Full-tensor Permeability Fields Research Articles

An online generalized multiscale approximation of the multipoint flux mixed finite element method

In this paper, we develop an online generalized multiscale approximation of the multipoint flux mixed finite element method (MFMFE) for Darcy flow in highly heterogeneous porous media, which can handle full tensor permeability fields and unstructured grids with local velocity elimination. Here, for illustration of the proposed method, we will mainly consider the situation that the porous medium is filled with fractures, which usually results in unstructured grids, and the extension to the situation that the permeability is represented as a full tensor is straightforward. From the fine-grid discretization by the MFMFE method, we derive the symmetric and positive definite discrete bilinear form for both the matrix and fracture pressure, and thereby obtain the corresponding discrete weak formulation that only related to pressure. In the offline stage, we compute offline basis functions which contain the important local multiscale information of each coarse-grid block to form the initial multiscale space. In the online stage, we construct and add online basis functions to enrich the multiscale space and consequently improve the accuracy of the multiscale solution. Each offline basis function and each online basis function contains multiscale information both for matrix and fracture. We give the theoretical analysis for the convergence of the online enrichment, which shows that more sufficient initial basis functions lead to faster convergence rates. A series of numerical examples that either the permeability is a full tensor or the porous medium is filled with fractures are presented to show the performance of the multiscale method.

A hybrid-dimensional discrete fracture model for non-isothermal two-phase flow in fractured porous media

We present a hybrid-dimensional numerical model for non-isothermal two-phase flow in fractured porous media, in which the fractures are modeled as entities of codimension one embedded in a bulk domain. Potential fields of applications of the model could be radioactive waste disposal or geothermal energy production scenarios in which a two-phase flow regime develops or where $$\hbox {CO}_2$$ is used as working fluid. We test the method on synthetic test cases involving compressible fluids and strongly heterogeneous, full tensor permeability fields by comparison with a reference solution obtained from an equi-dimensional discretization of the domain. The results reveal that especially for the case of a highly conductive fracture, the results are in good agreement with the reference. While the model qualitatively captures the involved phenomena also for the case of a fracture acting as both hydraulic and capillary barrier, it introduces larger errors than in the highly-conductive fracture case, which can be attributed to the lower-dimensional treatment of the fracture. Finally, we apply the method to a three-dimensional showcase that resembles setups for the determination of upscaled parameters of fractured blocks.

A Cell-Centred CVD-MPFA Finite Volume Method for Two-Phase Fluid Flow Problems with Capillary Heterogeneity and Discontinuity

A novel finite volume method is presented that is applicable to discontinuous capillary pressure fields. The method is developed within the control-volume distributed multi-point flux approximation (CVD-MPFA) framework (Edwards and Rogers in Comput Geosci 02(04):259–290, 1998; Friis et al. in SIAM J Sci Comput 31(02):1192–1220, 2008). Results are computed on structured and unstructured grids that demonstrate the ability of the method to resolve flow in the presence of a discontinuous capillary pressure field for diagonal and full-tensor permeability fields. In addition to an upwind approximation for the saturation equation flux, the importance of upwinding on capillary pressure flux via a hybrid formulation is shown.

CVD-MPFA full pressure support, coupled unstructured discrete fracture–matrix Darcy-flux approximations

Two novel control-volume methods are presented for flow in fractured media, and involve coupling the control-volume distributed multi-point flux approximation (CVD-MPFA) constructed with full pressure support (FPS), to two types of discrete fracture–matrix approximation for simulation on unstructured grids; (i) involving hybrid grids and (ii) a lower dimensional fracture model. Flow is governed by Darcy's law together with mass conservation both in the matrix and the fractures, where large discontinuities in permeability tensors can occur. Finite-volume FPS schemes are more robust than the earlier CVD-MPFA triangular pressure support (TPS) schemes for problems involving highly anisotropic homogeneous and heterogeneous full-tensor permeability fields. We use a cell-centred hybrid-grid method, where fractures are modelled by lower-dimensional interfaces between matrix cells in the physical mesh but expanded to equi-dimensional cells in the computational domain. We present a simple procedure to form a consistent hybrid-grid locally for a dual-cell. We also propose a novel hybrid-grid for intersecting fractures, for the FPS method, which reduces the condition number of the global linear system and leads to larger time steps for tracer transport. The transport equation for tracer flow is coupled with the pressure equation and provides flow parameter assessment of the fracture models. Transport results obtained via TPS and FPS hybrid-grid formulations are compared with the corresponding results of fine-scale explicit equi-dimensional formulations. The results show that the hybrid-grid FPS method applies to general full-tensor fields and provides improved robust approximations compared to the hybrid-grid TPS method for fractured domains, for both weakly anisotropic permeability fields and very strong anisotropic full-tensor permeability fields where the TPS scheme exhibits spurious oscillations. The hybrid-grid FPS formulation is extended to compressible flow and the results demonstrate the method is also robust for transient flow. Furthermore, we present FPS coupled with a lower-dimensional fracture model, where fractures are strictly lower-dimensional in the physical mesh as well as in the computational domain. We present a comparison of the hybrid-grid FPS method and the lower-dimensional fracture model for several cases of isotropic and anisotropic fractured media which illustrate the benefits of the respective methods.

