Mimetic Finite Difference Method Research Articles

Modeling acoustic wavefields from moving sources in the presence of a time-varying free surface

Marine vibrators are increasingly being recognized as a viable alternative to seismic airguns for ocean-bottom acquisition due to their ability to generate more low-frequency content and their less adverse impact on marine wildlife. However, their use introduces processing challenges, such as the Doppler effect and time-dependent source-receiver offsets, which are negligible in conventional airgun acquisition. In addition, the time-varying nature of the sea surface during the multisecond acquisition time introduces further challenges for processing and inversion. To accurately account for source motion and time-varying sea-surface effects in seismic data processing, we develop a reliable and robust numerical modeling tool. We use a mimetic finite-difference method in a generalized coordinate system to model the full acoustic wavefield triggered by a moving source in the presence of a time-varying sea surface. Our approach uses a time- and space-dependent coordinate transformation, which tracks the source movement and conforms to the irregular time-varying sea surface, to map an irregular physical domain in Cartesian coordinates to a regular computational domain in generalized coordinates. We formulate this coordinate transformation such that the coordinate systems conformally match below the ocean-bottom level. Numerical examples demonstrate that this approach is accurate and stable, even for an unrealistically exaggerated sea state. This computational tool is not limited to modeling but also could be used to develop advanced processing techniques for marine vibrator data, such as imaging and inversion.

Time-dependent model for two-phase flow in ultra-high water-cut reservoirs: Time-varying permeability and relative permeability

For the ultra-high water-cut reservoirs, after long-term water injection exploitation, the physical properties of the reservoir change and the heterogeneity of the reservoir becomes increasingly severe, which further aggravates the spatial difference of the flow field. In this study, the displacement experiments were employed to investigate the variations in core permeability, porosity, and relative permeability after a large amount of water injection. A relative permeability endpoint model was proposed by utilizing the alternating conditional expectation (ACE) transformation to describe the variation in relative permeability based on the experimental data. Based on the time dependent models for permeability and relative permeability, the traditional oil-water two-phase model was improved and discretized using the mimetic finite difference method (MFD). The two cases were launched to confirm the validation of the proposed model. The impact of time-varying physical features on reservoir production performance was studied in a real water flooding reservoir. The experimental results indicate that the overall relative permeability curve shifts to the right as water injection increases. This shift corresponds to a transition towards a more hydrophilic wettability and a decrease in residual oil saturation. The endpoint model demonstrates excellent accuracy and can be applied to time-varying simulations of reservoir physics. The impact of variations in permeability and relative permeability on the reservoir production performance yields two distinct outcomes. The time-varying permeability of the reservoir results in intensified water channeling and poor development effects. On the other hand, the time-varying relative permeability enhances the oil phase seepage capacity, facilitating oil displacement. The comprehensive time-varying behavior is the result of the combined influence of these two parameters, which closely resemble the actual conditions observed in oil field exploitation. The time-varying simulation technique of reservoir physical properties proposed in this paper can continuously and stably characterize the dynamic changes of reservoir physical properties during water drive development. This approach ensures the reliability of the simulation results regarding residual oil distribution.

A mimetic interpolation-free cell-centered finite volume scheme for the 2D and 3D heterogeneous anisotropic diffusion equations

We propose a mimetic interpolation-free cell-centered finite volume scheme for diffusion equations on arbitrary two-dimensional (2D) and three dimensional (3D) unstructured, possibly non star-shaped or nonplanar meshes. This scheme borrows ideas from the well known mimetic finite difference methods. This interpolation-free means that we do not need to introduce auxiliary unknowns in the process of constructing the scheme, and the final scheme has only cell-centered unknowns. After carefully handling continuity of solutions and normal fluxes, we construct piecewise linear approximations on each cell. This makes our scheme applicable to arbitrary discontinuities and arbitrary mesh topologies. Interpolation-free makes the program easy to implement and the 2D and 3D programs have a unified framework. It is worth mentioning that the traditional star-shaped mesh assumption has been abolished, and the new scheme has more relaxed restrictions on meshes. Some representative and even challenging numerical examples verify that the scheme is robust and almost has the optimal convergence order on some benchmark polygonal and polyhedral meshes.

