Mimetic Finite Difference Research Articles

Alternating direction implicit time integrations for finite difference acoustic wave propagation: Parallelization and convergence

This work studies the parallelization and empirical convergence of two finite difference acoustic wave propagation methods on 2-D rectangular grids, that use the same alternating direction implicit (ADI) time integration. This ADI integration is based on a second-order implicit Crank-Nicolson temporal discretization that is factored out by a Peaceman-Rachford decomposition of the time and space equation terms. In space, these methods highly diverge and apply different fourth-order accurate differentiation techniques. The first method uses compact finite differences (CFD) on nodal meshes that requires solving tridiagonal linear systems along each grid line, while the second one employs staggered-grid mimetic finite differences (MFD). For each method, we implement three parallel versions: (i) a multithreaded code in Octave, (ii) a C++ code that exploits OpenMP loop parallelization, and (iii) a CUDA kernel for a NVIDIA GTX 960 Maxwell card. In these implementations, the main source of parallelism is the simultaneous ADI updating of each wave field matrix, either column-wise or row-wise, according to the differentiation direction. In our numerical applications, the highest performances are displayed by the CFD and MFD CUDA codes that achieve speedups of 7.21x and 15.81x, respectively, relative to their C++ sequential counterparts with optimal compilation flags. Our test cases also allow to assess the numerical convergence and accuracy of both methods. In a problem with exact harmonic solution, both methods exhibit convergence rates close to 4 and the MDF accuracy is practically higher. Alternatively, both convergences decay to second order on smooth problems with severe gradients at boundaries, and the MDF rates degrade in highly-resolved grids leading to larger inaccuracies. This transition of empirical convergences agrees with the nominal truncation errors in space and time.

A preconditioned, multilevel Mimetic Finite Difference method for solving the pressure-velocity system

A Mimetic Finite Difference (MFD) implementation is proposed to solve the pressure-velocity system iteratively. The recursive procedure only solves for the velocity variable, while the pressure is calculated by direct substitution, simplifying the problem. The symmetric, positive definite system obtained is solved with an innovative implementation of a pre-conditioned, multilevel, conjugate gradient algorithm, where coarse algebraic objects are easily computed from the corresponding fine level ones, based on fundamental geometric properties of the MFD technology. A succinct eigen-analysis shows second order convergence on the velocity variable for Cartesian grids, which is corroborated numerically. Also, results consider different topologies and different medium properties, showing the applicability of the method and optimal convergence rates for the velocity on Cartesian and h2-uniform curved grids.

Flux-Corrected Transport with MT3DMS for Positive Solution of Transport with Full-Tensor Dispersion.

Solute transport is usually modeled by the advection-dispersion-reaction equation. In the standard approach, mechanical dispersion is a tensor with principal directions parallel and perpendicular to the flow vector. Since realistic scenarios include nonuniform and unsteady flow fields, the governing equation has full tensor mechanical dispersion. When conventional grid-based numerical methods are used, approximation of the cross terms arising from the off-diagonal terms cause nonphysical solution with oscillations. As an example, for the common scenario of contaminant input into a domain with zero initial concentration, the cross-dispersion terms can result in negative concentrations that can wreak havoc in reactive transport applications. To address this issue, we use the well-known flux-corrected-transport (FCT) technique for a standard finite volume method. Although FCT has most often been used to eliminate oscillations resulting from discretization of the advection term for explicit time stepping, we show that it can be adapted for full-tensor dispersion and implicit time stepping. Unlike other approaches based on new discretization techniques (e.g., mimetic finite difference, nonlinear finite volume), FCT has the advantage of being flexible and widely applicable. Implementation of FCT requires solving an additional system of equations at each time step, using a modified "low order" matrix and a modified right-hand-side vector. To demonstrate the flexibility of FCT, we have modified the well-known and widely used groundwater solute transport simulator, MT3DMS. We apply the new simulator, MT3DMS-FCT, to several benchmark problems that suffer from negative concentrations when using MT3DMS. The new results are mass conservative and strictly nonnegative.