Higher order cell-based multidimensional upwind schemes for flow in porous media on unstructured grids

Standard reservoir simulation schemes employ single-point upstream weighting for convective flux approximation. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent.New locally conservative cell-based multi-dimensional upwind schemes and higher-order cell-based multi-dimensional upwind schemes that reduce both directional and cross-wind diffusion are presented for convective flow approximation. The new higher-order schemes are comprised of two steps; (a) Higher-order approximation that corrects the directional diffusion of the approximation. (b) Truly multi-dimensional upwind approximation, which involves flux approximation using upwind information obtained by upstream tracing along multi-dimensional flow paths. This approximation reduces cross-wind diffusion. Conditions on tracing direction and CFL number lead to a local maximum principle that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations.Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve a range of unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical cross-wind diffusion effects, leading to improved resolution of concentration and saturation fronts.

A Mimetic Finite Volume Discretization Method for Reservoir Simulation

Summary Accurate representation of complex reservoir geology using nonorthogonal meshes, upscaling of high-resolution geostatistical reservoir models including cross-flow effects, and strongly heterogeneous anisotropic permeable media such as cross-bedded sands and thin-bedded turbidite channels all individually or in combination give rise to simulation models with full-tensor permeability fields. Such anisotropic multiphase flow problems can be solved accurately on arbitrary nonorthogonal structured and unstructured meshes using the mimetic finite volume method. The discretization operator of the mimetic finite volume method satisfies conservation laws and theorems of vector and tensor calculus on nonorthogonal, smoothly and nonsmoothly distorted, structured and unstructured computational meshes. In this paper, we demonstrate the formulation of a 3D variant of the mimetic finite volume method for corner-point geometry hexahedral meshes. We also discuss the results and issues associated with the implementation of the mimetic finite volume discretization operator in a parallel, fully implicit reservoir simulator developed using a general compositional formulation. Simulation results are presented for a variety of cases involving flow through geologically complex systems.

Higher Order Multidimensional Upwind Convection Schemes for Flow in Porous Media on Structured and Unstructured Quadrilateral Grids

Standard reservoir simulation schemes employ first order upwind schemes for approximation of the convective fluxes when multiple phases or components are present. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent. New locally conservative higher-order multidimensional upwind schemes that significantly reduce both directional and cross-wind diffusion are presented for convective flow approximation. The new schemes are composed of two steps; (a) higher order approximation that corrects the directional diffusion of the approximation and (b) truly multidimensional upwind approximation, which involves flux approximation using upwind information obtained by upstream tracing along wave-vector paths where wave information travels in multiple dimensions. This approximation reduces cross-wind diffusion. Conditions on the tracing direction and Courant–Friedrichs–Levy number lead to a local maximum principle that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations. Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation, including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve structured and unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical cross-wind diffusion effects, leading to improved resolution of concentration and saturation fronts.

Multidimensional upwind convection schemes for flow in porous media on structured and unstructured quadrilateral grids

Standard reservoir simulation schemes employ first order upwind schemes for approximation of the convective fluxes when multiple phases or components are present. These convective flux schemes rely upon upwind information that is determined according to grid geometry. As a consequence directional diffusion is introduced into the solution that is grid dependent. The effect can be particularly important for cases where the flow is across grid coordinate lines and is known as cross-wind diffusion. Truly higher dimensional upwind schemes that minimize cross-wind diffusion are presented for convective flow approximation on quadrilateral unstructured grids. The schemes are locally conservative and yield improved results that are essentially free of spurious oscillations. The higher dimensional schemes are coupled with full tensor Darcy flux approximations. The benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full tensor permeability fields. The test cases involve a range of structured and unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce the numerical cross-wind diffusion effect, leading to improved resolution of concentration and saturation fronts.

Higher‐resolution convection schemes for flow in porous media on highly distorted unstructured grids

AbstractHigher‐resolution schemes are presented for convective flow approximation on highly distorted unstructured grids. The schemes are coupled with continuous full‐tensor Darcy‐flux approximations. A sequence of non‐uniform and distorted grid formulations are developed and compared for a range of unstructured meshes with variable grid spacing. The higher‐order schemes are constructed using non‐uniform grid slope limiters such that they are stable with a local maximum principle, ensuring that solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full‐tensor permeability fields. The test cases involve a range of unstructured grids with variations in grid spacing, orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher‐order formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts. Copyright © 2008 John Wiley & Sons, Ltd.

Upscaling of channel systems in two dimensions using flow-based grids

A methodology for the gridding and upscaling of geological systems characterized by channeling is presented. The overall approach entails the use of a flow-based gridding procedure for the generation of variably refined grids capable of resolving the channel geometry, a specialized full-tensor upscaling method to capture the effects of permeability connectivity, and the use of a flux-continuous finite volume method applicable to full tensor permeability fields and non-orthogonal grids. The gridding and upscaling procedures are described in detail and then applied to several two-dimensional systems. Significant improvement in the accuracy of the coarse scale models, relative to that obtained using uniform Cartesian coarse scale models, is achieved in all cases. It is shown that, for some systems, improvement results from the use of the flow-based grid, while in other cases the improvement is mainly due to the new upscaling method.

Use of Border Regions for Improved Permeability Upscaling

A procedure for the improved calculation of upscaled grid block permeability tensors on Cartesian grids is described and applied. The method entails the use of a border region of fine-scale cells surrounding the coarse block for which the upscaled permeability is to be computed. The implementation allows for the use of full-tensor permeability fields on the fine and coarse scales. Either periodic or pressure–no flow boundary conditions are imposed over the extended local domain (target block plus border regions) though averaged quantities, used to compute the upscaled permeability tensor, are computed only over the target block region. Flow and transport results using this procedure are compared to those from standard methods for different types of geological and simulation models. Improvement using the new approach is consistently observed for the cases considered, though the degree of improvement varies for different models and flow quantities.