Hybrid Mimetic Finite Difference and Streamline Methods for Numerical Simulation of Two-phase Flow in Fractured Reservoirs

In this work, we present a novel numerical formulation for efficiently modeling two-phase flow in fractured reservoirs. The novel formulation is developed by hybridizing mimetic finite difference and streamline (MFD-SL) methods, in which mimetic finite difference (MFD) is employed to discretize the pressure equation, and the streamline (SL) method is adopted to solve the saturation equation along 1D streamlines. The hybrid formulation is implemented on discrete fracture model (DFM) and is operated in an IMPES-like manner. A simple and practical streamline tracing method is developed on 2D triangular and 3D tetrahedral grids that are commonly implemented in the DFM framework. We benchmark the hybrid MFD-SL formulation with others using various cases with different complexity. Results show the proposed approach could achieve promising results yet with lower computation costs compared to other classical formulations. These achievements benefit from utilizing streamline tracing, which significantly alleviates the numerical diffusion error and improves the computation efficiency. This work, for the first time, incorporates hybrid MFD-SL formulation into DFM in case of triangular and tetrahedral grids and has great potential to be deployed on reservoir-scale simulation workflow.

A novel streamline simulation method for fractured reservoirs with full-tensor permeability

In this work, we develop a novel streamline (SL) simulation method that integrates seamlessly within the embedded discrete fracture model (EDFM). The novel SL-based method is developed based on a hybrid of two-point flux approximation (TPFA) and mimetic finite difference (MFD) methods, which is applicable to a two-phase anisotropic flow in fractured reservoirs. We refer to this novel approach as EDFM-TPFA-MFD-SL. The approach is operated in an IMplicit Pressure Explicit Saturation (IMPES) manner. First, this work establishes a novel EDFM utilizing a hybrid TPFA-MFD scheme to solve the pressure equation for phase flux approximation. Subsequently, we introduce a practical streamline tracing workflow designed to derive the distribution of streamlines within the reservoir domain and the time-of-flight distribution along each individual streamline. This feature allows for the parallel computation of water saturation along the streamlines. Two numerical examples are presented to validate the superiority of the proposed EDFM-TPFA-MFD-SL method compared to the existing streamline-based EDFM on cases with full-tensor permeability. The results show that the proposed method could significantly mitigate the numerical dissipation and reduce the computation costs. Another numerical example demonstrates the effectiveness of the proposed method in dealing with complex fracture networks and providing rapid flow diagnostics, indicating its significant potential for real-world field applications.

Adaptive Dynamic Grids and Mimetic Finite Difference Method for Miscible Displacement Problem

Adaptive Dynamic Grids and Mimetic Finite Difference Method for Miscible Displacement Problem

A Novel Projection-based Embedded Discrete Fracture Model (pEDFM) for Anisotropic Two-phase Flow Simulation Using Hybrid of Two-point Flux Approximation and Mimetic Finite Difference (TPFA-MFD) Methods

This paper presents a novel projection-based embedded discrete fracture model (pEDFM) applicable to general two-phase flow simulation in anisotropic reservoirs for the first time. The novel pEDFM is developed based on hybrid two-point flux approximation and mimetic finite difference (TPFA-MFD) methods, which retains the basic features of two-point flux approximation (TPFA) based generic pEDFM workflow and incorporates new features of mimetic finite difference (MFD) to handle full-tensor permeabilities. This is achieved by introducing the concept of effective connection, formulating additional pressure degrees of freedom for effective connections, assigning effective matrix-matrix (m-m) and matrix-fracture (m-f) connections to the corresponding sides of matrix-cell interfaces, presenting methods corresponding to different types of connections to handle low-conductivity fractures, and further deriving formulas of numerical fluxes for effective connections. The resulting global equation system is solved by a Newton-Raphson nonlinear solver with the temporary discretization of the implicit backward Euler scheme. The 2D and 2.5D numerical examples show the novel pEDFM could achieve the same level of computation accuracy and convergence as the reference discrete fracture model (DFM), while avoiding the gridding difficulty of complex fracture networks. It also significantly outperforms classical EDFM based on MFD and generic pEDFM workflow in the computational accuracy for cases with anisotropic full tensor permeabilities. The novel pEDFM, to our best knowledge, is for the first time proposed and developed for general anisotropic two-phase flow simulation and maybe the numerical simulation framework for multi-phase flow in fractured reservoirs with the best comprehensive performance so far.

The curved mimetic finite difference method: Allowing grids with curved faces

We present a new mimetic finite difference method for diffusion problems that converges on grids with curved (i.e., non-planar) faces. Crucially, it gives a symmetric discrete problem that uses only one discrete unknown per curved face. The principle at the core of our construction is to abandon the standard definition of local consistency of mimetic finite difference methods. Instead, we exploit the novel and global concept of P0-consistency. Numerical examples confirm the consistency and the optimal convergence rate of the proposed mimetic method for cubic grids with randomly perturbed nodes as well as grids with curved boundaries.

