A multipoint flux approximation with diamond stencil finite volume scheme for the two‐dimensional simulation of fluid flows in naturally fractured reservoirs using a hybrid‐grid method

Abstract Mechanical deformation induced by injection and withdrawal of fluids from the subsurface can significantly alter the flow paths in naturally fractured reservoirs. Modeling coupled fluid-flow and mechanical deformation in fractured reservoirs relies on either sophisticated gridding techniques, or enhancing the variables (degrees-of-freedom) that represent the physics in order to describe the behavior of fractured formation accurately. The objective of this study is to develop a spatial discretization scheme that cuts the "matrix" grid with fracture planes and utilizes traditional formulations for fluid flow and geomechanics. The flow model uses the standard low-order finite-volume method with the Compartmental Embedded fracture Model (cEDFM). Due to the presence of non-standard polyhedra in the grid after cutting/splitting, we utilize numerical harmonic shape functions within a Polyhedral finite-element (PFE) formulation for mechanical deformation. In order to enforce fracture-contact constraints, we use a penalty approach. We provide a series of comparisons between the approach that uses conforming Unstructured grids and a Discrete Fracture Model (Unstructured DFM) with the new cut-cell PFE formulation. The manuscript analyzes the convergence of both methods for linear elastic, single-fracture slip, and Mandel’s problems with tetrahedral, Cartesian, and PEBI-grids. Finally, the paper presents a fully-coupled 3D simulation with multiple inclined intersecting faults activated in shear by fluid injection, which caused an increase in effective reservoir permeability. Our approach allows for great reduction in the complexity of the (gridded) model construction while retaining the solution accuracy together with great saving in the computational cost compared with UDFM. The flexibility of our model with respect to the types of grid polyhedra allows us to eliminate mesh artifacts in the solution of the transport equations typically observed when using tetrahedral grids and two-point flux approximation.

Download Full-text

Very high-order accurate finite volume scheme on curved boundaries for the two-dimensional steady-state convection–diffusion equation with Dirichlet condition

Applied Mathematical Modelling ◽

10.1016/j.apm.2017.10.016 ◽

2018 ◽

Vol 54 ◽

pp. 752-767 ◽

Cited By ~ 6

Author(s):

Ricardo Costa ◽

Stéphane Clain ◽

Raphaël Loubère ◽

Gaspar J. Machado

Keyword(s):

Steady State ◽

Diffusion Equation ◽

Finite Volume ◽

Dirichlet Condition ◽

Finite Volume Scheme ◽

Two Dimensional ◽

Curved Boundaries ◽

Convection Diffusion ◽

Convection Diffusion Equation ◽

Very High

Download Full-text

A General Modeling Framework for Simulating Complex Recovery Processes in Fractured Reservoirs at Different Resolutions

SPE Journal ◽

10.2118/182621-pa ◽

2018 ◽

Vol 23 (02) ◽

pp. 598-613 ◽

Cited By ~ 14

Author(s):

Mun-Hong (Robin) Hui ◽

Mohammad Karimi-Fard ◽

Bradley Mallison ◽

Louis J. Durlofsky

Keyword(s):

Matrix Models ◽

Fractured Reservoirs ◽

Input Parameter ◽

Grid Cell ◽

Naturally Fractured Reservoirs ◽

Fine Scale ◽

Modeling Framework ◽

Fine Grid ◽

Flux Approximation ◽

Discrete Fracture

Summary A comprehensive methodology for gridding, discretizing, coarsening, and simulating discrete-fracture-matrix models of naturally fractured reservoirs is described and applied. The model representation considered here can be used to define the grid and transmissibilities, either at the original fine scale or at coarser scales, for any connectivity-list-based finite-volume flow simulator. For our fine-scale mesh, we use a polyhedral-gridding technique to construct a conforming matrix grid with adaptive refinement near fractures, which are represented as faces of grid cells. The algorithm uses a single input parameter to obtain a suitable compromise between fine-grid cell quality and the fidelity of the fracture representation. Discretization using a two-point flux approximation is accomplished with an existing procedure that treats fractures as lower-dimensional entities (i.e., resolution in the transverse direction is not required). The upscaling method is an aggregation-based technique in which coarse control volumes are aggregates of fine-scale cells, and coarse transmissibilities are computed with a general flow-based procedure. Numerical results are presented for waterflood, sour-gas injection, and gas-condensate primary production for fracture models with matrix and fracture heterogeneities. Coarse-model accuracy is shown to generally decrease with increasing levels of coarsening, as would be expected. We demonstrate, however, that with our methodology, two orders of magnitude of speedup can typically be achieved with models that introduce less than approximately 10% error (with error appropriately defined). This suggests that the overall framework may be very useful for the simulation of realistic discrete-fracture-matrix models.

Download Full-text

Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms

IMA Journal of Numerical Analysis ◽

10.1093/imanum/draa040 ◽

2020 ◽

Author(s):

Esther S Daus ◽

Ansgar Jüngel ◽

Antoine Zurek

Keyword(s):

Finite Volume ◽

System Modeling ◽

Gradient Flow ◽

Continuous Model ◽

Diffusion System ◽

Finite Volume Scheme ◽

Biofilm Growth ◽

Volume Scheme ◽

Cross Diffusion ◽

Flux Approximation

Abstract An implicit Euler finite-volume scheme for a cross-diffusion system modeling biofilm growth is analyzed by exploiting its formal gradient-flow structure. The numerical scheme is based on a two-point flux approximation that preserves the entropy structure of the continuous model. Assuming equal diffusivities the existence of non-negative and bounded solutions to the scheme and its convergence are proved. Finally, we supplement the study by numerical experiments in one and two space dimensions.

Download Full-text