Design of coupled finite volume schemes minimizing the grid orientation effect in reservoir simulation (original) (raw)
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Computers & Mathematics with Applications, 2019
In this paper, flow of multi-component two-phase fluids in highly heterogeneous anisotropic twodimensional porous media is studied using computational methods suitable for unstructured triangular and/or quadrilateral grids. The physical model accounts for miscibility and compressibility of fluids while gravity and capillary effects are neglected. The governing equations consist of a pressure equation together with a system of mass conservation equations. For solving pressure equation, a new method called Control Volume Distributed Finite Element Method (CVDFEM) is introduced which uses Control Volume Distributed (CVD) vertex-centered grids. It is shown that the proposed method is able to approximate the pressure field in highly anisotropic and heterogeneous porous media fairly accurately. The system of mass conservation equations is solved using various upwind and central schemes. These schemes are extended from one-dimensional to two-dimensional unstructured grids. Using a series of numerical test cases, comparison are made between different approaches for approximation of the hyperbolic flux function. Semi one-dimensional high-order data reconstruction procedures are employed to decrease stream-wise numerical diffusion. The results suggest that the Modified Dominant Wave (MDW) scheme outperforms other hyperbolic schemes studied in this paper from both accuracy and computational cost points of view.
SPE Reservoir Evaluation & Engineering, 2010
To apply upscaling techniques is an undeniable demand in reservoir simulation, when one considers the difference between the level of detail in a geological model and the level of details that can be handled by a reservoir simulator. Upscaling the reservoir model involves first constructing a coarse grid by using gridding algorithms and then computing average properties for coarse gridblocks. Although various techniques have been proposed for each of these steps, one needs to be aware of strengths and weaknesses of each technique before attempting to apply them. In this paper, we focus on different gridding methods and evaluate their performances. Three main grid-generation techniques are considered: permeability-based (PB), flow-based (FB), and vorticity-based (VB) methods. We apply all three methods to a number of 2D heterogeneous models and simulate two-phase flow on the constructed grids. Then we compare their obtained global and local results. Fluid cuts at the producer is employed as the global performance indicator and saturation-distribution error as the local indicator. We show that FB and VB gridding, which are dynamic methods, are superior to PB gridding, which is a static method. On the basis of this analysis, we then concentrate on FB and VB gridding and investigate their performance in greater detail. While FB gridding uses fluid velocity as gridblock density indicator, VB gridding combines velocity and permeability variation in gridding according to its definition and takes advantage of both. Therefore, although performance of FB and VB gridding is comparable in many cases, VB has the benefit of producing coarse gridblocks with more-uniform permeability and fluid-properties distribution. This in turn yields more-accurate global and local results and reduces application of sophisticated upscaling techniques and fulltensor permeability upscaling.
Computer Methods in Applied Mechanics and Engineering, 2013
Standard reservoi r simulation schemes employ single-point upstream weighting for convective flux approximation. These schemes introduce both coordinate-line numerical diffusion and crosswind diffusion into the solution that is grid and geometry dependent. New locally conservative cell-bas ed multi-dimensional upwind schemes and higher-order cell-based multi-dimensional upwind schemes that reduce both directional and crosswind diffusion are presented for convective flow approximation. The new higher-o rder schemes are comprised of two steps ; (a) Higher-order approximation that corrects the directional diffusion of the app roximation. (b) Truly multi-dimensional upwind approximation, which involves flux app roximation using upwind information obtained by upstream tracing along multi-dimensional flow paths. This approximation reduces crosswind diffusion. Conditions on tracing direction and CFL number lead to a local maximum princ iple that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations. Benefits of the resulting sch emes 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 crosswind diffusion effects, leading to improved resolution of concentration and saturation fronts.
Finite volume schemes for two-phase flow in porous media
Computing and Visualization in Science, 2004
The system of equations obtained from the conservation of multiphasic fluids in porous media is usually approximated by finite volume schemes in the oil reservoir simulation setting. The convergence properties of these schemes are only known in a few simplified cases. The aim of this paper is to present some new results of convergence in more complex cases. These results are based on an adaptation of the H-convergence notion to the limit of discrete approximates.
2019
This thesis studies characteristic-based numerical schemes for a mathematical model of miscible fluid flow in porous media, a coupled and advection-dominated system of partial differential equations describing an oil recovery process. In particular, we analyse characteristic-based schemes on generic polygonal meshes, with diffusive component discretised using modern finite volume methods. We also develop a novel algorithm for reconstructing velocity fields suitable for characteristic tracking. The proposed scheme mixes two characteristic-based discretisations, and implements an original post-processing algorithm to ensure local and global mass conservation. Extensive numerical tests are provided, on various polygonal meshes. A rigorous convergence analysis is also performed.
High Resolution Schemes in Curvilinear Grids for Reducing the Grid Orientation Effects
One of the most important tasks in the numerical simulation of fluid flow problems is the reduction of numerical diffusion contained in the solution. Numerical diffusion is caused by the use of first order interpolation schemes in the approximation of the convective terms in the momentum equations. In petroleum reservoir simulation, a similar problem arises when the mobility is interpolated at the interfaces of the control volumes for the mass fluxes calculation. Depending on the interpolation function used for the mobility, the solution may be contaminated with the so-called grid orientation effect, a similar error as the wellknown numerical diffusion. The grid orientation effect in petroleum reservoir simulation causes different breakthroughs, even when the production wells are symmetrically located with respect to the injection well in a homogeneous media. Usually, the upwind scheme is employed, due to its ability of promoting numerical stability. The price one pays is the introd...
A finite volume scheme with improved well modeling in subsurface flow simulation
Computational Geosciences, 2017
We present the latest enhancement of the nonlinear monotone finite volume method for the near-well regions. The original nonlinear method is applicable for diffusion, advection-diffusion, and multiphase flow model equations with full anisotropic discontinuous permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional two-point flux approximation (TPFA) on cubic meshes but has much better accuracy for the general case of non-orthogonal grids and anisotropic media. The latest modification of the nonlinear method takes into account the nonlinear (e.g., logarithmic) singularity of the pressure in the near-well region and introduces a correction to improve accuracy of the pressure and the flux calculation. In this paper, we consider a linear version of the nonlinear method waiving its monotonicity for sake of better accuracy. The new method is generalized for anisotropic media, polyhedral grids and nontrivial cases such as slanted, partially perforated wells or wells Kirill Nikitin