The tensor-train mimetic finite difference method for three-dimensional Maxwell’s wave propagation equations

Coupling the mimetic finite difference method with the tensor-train format results in a very effective method for low-rank numerical approximations of the solutions of the time-dependent Maxwell wave propagation equations in three dimensions. To this end, we discretize the curl operators on the primal/dual tensor product grid complex and we couple the space discretization with a staggered-in-time second-order accurate time-marching scheme. The resulting solver is accurate to the second order in time and space, and still compatible, so that the approximation of the magnetic flux field has zero discrete divergence with a discrepancy close to the machine precision level. Our approach is not limited to the second-order of accuracy. We can devise higher-order formulations in space through suitable extensions of the tensor-train stencil to compute the derivatives of the mimetic differential operators. Employing the tensor-train format improves the solver performance by orders of magnitude in terms of CPU time and memory storage. A final set of numerical experiments confirms this expectation.

Accurate Two-Phase Flow Simulation in Faulted Reservoirs by Combining Two-Point Flux Approximation and Mimetic Finite Difference Methods

Summary Faulted reservoirs are commonly modeled by corner-point grids (CPGs). Because the two-point flux approximation (TPFA) method is not consistent on non-K-orthogonal grids, multi-phase flow simulation using TPFA on CPGs may have significant discretization errors if grids are not K-orthogonal. We developed a novel method to improve the simulation accuracy where the faults are modeled by polyhedral cells, and mimetic finite difference (MFD) methods are used to solve flow equations. We use a cut-cell approach to build the mesh for faulted reservoirs. A regular K-orthogonal grid is first constructed, and then cells are divided where fault planes are present. Most cells remain K-orthogonal while irregular non-K-orthogonal polyhedral cells can be formed with multiple cell divisions. We investigated three spatial discretization methods for solving the pressure equation on general polyhedral grids, including the TPFA, MFD, and TPFA-MFD hybrid methods. In the TPFA-MFD hybrid method, the MFD method is only applied to the part of the domain with severe grid non-K-orthogonality, while the TPFA method is applied to the rest of the domain. We compare flux accuracy between TPFA and MFD methods by solving a single-phase flow problem. The reference solution is obtained on a rectangular grid, while the same problem is solved by TPFA and MFD methods on a grid with non-K-orthogonal cells near a fault. Fluxes computed using TPFA exhibit larger errors in the vicinity of the fault, while fluxes computed using MFD are still as accurate as the reference solution. We also compare saturation accuracy for two-phase (oil and water) flow in faulted reservoirs when the pressure equation is solved by different discretization methods. Compared with the reference saturation solution, saturation exhibits non-physical errors near the fault when the pressure equation is solved by the TPFA method. Because the MFD method yields accurate fluxes over general polyhedral grids, the resulting saturation solutions agree with reference saturation solutions with an enhanced accuracy when the pressure equation is solved by the MFD method. Based on the results of our simulation studies, we observe that the accuracy of the TPFA-MFD hybrid method is very close to the accuracy of the MFD method, while the TPFA-MFD hybrid method is computationally cheaper than the MFD method.

Flow and transport in three-dimensional discrete fracture matrix models using mimetic finite difference on a conforming multi-dimensional mesh

We present a comprehensive workflow to simulate single-phase flow and transport in fractured porous media using the discrete fracture matrix approach. The workflow has three primary parts: (1) a method for conforming mesh generation of and around a three-dimensional fracture network, (2) the discretization of the governing equations using a second-order mimetic finite difference method, and (3) implementation of numerical methods for high-performance computing environments. A method to create a conforming Delaunay tetrahedralization of the volume surrounding the fracture network, where the triangular cells of the fracture mesh are faces in the volume mesh, that addresses pathological cases which commonly arise and degrade mesh quality is also provided. Our open-source subsurface simulator uses a hierarchy of process kernels (one kernel per physical process) that allows for both strong and weak coupling of the fracture and matrix domains. We provide verification tests based on analytic solutions for flow and transport, as well as numerical convergence. We also provide multiple expositions of the method in complex fracture networks. In the first example, we demonstrate that the method is robust by considering two scenarios where the fracture network acts as a barrier to flow, as the primary pathway, or offers the same resistance as the surrounding matrix. In the second test, flow and transport through a three-dimensional stochastically generated network containing 257 fractures is presented.

