Mimetic Finite Difference Research Articles

Numerical Analysis for the Pure Neumann Control Problem Using the Gradient Discretisation Method

Abstract The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low-order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, non-conforming and mimetic finite difference methods confirm the theoretical rates of convergence.

Bracket formulations and energy- and helicity-preserving numerical methods for incompressible two-phase flows

A diffuse interface model for three-dimensional viscous incompressible two-phase flows is formulated within a bracket formalism using a skew-symmetric Poisson bracket together with a symmetric negative semi-definite dissipative bracket. The budgets of kinetic energy, helicity, and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method. The Cahn–Hilliard and Allen–Cahn equations are employed as diffuse interface models, in which the equalities of densities and viscosities of two different phases are assumed. Numerical experiments on the motion of periodic arrays of tubes and those of droplets have been conducted to examine the properties and usefulness of the proposed method.

Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes

In this work we develop arbitrary-order Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomials on the mesh skeleton. The key ingredient is a high-order gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high-order residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.

Coupled fluid flow and geomechanics modeling of stress-sensitive production behavior in fractured shale gas reservoirs

Compared with conventional reservoirs, gas flow in shale formation is affected by additional nonlinear coupled processes such as matrix/fracture deformations. We present a fully-coupled fluid flow and geomechanics model to accurately characterize the complex production behaviors of fractured shale gas reservoirs. The flow equations are discretized using a mimetic finite difference method, and poro-elasticity equations by a Galerkin finite-element approximation. A unified apparent-permeability model is implemented to quantify the combined impacts of the non-Darcy flow, adsorbed layer and pore-structure alterations on matrix permeability.The discrete fracture-matrix (DFM) model based on a conformal unstructured grid is employed to explicitly represent fractures. The nonlinear contact problem between the two fracture surfaces is introduced to describe the fracture mechanics behavior. A splitting-node technique is used to deal with the discontinuities in the displacement field across the fracture interface. Under the effects of pressure decline and high confining stresses on the fracture faces, proppant compaction and embedment may occur, causing fracture closure and thus substantial production loss. Hence we also develop a comprehensive proppant-fracture model which is based on the theories of elasto-plastic contact mechanics, to capture the complex interactions between proppant and fracture.The multiphysics numerical model enables us to investigate which factors have the most influential effect on the gas recovery of shale formations. High fidelity numerical solutions are provided to characterize the rate-transient signatures in the presence of the different flow and geomechanical mechanisms.

Efficiency and accuracy of equivalent fracture models for predicting fractured geothermal reservoirs: the influence of fracture network patterns

We evaluate the accuracy and efficiency of equivalent fracture models with a Cartesian grid regarding the variation of fracture network patterns. We apply both the standard finite volume method and the mimetic finite difference method for discretizing the flow equation in the equivalent fracture models. The study underlines that the equivalent fracture models yield accurate and efficient results for predicting the temperature in geothermal reservoirs. The equivalent fracture models based on the mimetic finite difference method yields comparable results with those based on the standard finite volume method. Whereas their accuracy is influenced by the orientations of sparsely distributed fractures.

The arbitrary order mimetic finite difference method for a diffusion equation with a non-symmetric diffusion tensor

We present the arbitrary order mimetic finite difference (MFD) discretization for the diffusion equation with non-symmetric tensorial diffusion coefficient in a mixed formulation on general polygonal meshes. The diffusion tensor is assumed to be positive definite. The asymmetry of the diffusion tensor requires changes to the standard MFD construction. We present new approach for the construction that guarantees positive definiteness of the non-symmetric mass matrix in the space of discrete velocities. The numerically observed convergence rate for the scalar quantity matches the predicted one in the case of the lowest order mimetic scheme. For higher orders schemes, we observed super-convergence by one order for the scalar variable which is consistent with the previously published result for a symmetric diffusion tensor. The new scheme was also tested on a time-dependent problem modeling the Hall effect in the resistive magnetohydrodynamics.