Modeling full-wavefield time-varying sea-surface effects on seismic data: A mimetic finite-difference approach

Seismic-data processing flows often ignore spatial and temporal variations in the sea surface during marine seismic acquisition by assuming a flat free surface. However, weather patterns during data acquisition can generate rough sea conditions, which can significantly influence seismic full-wavefield source behavior, including ghost reflections and surface-related multiples, by introducing spatial and temporal distortions of the seismic wavelet. To investigate the effects of rough seas on seismic wave propagation, we have developed and solved a new acoustic wave equation using a mimetic finite-difference time-domain (MFDTD) scheme that uses a dynamic (i.e., moving) generalized coordinate system defined to be conformal to the assumed known time-varying free surface. This “sea-surface” coordinate system allows us to model the full dynamic effects associated with this complex boundary condition. Numerical examples demonstrate that the developed MFDTD method can accurately simulate seismic wavefield propagation on a moving mesh for significant wave heights of 5 m and beyond, and it is thus a reliable tool for applications involving modeling, processing, imaging, and inversion of seismic data acquired in rough seas.

A fully implicit mimetic finite difference scheme for general purpose subsurface reservoir simulation with full tensor permeability

In the previous article Abushaikha et al. (2017) [1], we presented a fully-implicit mixed hybrid finite element (MHFE) method for general-purpose compositional reservoir simulation. The present work extends the implementation for mimetic finite difference (MFD) discretization method. The new approach admits fully implicit solution on general polyhedral grids. The scheme couples the momentum and mass balance equations to assure conservation and applies a cubic equation-of-state for the fluid system. The flux conservativity is strongly imposed for the fully implicit approach and the Newton-Raphson method is used to linearize the system. We test the method through extensive numerical examples to demonstrate the convergence and accuracy on various shapes of polyhedral. We also compare the method to other discretization schemes for unstructured meshes and tensor permeability. Finally, we apply the method through applied computational cases to illustrate its robustness for full tensor anisotropic, highly heterogeneous and faulted reservoirs using unstructured grids.

Correction to: The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

Correction to: The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

An Efficient Hybrid Model for Nonlinear Two-Phase Flow in Fractured Low-Permeability Reservoir

The staged fracturing horizontal well has proven to be an attractive alternative for improving the development effect of a low permeability waterflood reservoir. Due to the coexistence of matrix, fracture, and horizontal wellbore, it remains a great challenge to accurately simulate the nonlinear flow behaviors in fractured porous media. Using a discrete fracture model to reduce the dimension of the fracture network, a two-parameter model is used to describe the nonlinear two-phase flow behavior, and the equivalent pipe flow equation is selected to estimate the horizontal wellbore pressure drop in the fractured low-permeability reservoir. A hybrid mathematical model for the nonlinear two-phase flow, including the effect of horizontal wellbore pressure drop in fractured porous media, is developed. A numerical scheme of the hybrid model is derived using the mimetic finite difference method and finite volume method. With a staggered five-spot flood system, the accuracy of the proposed model and the effect of fracture properties on nonlinear two-phase flow behaviors are further investigated. The results also show that with an increase of fracture length near injectors, the breakthrough time of injected water into the horizontal wellbore will be shorter, indicating a faster rise of the water cut, and a worse development effect. The impact of shortening fracture spacing is consistent with that of enlarging fracture length. Successful practice in modeling the complex waterflood behaviors for a 3-D heterogeneous reservoir provides powerful evidence for the practicability and reliability of our model.

The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of time-dependent diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. The method of lines (MOL) is used to combine spatial and temporal discretizations. The spatial scheme requires the definition of a high-order approximation of the divergence and gradient operators and the two inner products for the discrete analogs of fluxes and scalar unknowns. The discrete divergence and gradient operators are built according to a discrete duality relation. The inner product for the flux grid functions is built by explicitly imposing the conditions of consistency and stability. The family of semi-discrete mimetic methods is proved theoretically to be energy-stable as the corresponding continuous problem. Then, a full discretization is derived by combining via MOL the MFD method of order k with time marching schemes from the backward differentiation formula of order $$k+2$$ . Optimal order of accuracy is demonstrated for the scalar variable and verified numerically by solving the time-dependent diffusion problems with a variable diffusion tensor for k from 0 to 3 on three different unstructured mesh families.

High-order mimetic finite-difference operators satisfying the extended Gauss divergence theorem

We present high-order mimetic finite-difference operators that satisfy the extended Gauss theorem. These operators have the same order of accuracy in the interior and at the boundary, no free parameters and optimal bandwidth. They are defined over staggered grids, using weighted inner products with a diagonal norm. We present several examples to demonstrate that mimetic finite-difference schemes using these operators produce excellent results.