A New Approach to Reduce Transverse Shear Locking of Reissner–Mindlin Plate Based on The Cell-Centered Finite Element Method

This study investigates a new approach to solve a shear locking free on a low-order cell-centered finite element method (FECC) of the Reissner–Mindlin plate. The proposed method pursues the Lagrange piecewise of linear polynomial functions to attain the transverse displacement and rotations on the dual sub-mesh derived from the general one. The suggested theory is employed in numerical computation to verify the effectiveness and accurate performance. Detailed studies are carried out on several mesh types to compare with the analytical solutions reported by the low-order mimetic finite difference method [Beir ao Da Veiga, L., Lovadina, C. and Mora, D. [2013] “Numerical results for mimetic discretization of Reissner–Mindlin plate problems,” Calcolo 50, 209–237] and high-order mixed finite element methods [Chinosi, C. and Lovadina, C. [1995] “Numerical analysis of some mixed finite element methods for Reissner–Mindlin plates,” Comput. Mech. 16, 36–44]. The numerical results indicated that the present scheme based on the low-order FECC approximation can effectively predict on even the distorted meshes, satisfy the optimized convergence and be uniformly stable to the thin plate thicknesses.

Novel numerical techniques based on mimetic finite difference method for pricing two dimensional options

The Black–Scholes differential operator which underlies the option pricing of European and American options is known to be degenerate close to the boundary at zero. At this singularity, important properties of the differential operator are lost and the classical finite difference scheme applied to this problem fails to give accurate approximations as it is no longer monotone. In this paper novel numerical techniques based on mimetic finite difference method are proposed for accurately pricing European and American options. More precisely, we propose the mimetic and fitted mimetic finite difference methods, which are techniques that preserve and conserve general properties of the continuum operator in the discrete case. The fitted method further handles the degeneracy of the underlying partial differential equations (PDE). Those spatial discretization methods are coupled with the Euler implicit method for time discretization. Several numerical simulations are performed to demonstrate the robustness of our methods comparing to standard fitted finite volume method for both European and American put options.

An implementation of mimetic finite difference method for fractured reservoirs using a fully implicit approach and discrete fracture models

In this paper, we present a fully implicit mimetic finite difference method (MFD) for general fractured reservoir simulation. The MFD is a novel numerical discretization scheme that has been successfully applied to many fields and it is characterized by local conservation properties and applicability to complex grids. In our work, we extend this method to the numerical simulation of fractured reservoirs using discrete fracture models. The MFD scheme supports general polyhedral meshes and full tensor properties which improves the modeling and simulation of subsurface reservoirs. Furthermore, we describe in detail the principle of our MFD approach and the corresponding numerical formulations of the discrete fracture model. In our tests, we use a fully implicit scheme that assures flux conservation and simulation efficiency. Several case studies are conducted to show the accuracy and the robustness of the proposed numerical scheme.

Un análisis comparativo de los métodos: miméticos, diferencias finitas y elementos finitos para problemas estacionarios 1-dimensional

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

Mimetic finite difference methods for restoration of fundus images for automatic detection of glaucoma suspects

ABSTRACT Glaucoma is one of the leading causes of irreversible blindness in people over 40 years old. Previously, it was developed acomputational tool for automatic glaucoma detection, which implements anovel method that has shown improvement in the accuracy of the detection compared to other classical methods. However, this method is sensitive to the quality of the acquired image. For this reason, automatic image restoration of the source images is needed to improve the quality of glaucoma suspect detection. We propose to use the Perona–Malik anisotropic diffusion filter as part of the pre-processing step because apart from attenuating noise and brilliance in the images, it preserves essential information such as the borders of the optic disc and the cup boundaries, which are of particular interest in this work. We solve numerically the problem in partial differential equations, which represent the Perona–Malik diffusion filter, using explicit finite mimetic differences methods, which have the advantage of preserving the continuum properties of the mathematical operators often encountered in image processing and analysis equations, ensuring better orders of convergence. By guaranteeing these mathematical properties, the original structure of the source image is maintained, improving the diagnosis of the patient.

Modeling the effects of capillary pressure with the presence of full tensor permeability and discrete fracture models using the mimetic finite difference method

Capillary dominated flow or imbibition—whether spontaneous or forced—is an important physical phenomena in understanding the behavior of naturally fractured water-driven reservoirs (NFR’s). When the water flows through the fractures, it imbibes into the matrix and pushes the oil out of the pores due to the difference in the capillary pressure. In this paper, we focus on modeling and quantifying the oil recovered from NFR’s through the imbibition processes using a novel fully implicit mimetic finite difference (MFD) approach coupled with discrete fracture/discrete matrix (DFDM) technique. The investigation is carried out in the light of different wetting states of the porous media (i.e., varying capillary pressure curves) and a full tensor representation of the permeability. The produced results proved the MFD to be robust in preserving the physics of the problem, and accurately mapping the flow path in the investigated domains. The wetting state of the rock affects greatly the oil recovery factors along with the orientation of the fractures and the principal direction of the permeability tensor. We can conclude that our novel MFD method can handle the fluid flow problems in discrete-fractured reservoirs. Future works will be focused on the extension of MFD method to more complex multi-physics simulations.