Multiscale mimetic method for two-phase flow in fractured media using embedded discrete fracture model

A multiscale mimetic method is developed for the simulation of multiphase flow in fractured porous media in the context of an embedded discrete fracture model (EDFM). The EDFM constructs independent grids for matrix and fracture system. Therefore, it is an efficient and practical flow model as it avoids the complicated unstructured grid subdivision and computing process. In order to extend the EDFM to field-scale applications, we integrate EDFM into a multiscale mimetic method. In this work, we use the multiscale basis functions to capture the detailed interactions between the fractures and the background. The multiscale basis functions are calculated numerically by solving EDFM on the local fine-grid with mimetic finite difference (MFD) method. The MFD method is conservative and robust, which makes it possible to deal with highly complex grid systems. Through combination of multiscale mimetic method and EDFM, this formulation can generate accurate velocity field and pressure field on the fine-scale grid more efficiently than the traditional methods. Numerical results are presented for verification of this multiscale mimetic approach for embedded discrete fracture media, and demonstrate its computational efficiency. The results show that this method is an accurate and efficient method for flow simulation in real-field fractured heterogeneous reservoirs.

Solving the tensorial 3D acoustic wave equation: A mimetic finite-difference time-domain approach

Generating accurate numerical solutions of the acoustic wave equation (AWE) is a key computational kernel for many seismic imaging and inversion problems. Although finite-difference time-domain (FDTD) approaches for generating full-wavefield solutions are well-developed for Cartesian computational domains, several challenges remain when applying FDTD approaches to scenarios arguably best described by more generalized geometry. In particular, how best to generate accurate and stable FDTD solutions for scenarios involving grids conforming to complex topography or internal surfaces. We address these issues by developing a mimetic FDTD (MFDTD) approach that combines four key components: a tensorial 3D AWE, mimetic finite-difference (MFD) operators, fully staggered grids (FSGs), and MFD Robin boundary conditions (RBC). The tensorial formulation of the 3D AWE permits wave propagation to be specified on (semi-) analytically defined coordinate meshes designed to conform to complex domain boundaries. MFD operators allow for higher order FD stencils to be applied throughout the model domain, including the boundary region where implementing centered FD stencils can be problematic. The FSG approach combines wavefield information propagated on four complementary subgrids to ensure the existence of all wavefield gradients required for computing the tensorial Laplacian operator, and thereby avoids interpolation approximations. The RBCs are implemented with a flux-preserving mimetic boundary operator that forestalls introduction of nonphysical energy into the grid by enforcing underlying flux-conservation laws. After validating the 3D MFDTD scheme on a sheared Cartesian mesh, we generate 3D wavefield simulation examples for internal boundary (IB) and topographic coordinate systems. The numerical examples demonstrate that the MFDTD scheme is capable of providing accurate and low-dispersion impulse responses for scenarios involving distorted IB meshes conforming to water-bottom surfaces and topographic coordinate systems exhibiting 2.5 km of topographic relief and including steep (65°) slope angles.

Mimetic finite difference methods for Hamiltonian wave equations in 2D

In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimensions. We construct a two step procedure in which we first discretize the space by the Mimetic Finite Difference (MFD) method and then we employ a standard symplectic scheme to integrate the semi-discrete Hamiltonian system derived. The main characteristic of the MFD methods, when applied to stationary problems, is to mimic important properties of the continuous system. This approach yields a full numerical procedure suitable to integrate Hamiltonian problems. A complete theoretical analysis of the method and some numerical simulations are developed in the paper.

Data assimilation method for fractured reservoirs using mimetic finite differences and ensemble Kalman filter

