Control-volume Distributed Multi-point Flux Approximation Research Articles

A nonlinear repair technique for the MPFA-D scheme in single-phase flow problems and heterogeneous and anisotropic media

A novel Flux Limited Splitting (FLS) non-linear Finite Volume (FV) method for families of linear Control Volume Distributed Multi Point Flux Approximation (CVD-MPFA) schemes is presented. The new formulation imposes a local discrete maximum principal (LDMP) which ensures that the discrete solution is free of spurious oscillations. The FLS scheme can be seen as a natural extension of the M-Matrix Flux Splitting method that splits the MPFA flux components in terms of the Two-Point Flux Approximation (TPFA) flux and Cross Diffusion Terms (CDT), with the addition of a dynamically computed relaxation parameter to the CDT that identifies and locally corrects the regions where the LDMP is violated. Moreover, the whole non-linear procedure was devised as a series of simple straightforward matrix operations. The methodology is presented considering the Multi-Point Flux Approximation with a Diamond (MPFA-D) in what we call the FLS + MPFA-D formulation which is tested using a series of challenging benchmark problems. For all test cases, the FLS repair technique imposes the LDMP and eliminates the spurious oscillations induced by the original MPFA-D method.

An algebraic multiscale solver for the simulation of two-phase flow in heterogeneous and anisotropic porous media using general unstructured grids (AMS-U)

We introduce a new multiscale finite volume framework for the simulation of multi-phase flow in heterogenous and anisotropic porous media on quite general unstructured grids that enable geophysical grid defined properties to be used directly on a high definition grid.A novel background grid strategy is presented, where an auxiliary mesh aids the creation of multiscale primal and dual coarse grids. Due to the unstructured grid connectivity, the dual grids do not retain the natural structured grid decomposition. This may induce errors in the computed basis functions, and contribute to loss of mass conservation. A generalization of the Algebraic Multiscale Solver (AMS) is proposed to prevent the basis function induced leakage outside the support region enabling coupling with a Control Volume Distributed MultiPoint Flux Approximation (CVD-MPFA), ensuring consistency on non k-orthogonal unstructured grids.We present three different problems in which the accuracy of the framework is validated by comparing the multiscale solver with the direct simulation on the fine-scale. The results obtained show that the method can produce well resolved solutions for two-phase flow in highly heterogeneous and anisotropic porous media using general unstructured grids on all scales. As a consequence, the method captures the most important flow features that result on high-resolution geophysical models while providing suitable discretization for complex geological formations found in current petroleum reservoir problems.The novelty of this scheme involves three components which are combined to enable consistent multiscale simulation on unstructured grids: i) a new approach to primal and dual coarse grid generation, ii) a novel technique to prevent basis function induced leakage and iii) the coupling of the Algebraic Multiscale Solver (AMS) with a Multipoint Flux approximation with a Diamond Stencil (MPFA-D).

Sequential coupling of thermal module into the reservoir simulator DMflow

In this work, we present a numerical framework for non-isothermal compositional compressible gas-water phase flow simulations in non-fractured and fractured porous media on structured and unstructured meshes. The framework is implemented as the simulator “DMflow”. The discretization of the mass conservation equations is based on the fully-implicit scheme and cell-centered finite-volume control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a lower dimensional fractured model. Similarly, the discretization of the energy equation is also based on the same method. The recent newly developed enthalpy model of CO2−CH4-H2S-N2-O2-brine system (Guo et al., 2019), which is validated within a wide temperature and pressure range, is used in the energy equation. Here, we present a sequential implicit method for the non-isothermal flow and avoid the simultaneous solution of the fluid flow primary variables and the temperature. We present the validation cases where the 1D and 2D results for non-isothermal carbon dioxide (CO2) injection into water obtained by the new implemented numerical framework are in agreement with the CMG-GEM. The simulation results of the injection of the CO2 into an aquifer shows the robustness of the method to produce gas phase behavior and temperature distribution in agreement with the literature. We also present a simulation case to demonstrate the capability of our simulator to model non-isothermal CO2 injection process into a fractured water saturated reservoir using an unstructured grid. The method is also then applied to simulate and compare the isothermal and the non-isothermal cases of injection of pure CO2 and impure CO2 into water reservoir with heterogeneity in porosity and permeability.

