of a rock- and geometry-dependent transmissibility, a nonlinear diffusion coefficient, and a saturation gradient. We propose a discretization that yields a consistent, bounded, and differentiable numerical flux. Importantly, the numerical flux in the presence of capillary forces is monotone. We show that the monotonicity of the numerical flux is critical for the robustness of the scheme when applied to heterogeneous porous media. Our numerical examples, which range from buoyancy-driven flow with capillary barriers to viscous-dominated flow, demonstrate that the Implicit Hybrid Upwinding scheme leads to significant reductions in the number of nonlinear iterations compared with the standard phase-based upwinding schemes.

Asynchronous Time Integration of Flux-conservative Transport

We investigate the potential of a flux-conservative, asynchronous method for the explicit time integration of subsurface transport equations. This method updates each discrete unknown using a local time step, chosen either in accordance with local stability conditions, or based on predicted change of the solution, as in Omelchenko and Karimabadi (Self-adaptive time integration of flux-conservative equations with sources. J. Comput. Phys. 216 (2006)). We show that the scheme offers the advantage of avoiding the overly-restrictive global CFL conditions. This makes it attractive for transport problems with localized time scales, such as those encountered in reservoir simulation where localized time scales can arise due to well singularities, spatial heterogeneities, moving fronts, or localized kinetics, amongst others. We conduct an analysis of the accuracy properties of the method for one-dimensional linear transport and first order discretization in space. The method is found to be locally inconsistent when the temporal step sizes in two adjacent cells differ significantly. We show numerically, however, that these errors do not destroy the order of accuracy when localized. The asynchronous time stepping compares favorably with a traditional first order explicit method for both linear and nonlinear problems, giving similar accuracy for much reduced computational costs. The computational advantage is even more striking in two dimensions where we combine our integration strategy with an IMPES treatment of multiphase flow and transport. Global time steps between pressure updates are determined using a strategy often used for adaptive implicit methods. Numerical results are given for immiscible and miscible displacements using a structured grid with nested local refinements around wells.

Nonlinear Two-Point Flux Approximations for Simulating Subsurface Flows with Full-Tensor Anisotropy

Full-tensor anisotropy effects are often encountered in subsurface flow due to either grid nonorthogonality or permeability anisotropy. A multipoint flux approximation (MPFA) is generally needed to accurately simulate flow for such systems, though the resulting discrete system is more complex and may be less robust than that resulting from a two-point flux approximation (TPFA). In this paper, we present and apply a different approach, nonlinear two-point flux approximation (NTPFA), for modeling systems with full-tensor effects and grid nonorthogonality. NTPFA incorporates global flow into the determination of the two-point flux transmissibilities, which is analogous to the use of two-point flux transmissibility in global and local-global upscaling procedures. For a given flow scenario, the global MPFA solution can be used to compute the two-point flux transmissibility, leading to a global NTPFA scheme. To avoid solving the full global MPFA system, we have also developed a local-global NTPFA procedure, in which the global flow is approximated by local MPFA solutions iteratively coupled with a global TPFA solution. The NTPFA methods described here are applied to 2D parallelogram grids, heterogeneous full-tensor permeability fields, and a 3D multiblock model with grid nonorthogonality. Results from both NTPFA schemes are generally in close agreement with the MPFA solution and provide considerable improvement over the standard TPFA scheme.