Optimal management of subsurface processes requires the characterization of the uncertainty in reservoir description and reservoir performance prediction. For fractured reservoirs, the location and orientation of fractures are crucial for predicting production characteristics. With the help of accurate and comprehensive knowledge of fracture distributions, early water/CO 2 breakthrough can be prevented and sweep efficiency can be improved. However, since the rock property fields are highly non-Gaussian in this case, it is a challenge to estimate fracture distributions by conventional history matching approaches. In this work, a method that combines vector-based level-set parameterization technique and ensemble Kalman filter (EnKF) for estimating fracture distributions is presented. Performing the necessary forward modeling is particularly challenging. In addition to the large number of forward models needed, each model is used for sampling of randomly located fractures. Conventional mesh generation for such systems would be time consuming if possible at all. For these reasons, we rely on a novel polyhedral mesh method using the mimetic finite difference (MFD) method. A discrete fracture model is adopted that maintains the full geometry of the fracture network. By using a cut-cell paradigm, a computational mesh for the matrix can be generated quickly and reliably. In this research, we apply this workflow on 2D two-phase fractured reservoirs. The combination of MFD approach, level-set parameterization, and EnKF provides an effective solution to address the challenges in the history matching problem of highly non-Gaussian fractured reservoirs.

A new equi-dimensional fracture model using polyhedral cells for microseismic data sets

We present a method for modeling flow in porous media in the presence of complex fracture networks. The approach utilizes the Mimetic Finite Difference (MFD) method. We employ a novel equi-dimensional approach for meshing fractures. By using polyhedral cells we avoid the common challenge in equi-dimensional fracture modeling of creating small cells at the intersection point. We also demonstrate how polyhedra can mesh complex fractures without introducing a large number of cells. We use polyhedra and the MFD method a second time for embedding fracture boundaries in the matrix domain using a “cut-cell” paradigm. The embedding approach has the advantage of being simple and localizes irregular cells to the area around the fractures. It also circumvents the need for conventional mesh generation, which can be challenging when applied to complex fracture geometries. We present numerical results confirming the validity of our approach for complex fracture networks and for different flow models. In our first example, we compare our method to the popular dual-porosity technique. Our second example compares our method with directly meshed fractures (single-porosity) for two-phase flow. The third example demonstrates two-phase flow for the case of intersecting ellipsoid fractures in three-dimensions, which are typical in microseismic analysis of fractures. Finally, we demonstrate our method on a two-dimensional fracture network produced from microseismic field data.

ArcFVDSL, a DSEL Combined to HARTS, a Runtime System Layer to Implement Efficient Numerical Methods to Solve Diffusive Problems on New Heterogeneous Hardware Architecture

Nowadays, some frameworks like Arcane and Dune offer a number of advanced tools to deal with the complexity related to parallelism, meshes and linear solvers. However, they do not handle the high level complexity related to discretization methods and physical models. Generative programming and Domain Specific Languages (DSL) are key technologies allowing to write code with a high level expressive language and take advantage of the efficiency of generated code with low level services. DSL may be embedded in host languages like Python or C++ . Such languages, named in that case Domain Specific Embedded Languages (DSEL), are applied for instance in frameworks like Fenics or Feel++ which are dedicated to the domain of Finite Element (FE) methods and Galerkin methods. ArcFVDSL is a DSEL developed on top of the Arcane framework, aiming to implement various lowest order methods (Finite-Volume (FV), Mimetic Finite Difference (MFD), Mixed Hybrid Finite Volume (MHFV), etc.) for diffusive problems on general meshes. In this paper, we present various implementations of different complex academic problems. We focus on the capability of the language to allow the description and the resolution of these problems with several lowest-order methods. We illustrate the benefits of such technology combined to runtime system tools like Heterogeneous Abstract RunTime System (HARTS) and its ability to handle seamlessly new heterogeneous architectures with multi-core processors enhanced by General Purpose computing on Graphics Processing Units (GP-GPU). We present the performance results of each implementation on different kinds of heterogeneous hardware architecture.

A generalized Mimetic Finite Difference method and Two-Point Flux schemes over Voronoi diagrams

We develop a generalization of the mimetic finite difference (MFD) method for second order elliptic problems that extends the family of convergent schemes to include two-point flux approximation (TPFA) methods over general Voronoi meshes, which are known to satisfy the discrete maximum principle. The method satisfies a modified consistency condition, which utilizes element and face weighting functions. This results in shifting the points on the elements and faces where the pressure and the flux are most accurately approximated. The flux bilinear form is non-symmetric in general, although it reduces to a symmetric form in the case of TPFA. It can be defined as the L 2 -inner product of vectors in two H ( Ω ;div) discrete spaces, which are constructed via suitable lifting operators. A specific construction of such lifting operators is presented on rectangles. We note that a different choice is made for test and trial spaces, therefore the method can be viewed as a H ( Ω ;div)-conforming Petrov–Galerkin Mixed Finite Element method. We prove first-order convergence in pressure and flux, and superconvergence of the pressure under further restrictions. We present numerical results that support the theory.