Three-dimensional unstructured gridding for complex wells and geological features in subsurface reservoirs, with CVD-MPFA discretization performance

Grid generation for reservoir simulation, must honour classical key geological features and multilateral wells. The features to be honoured are classified into two groups; (1) involving layers, faults, pinchouts and fractures, and (2) involving well distributions. In the former, control-volume boundary aligned grids (BAGs) are required, while in the latter, control-point (defined as the centroid of the control-volume) well aligned grids (WAGs) are required. Depending on discretization method type and formulation, a choice of control-point and control-volume type is made, i.e. for a cell-centered method the primal grid cells act as control-volumes, otherwise for a vertex-centered method the dual-grid cells act as control-volumes. Novel three-dimensional unstructured grid generation methods are proposed that automate control-volume boundary alignment to geological features and control point alignment to complex wells, yielding essentially perpendicular bisector (PEBI) meshes either with respect to primal or dual-cells depending on grid type. Both grid types use tetrahedra, pyramids, prisms and hexahedra as grid elements. Primal-cell feature aligned grids are generated using special boundary surface protection techniques together with constrained cell-centered well trajectory alignment. Dual-cell feature aligned grids are generated from underlying primal-meshes, whereby features are protected such that dual-cell control-volume faces are aligned with interior feature boundaries, together with protected vertex-centered (control point) well trajectory alignment. The novel methods of grid generation presented enable practical application of both method types in 3-D for the first time. The primal and dual grids generated here demonstrate the gridding methods, and enable the first comparative performance study of cell-vertex versus cell-centered control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent mesh resolution on challenging problems in 3-D. Pressure fields computed by the cell-centered and vertex-centered CVD-MPFA schemes are compared and contrasted relative to the respective degrees of freedom employed, and demonstrate the relative benefits of each approximation type. Stability limits of the methods are also explored. For a given mesh the cell-vertex method uses approximately a fifth of the unknowns used by a cell-centered method and proves to be the most beneficial with respect to accuracy and efficiency. Numerical results show that vertex-centered CVD-MPFA methods outperform cell-centered CVD-MPFA method.

A numerical framework for two-phase flow of CO2 injection into fractured water-saturated reservoirs

A novel numerical framework is presented for simulation of compositional compressible gas-water phase flow in fractured porous media based on a fully-implicit cell-centred finite-volume method on unstructured grids. We employ a discrete-fracture model where fractures are modelled lower-dimensionally, within a rock matrix. A mass balance equation is solved over the rock-matrix while a lower-dimensional flow equation is solved over the lower-dimensional fractures, with coupling terms to account for the flux transfer from the surrounding rock-matrix. The discretisation of the Darcy-fluxes is based on the control-volume distributed multi-point flux approximation (CVD-MPFA) coupled with a lower-dimensional fracture model. We solve a non-linear system for the primary variables that are the phase pressure and the component molar densities while the secondary variables are updated depending on the updated values of the primary variables at each iteration of the non-linear solver. We use the cubic type equation-of-state (EOS) to model the physical properties of gas components, e.g. CO2. The water phase is multi-component as we allow solubility of gas components in the water phase. Solubility of H2O in the gaseous phase is also included in the developed method. We compare compositional simulation results, using unstructured grids, obtained via the lower-dimensional fracture model and the results obtained by the equi-dimensional fracture model for a case of CO2 injection into a water saturated reservoir with intersecting fractures. The results show good agreement between the two models while the lower-dimensional fracture model is also computationally cost effective. Results are also presented for the evolution of injected CO2 under gravity through a water saturated reservoir with and without fractures that demonstrate the profound effects of discrete-fractures that can lead to CO2 leakage. The presented method is also applied to a 3D simulation of injection of a multi-component gas with CO2 and CH4 into a saline water saturated reservoir with a surface 2D fracture network.

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.

Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Grid generation for reservoir simulation must honor classical key constraints and be boundary aligned such that control-volume boundaries are aligned with geological features such as layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. An unstructured grid generation procedure is proposed that automates control-volume and/or control point boundary alignment and yields a PEBI-mesh both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles, and to generate a dual-cell feature based grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom. The formulation of CVD-MPFA schemes in cell-centred and cell-vertex modes is analogous and requires switching control volume from primal to dual or vice versa together with appropriate data structures and boundary conditions. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are made clear in terms of flow resolution and degrees of freedom required.

Higher resolution total velocity Vt and Va finite-volume formulations on cell-centred structured and unstructured grids

Novel cell-centred finite-volume formulations are presented for incompressible and immiscible two-phase flow with both gravity and capillary pressure effects on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity, and capillary pressure flux operators. Two IMPES formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both Vt and Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However, when gravity is present, flow regimes are identified where the Va formulation becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first-order single-point upstream weighting.

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.

Three-dimensional control-volume distributed multi-point flux approximation coupled with a lower-dimensional surface fracture model

A novel cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture–matrix simulations on unstructured grids in three-dimensions (3D). The grid is aligned with fractures and barriers which are then modelled as lower-dimensional surface interfaces located between the matrix cells in the physical domain. The three-dimensional pressure equation is solved in the matrix domain coupled with a two-dimensional (2D) surface pressure equation solved over fracture networks via a novel surface CVD-MPFA formulation. The CVD-MPFA formulation naturally handles fractures with anisotropic permeabilities on unstructured grids. Matrix–fracture fluxes are expressed in terms of matrix and fracture pressures and define the transfer function, which is added to the lower-dimensional flow equation and couples the three-dimensional and surface systems. An additional transmission condition is used between matrix cells adjacent to low permeable fractures to couple the velocity and pressure jump across the fractures. Convergence and accuracy of the lower-dimensional fracture model is assessed for highly anisotropic fractures having a range of apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approximation for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretisation. Lower-dimensional fracture model results are compared with equi-dimensional model results. Fractures and barriers are efficiently modelled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Pressure continuity is built into the model across highly conductive fractures, leading to reduced local degrees of freedom in the CVD-MPFA approximation. The formulation is applied to geologically complex fracture networks in three-dimensions. The effects of the fracture permeability, aperture and grid resolution are also assessed with respect to convergence and computational cost.

Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model

A cell-centered control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture-matrix simulations. The grid is aligned with the fractures and barriers which are then modeled as lower-dimensional interfaces located between the matrix cells in the physical domain. The nD pressure equation is solved in the matrix domain coupled with an (n−1)D pressure equation solved in the fractures. The CVD-MPFA formulation naturally handles fractures with anisotropic permeabilities on unstructured grids. Matrix-fracture fluxes are expressed in terms of matrix and fracture pressures, and must be added to the lower-dimensional flow equation (called the transfer function). An additional transmission condition is used between matrix cells adjacent to low permeable fractures to link the velocity and pressure jump across the fractures. Numerical tests serve to assess the convergence and accuracy of the lower-dimensional fracture model for highly anisotropic fractures having different apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approach for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretization. Lower-dimensional fracture model results are compared with hybrid-grid and equi-dimensional model results. Fractures and barriers are efficiently modeled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Highly conductive fractures are modeled as lower-dimensional entities without the use of locally refined grids that are required by the equi-dimensional model, while pressure continuity across fractures is built into the model, without depending on the extra degrees of freedom which must be added locally by the hybrid-grid method. The lower-dimensional fracture model also yields improved results when compared to those of the hybrid-grid model for fractures with low-permeability in the normal direction to the fracture. In addition, a transient pressure simulation involving geologically representative complex fracture networks is presented.

A family of multi‐point flux approximation schemes for general element types in two and three dimensions with convergence performance

SUMMARYA family of flux‐continuous, locally conservative, control‐volume‐distributed multi‐point flux approximation (CVD‐MPFA) schemes has been developed for solving the general geometry‐permeability tensor pressure equation on structured and unstructured grids. These schemes are applicable to the full‐tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full‐tensor flow approximation. The family of flux‐continuous schemes is characterized by a quadrature parameterization. Improved numerical convergence for the family of CVD‐MPFA schemes using the quadrature parameterization has been observed for structured and unstructured grids in two dimensions.The CVD‐MPFA family cell‐vertex formulation is extended to classical general element types in 3‐D including prisms, pyramids, hexahedra and tetrahedra. A numerical convergence study of the CVD‐MPFA schemes on general unstructured grids comprising of triangular elements in 2‐D and prismatic, pyramidal, hexahedral and tetrahedral shape elements in 3‐D is presented. Copyright © 2011 John Wiley & Sons, Ltd.

The effects of control‐volume distributed multi‐point flux approximation (CVD‐MPFA) on upscaling—A study using the CVD‐MPFA schemes

AbstractA family of flux‐continuous, locally conservative, finite‐volume schemes has been developed for solving the general geometry‐permeability tensor (petroleum reservoir‐simulation) pressure equation on structured and unstructured grids and are control‐volume distributed (textit Comput. Geo. 1998; 2:259–290; Comput. Geo. 2002; 6:433–452). The schemes are applicable to diagonal and full tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir‐simulation schemes (two‐point flux approximation) when applied to full tensor flow approximation. The family of flux‐continuous schemes is quantified by a quadrature parameterization (Int. J. Numer. Meth. Fluids 2006; 51:1177–1203). Improved convergence (for two‐ and three‐dimensional formulation) using the quadrature parameterization has been observed for the family of flux‐continuous control‐volume distributed multi‐point flux approximation (CVD‐MPFA) schemes (Ph.D. Thesis, University of Wales, Swansea, U.K., 2007).In this paper family of flux‐continuous (CVD‐MPFA) schemes are used as a part of numerical upscaling procedure for upscaling the fine‐scale grid information (permeability) onto a coarse grid scale. A series of data‐sets (SPE, 2001) are tested where the upscaled permeability tensor is computed on a sequence of grid levels using the same fixed range of quadrature points in each case. The refinement studies presented involve: Refinement comparison study: In this study, permeability distribution for cells at each grid level is obtained by upscaling directly from the fine‐scale permeability field as in standard simulation practice. Refinement study with renormalized permeability: In this refinement comparison, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing only the effect of increased grid resolution on upscaled permeability, compared with that obtained directly from the fine‐scale solution. Refinement study with invariant permeability distribution: In this study, a classical mathematical convergence test is performed. The same coarse‐scale underlying permeability map is preserved on all grid levels including the fine‐scale reference solution. The study is carried out for the discretization of the scheme in physical space. The benefit of using specific quadrature points is demonstrated for upscaling in this study and superconvergence is observed. Copyright © 2010 John Wiley & Sons, Ltd.