A higher order approximate static condensation method for multi-material diffusion problems

The paper studies an approximate static condensation method for the diffusion problem with discontinuous diffusion coefficients. The method allows for a general polygonal mesh which is unfitted to the material interfaces. Moreover, the interfaces can be discontinuous across the mesh edges as typical for numerical reconstructions using the volume or moment-of-fluid methods. We apply a mimetic finite difference method to solve local diffusion problems and use P1 (mortar) edge elements to couple local problems into the global system. The condensation process and the properties of the resulting algebraic system are discussed. It is demonstrated that the method is second order accurate on smooth solutions and performs well for problems with high contrast in diffusion coefficients. Experiments also show the robustness with respect to position of the material interfaces against the underlying mesh.

A VEM–based mesh–adaptive strategy for potential problems

The Virtual Element Method (VEM) is an evolution of the mimetic finite difference method which overcomes many limitations affecting classic Finite Element Methods (FEM). VEM for 2D problems allows for exploiting meshes consisting of any polygonal elements. No limitations on their internal angles are needed. Hanging nodes are easily treated. Notably, VEM is well apt to mesh–adaptive algorithms. In this paper we detail an implementation of mesh–adaptive VEM for potential problems. We suggest a fresh, promising approach. We show on suitable test problems that a gain in efficiency can be obtained, respect to uniform, fine discretizations.

Convergence of Grid Boundary-Value Problems for Functions Defined on Grid Cells and Faces

For stationary diffusion-type equations, we study the convergence of grid inhomogeneous boundary-value problems of a version of the mimetic finite difference (MFD) technique in which grid scalars are defined inside grid cells and grid vectors are specified by their local normal coordinates on the plane faces of grid cells. Grid equations and boundary conditions are formulated in operator form using consistent grid analogs of invariant first-order differential operators and of boundary operators. Convergence is studied on the basis of the theory of operator difference schemes; i.e., a priori estimates for the norm of the solution error in terms of the norm of the approximation error are obtained that guarantee convergence of the first order under inhomogeneous boundary conditions of the first, second, and third kind in a domain with a curvilinear boundary. Grid analogs of embedding inequalities and approximation relations obtained earlier are used.

Numerical simulation of hydro-mechanical coupling in fractured vuggy porous media using the equivalent continuum model and embedded discrete fracture model

In this study, an efficient numerical model is proposed to simulate the hydro-mechanical coupling in the fractured vuggy porous media containing multi-scale fractures and vugs. Specifically, an improved Equivalent Continuum Model (ECM), which is obtained through the asymptotic homogenization method, is applied to model micro fractures and vugs, while macro fractures are modeled explicitly by using the Embedded Discrete Fracture Model (EDFM) and non-matching grids. As an important feature of the explicit representment, the effects of fillings and fluid pressure on preventing macro fractures from closing can be considered. After that, the new mixed space discretization (i.e., Mimetic Finite Difference (MFD) method for flow and stabilized eXtended Finite Element Method (XFEM) for geomechanics) and fully coupled methods are applied to solve the proposed model. The mixed space discretization can deal with complex heterogeneous properties (e.g., full tensor permeability), and yields the benefits of local mass conservation and numerical stability in space. Finally, we demonstrate the accuracy and application of the proposed model to capture the coupled hydro-mechanical impacts of multiscale fractures and vugs on fluid flow in fractured porous media.

Finite volume element approximation for nonlinear diffusion problems with degenerate diffusion coefficients

Some standard numerical methods, such as mimetic finite difference method, finite volume method and mixed finite element method, often fail in solving nonlinear diffusion problems with degenerate diffusion coefficients, since a harmonic average of diffusion coefficients is involved in these methods. To avoid such problem, we present some finite volume element schemes to solve a class of 1D degenerate nonlinear parabolic equations in this paper. Some fully discrete schemes are given by using linear and quadratic finite volume elements in space and a backward difference formulation in time. To deal with unphysical numerical oscillation, we apply two nonnegativity-preserving repair techniques based on a posteriori corrections to finite volume element solutions. One is a local approach, in which any negative energy corresponding to some mesh node is absorbed by the positive values near the current node. The other is a global strategy, in which the total negative energy is redistributed to each positive value in the numerical solution in proportion to its value. In addition, some monotonous finite volume element schemes with lumped-mass strategy are presented. Numerical examples are included to demonstrate the effectiveness and competitive behavior of the proposed methods.