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality system of the control problem. Error estimates for state, adjoint and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution is discussed. These super-convergence results are shown to apply to non-conforming $\mathbb{P}_1$ finite elements, and to the mixed/hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, nonconforming and mixed/hybrid mimetic finite difference schemes.

Convergence Analysis of the mimetic Finite Difference Method for Elliptic Problems with Staggered Discretizations of Diffusion Coefficients

We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [K. Lipnikov, G. Manzini, J. D. Moulton, and M. Shashkov, J. Comput. Phys, 305 (2016), pp. 111--126]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic operator, i.e., the discrete divergence, and the inner product in the space of gradients. The diffusion coefficient is therefore evaluated on different mesh locations, i.e., inside mesh cells and on mesh faces. Such a staggered discretization may provide the flexibility necessary for future development of efficient numerical schemes for nonlinear problems, especially for problems with degenerate coefficients. These new mimetic schemes preserve symmetry and positive-definiteness of the continuum problem, which allow us to use efficient algebraic solvers such as the preconditioned conjugate gradient method. We show that these schemes are inf-sup stable and establish a priori error...

Bracket formulations and energy- and helicity-preserving numerical methods for the three-dimensional vorticity equation

The vorticity equation for three-dimensional viscous incompressible fluid flows is formulated within different bracket formalisms using the Poisson or Nambu bracket together with a dissipative bracket. The budgets of kinetic energy, helicity, and enstrophy derived from the bracket formulations are properly inherited by the finite difference equations obtained by invoking the discrete variational derivative method combined with the mimetic finite difference method. In particular, energy and helicity are conserved precisely in inviscid flow computations. The energy and enstrophy dissipate properly owing to viscosity in viscous flow computations, and the enstrophy is appropriately produced by the vortex stretching effect in both inviscid and viscous flow computations. The relationships between the stream function, velocity, and vorticity as well as the solenoidal conditions on the velocity and vorticity fields are also inherited. Numerical experiments on a periodic array of rolls that permits analytical solutions have been done to examine the properties and usefulness of the proposed method.

Two-phase numerical simulation of discrete fracture model based on multiscale mixed finite element method

Fractured reservoirs are characterized by complex fractures on multiple scales and are quite difficult to model. Numerical simulation of fractured media is usually done based on dual-porosity model and equivalent continuum model. Dual-porosity model treats matrix system and fracture system as two parallel continuous systems coupled by crossflow function. This model is only valid for reservoirs with highly developed fractures. Equivalent continuum model treats fractured media as a continuum media and equivalent absolute permeability tensors for each grid block are calculated to describe the heterogeneity of the reservoir. This model is efficient only when there exist representative element volume and the equivalent permeability are difficult to decide. Both models treat the fractured media as a simplified model and cannot describe the multiscale flows exactly, because they cannot precisely consider the diversion effect of the fractures. Although the discrete fracture network (DFN) model can provide a detailed representation of flow characteristic, traditional numerical method does not suitable for DFN. The major difficulty is the size of the computation. A tremendous amount of computer memory and CPU time are required, and this can exceed limit of today’s computer resources. Upscaling methods are generally used to reduce the computational cost. However, it is not possible to have a priori estimates of the errors that are present when complex flow processes are investigated using coarse models constructed via simplified settings. In this paper, multiscale mixed finite element method (MsMFEM) is proposed to simulate water/oil two phase flow in discrete fracture media. By combining MsMFEM with the discrete fracture model, we aim towards a numerical scheme that facilitates fractured reservoir simulation without upscaling. The MsMFEM uses a standard Darcy model to approximate pressure and fluxes on a coarse grid. The multiscale basis functions are constructed numerically by solving local differential equations on the fine-scale grid. The advantage of MsMFEM is that the basis functions capable of reflecting information about fractures within elements. Therefore, this method can capture the fine-scale effects on the coarse grid, that is, multiscale method can reduce the computational cost and keep high calculation accuracy at the same time. Traditional numerical methods generally difficult to deal with complex grid element, in this paper, mimetic finite difference (MFD) method is used to construct the multiscale basis functions due to its local conservativeness and applicability of complex grids. Compared with traditional multiscale mixed finite element methods, this method is suitable for arbitrary complex grid system. This paper introduced fundamental principles of the multiscale mixed finite element method and described the numerical scheme of discrete fracture model based on MsMFEM in detail. Then we deduced discrete fracture model computing formulate for the multiscale basis function by using mimetic finite difference method. Oversampling technique is applied to get more accurate small-scale details. IMPES scheme is used in the two-phase flow simulation. Physical experiment is used to prove the validity the multiscale method. The numerical results show that compared with traditional numerical method, the MsMFEM can represent the fine-scale flow in fracture networks exactly and meanwhile has a higher computational efficiency.