Forward modelling of geophysical electromagnetic data on unstructured grids using an adaptive mimetic finite-difference method

While the mimetic finite-difference method shares many similarities with the finite-element and finite-volume methods, it has the advantage of naturally accommodating grids with arbitrary polyhedral elements. In this study, we use this attribute to develop an adaptive scheme for the solution of the geophysical electromagnetic modelling problem on unstructured grids. Starting with an initial tetrahedral grid, our mesh adaptivity implements an iterative h-refinement where a residual- and jump-based goal-oriented error estimator is used to mark a certain portion of the elements. The marked elements are decomposed into new tetrahedra by regular subdivision, creating an octree-like unstructured grid. Since arbitrary polyhedra are naturally permitted in the mimetic finite-difference method, the added nodes are not regarded as hanging nodes and hence any level of non-conformity can be implemented without a modification to the mimetic scheme. In this study, we consider 2-irregularity where two levels of non-conformity between the adjacent elements is permitted. We use a total field approach where the electric field is defined at the edges of the polyhedral elements and the electromagnetic source may have an arbitrary shape and location. The accuracy of the mimetic scheme and the effectiveness of the proposed mesh adaptivity are verified using benchmark and realistic examples that represent various magnetotelluric and controlled-source survey scenarios. The mesh adaptivity generates grids with refinement generally concentrated at the transmitter and receiver locations and the interfaces of materials with contrasting conductivities, and the mimetic finite-difference solutions have good agreement with the reference numerical and real data. We also demonstrate the practicality of our method using an example with an analytical solution and comparison with a standard mesh regeneration technique. The results show that our mesh adaptivity procedure can result in a higher accuracy, with similar numbers of elements, when compared with the mesh regeneration approach. Also, using a generic sparse direct solver, our method is found to be more efficient than the mesh regeneration approach in terms of computation time and memory usage. Moreover, a comparison between 1- and 2-irregularity shows the higher efficiency of the latter in terms of the number of elements required to reach a certain level of accuracy.

Convergence of the mimetic finite difference and fitted mimetic finite difference method for options pricing

We present in this paper two novel numerical spatial discretization techniques based on the mimetic finite difference method for a degenerated partial differential equation (PDE) in one dimension. This PDE is well known as the Black-Scholes PDE which govern option pricing. To handle the degeneracy of the PDE, a novel fitted mimetic finite difference scheme is proposed together with the standard mimetic finite difference method. The temporal discretization is performing using standard implicit scheme. Furthermore rigorous convergence proofs in appropriate normed spaces are proposed. We validate the theoretical results by presenting numerical results and simulations. Those numerical experiments show that our two novel schemes outperform the standard finite difference method and the standard fitted finite volume method in terms of accuracy.

Coupling surface flow and subsurface flow in complex soil structures using mimetic finite differences

We explore the coupling of surface and subsurface flows on fully unstructured meshes that conform to complex soil structures. To accommodate the distorted meshes that inevitably result from explicit representation of complex soil structures, we leverage the structure of the Mimetic Finite Difference (MFD) spatial discretization scheme to couple surface and subsurface flows. The MFD method achieves second-order accuracy and maintains local mass conservation on distorted meshes. We couple the diffusion wave approximation for surface flows to the Richards equation for subsurface flow, ensuring continuity of both pressure and flux between the surface and subsurface. The MFD method is particularly convenient for this coupling because it uses face-based constraints in the subsurface system that can be expressed as face-pressure unknowns. Those unknowns are coincident with surface cell-based unknowns, thus allowing the discrete surface system to be directly substituted into the subsurface system and solved implicitly as a global system. Robust representation of the transition between wet and dry surface conditions requires upwinding of the relative permeability and is facilitated by globalization in the nonlinear solver. The approach and its implementation in the Advanced Terrestrial Simulator (ATS) are evaluated by comparison to previously published benchmarks. Using runoff from soils with patchy groundcover (duff) as an example, we show that the new method converges significantly faster in mesh convergence tests than the commonly used two-point flux approximation.