Numerical simulation of two-phase flow in porous media based on mimetic Green element method

Based on mimetic Green element method (GEM), which synthesizes some typical advantages of boundary element method (BEM), finite element method (FEM) and mimetic finite difference method (mimetic FDM), This paper study the application of mimetic GEM in the system of partial differential equations (PDEs) of oil-water two-phase flow in porous media. It is the first time to apply GEM to solve numerically the system of PDEs, which is one of novelties in this paper. For numerical simulation of oil-water two-phase flow in porous media, this paper provides a novel numerical method based on mimetic GEM, which is the other novelty of this work.

Study of hydraulic fracturing in an anisotropic poroelastic medium via a hybrid EDFM-XFEM approach

In this paper, we investigate the effects of poroelastic properties and permeability on hydraulic fracturing in an anisotropic medium. A hybrid approach combining the EDFM (embedded discrete fracture model) and XFEM (extended finite element method) is proposed. Fractures are embedded into a nonconforming grid used to determine the fluid flow in a porous matrix by the mimetic finite difference method, accounting for an anisotropic permeability tensor. The stress-strain problem is solved by the extended finite element method with the same grid. The fluid flow and rock deformation as well as the fracture propagation are iteratively coupled. The proposed approach is validated against the analytical solutions for Mandel’s problem and the KDG model. A series of calculations are performed, and the obtained results are analyzed to investigate the effects of the anisotropy of permeability, that of the elastic modulus and Biot’s coefficient on the hydraulic fracturing process. It is found that the anisotropy of permeability has a significant influence on the geometrical parameter of a fracture, while the anisotropy of the elastic modulus has a dominating influence on the propagation direction of a fracture. Biot’s coefficient also has an influence on the fracture propagation kinetics.

Irregular surface seismic forward modeling by a body-fitted rotated–staggered-grid finite-difference method

Finite-difference (FD) methods are widely used in seismic forward modeling owing to their computational efficiency but are not readily applicable to irregular topographies. Thus, several FD methods based on the transformation to curvilinear coordinates using body-fitted grids have been proposed, e.g., stand staggered grid (SSG) with interpolation, nonstaggered grid, rotated staggered grid (RSG), and fully staggered. The FD based on the RSG is somewhat superior to others because it satisfies the spatial distribution of the wave equation without additional memory and computational requirements; furthermore, it is simpler to implement. We use the RSG FD method to transform the firstorder stress–velocity equation in the curvilinear coordinates system and introduce the highprecision adaptive, unilateral mimetic finite-difference (UMFD) method to process the freeboundary conditions of an irregular surface. The numerical results suggest that the precision of the solution is higher than that of the vacuum formalism. When the minimum wavelength is low, UMFD avoids the surface wave dispersion. We compare FD methods based on RSG, SEM, and nonstaggered grid and infer that all simulation results are consistent but the computational efficiency of the RSG FD method is higher than the rest.

Analysis of a mimetic finite difference approximation of flows in fractured porous media

We consider the mixed formulation for Darcy’s flow in fractured media. We give a well-posedness result that does not rely on the imposition of pressure in part of the boundary of the fracture network, thus including a fully immersed fracture network. We present and analyze a mimetic finite difference formulation for the problem, providing convergence results and numerical tests.

Properties of Consistent Grid Operators for Grid Functions Defined Inside Grid Cells and on Grid Faces

Using grids (meshes) formed from polyhedra (polygons in the two-dimensional case), we consider differential and boundary grid operators that are consistent in the sense of satisfying the grid analog of the integral identity – a corollary of the formula for the divergence of the product or a scalar by a vector. These operators are constructed and applied in the Mimetic Finite Difference (MFD) method, in which grid scalars are defined inside the grid cells and grid vectors are defined by their local normal coordinates on the planar faces of the grid cells. We show that the basic grid summation identity is a limit of an integral identity written for piecewise-smooth approximations of the grid functions. We also show that the MFD formula for the reconstruction of a grid vector field is obtained by approximation analysis of the summation identity. Grid embedding theorems are proved, analogous to well-known finite-difference embedding theorems that are used in finite-difference scheme theory to derive prior bounds for convergence analysis of the solutions of finite-difference nonhomogeneous boundary-value problems.

High Order Mimetic Finite Difference Operators Satisfying a Gauss Divergence Theorem

High order mimetic finite difference operators that satisfy a discrete extended Gauss Divergence theorem are presented. These operators have the same order of accuracy in the interior as well as the boundary, no free parameters and optimal bandwidth. They are constructed on staggered grids, using weighted inner products with a diagonal norm. We present several examples to demonstrate that mimetic finite difference schemes using these operators produce excellent results.