The mimetic finite difference method for the Landau–Lifshitz equation

The Landau–Lifshitz equation describes the dynamics of the magnetization inside ferromagnetic materials. This equation is highly nonlinear and has a non-convex constraint (the magnitude of the magnetization is constant) which poses interesting challenges in developing numerical methods. We develop and analyze explicit and implicit mimetic finite difference schemes for this equation. These schemes work on general polytopal meshes which provide enormous flexibility to model magnetic devices with various shapes. A projection on the unit sphere is used to preserve the magnitude of the magnetization. We also provide a proof that shows the exchange energy is decreasing in certain conditions. The developed schemes are tested on general meshes that include distorted and randomized meshes. The numerical experiments include a test proposed by the National Institute of Standard and Technology and a test showing formation of domain wall structures in a thin film.

An efficient hybrid model for fractured reservoirs

In this study, an efficient hybrid model is proposed to simulate the fluid flow in the reservoirs with multi-scale fractures, which cannot be easily modeled by neither the continuum models nor the discrete fracture models. In the proposed method, the small fractures are modeled by using an improved Multiple Sub-Region method, which can capture the strongly anisotropy of fracture elements and the effects of border region on the transmissibility and provide more accurate results, on the other hand, the large fractures are modeled explicitly as major fluid conduits by the Embedded Discrete Fracture Model. Then, an efficient numerical algorithm based on the Mimetic Finite Difference method is developed to solve the hybrid method. At the end, several numerical examples are carried out to verify the accuracy and applicability of the proposed numerical model.

Modeling anisotropic flow and heat transport by using mimetic finite differences

Modeling anisotropic flow in porous or fractured rock often assumes that the permeability tensor is diagonal, which means that its principle directions are always aligned with the coordinate axes. However, the permeability of a heterogeneous anisotropic medium usually is a full tensor. For overcoming this shortcoming, we use the mimetic finite difference method (mFD) for discretizing the flow equation in a hydrothermal reservoir simulation code, SHEMAT-Suite, which couples this equation with the heat transport equation. We verify SHEMAT-Suite-mFD against analytical solutions of pumping tests, using both diagonal and full permeability tensors. We compare results from three benchmarks for testing the capability of SHEMAT-Suite-mFD to handle anisotropic flow in porous and fractured media. The benchmarks include coupled flow and heat transport problems, three-dimensional problems and flow through a fractured porous medium with full equivalent permeability tensor. It shows firstly that the mimetic finite difference method can model anisotropic flow both in porous and in fractured media accurately and its results are better than those obtained by the multi-point flux approximation method in highly anisotropic models, secondly that the asymmetric permeability tensor can be included and leads to improved results compared the symmetric permeability tensor in the equivalent fracture models, and thirdly that the method can be easily implemented in existing finite volume or finite difference codes, which has been demonstrated successfully for SHEMAT-Suite.