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1.
We present a new nonlinear monotone finite volume method for diffusion equation and its application to two-phase flow model. We consider full anisotropic discontinuous diffusion or permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which provides the conventional seven-point stencil for the discrete diffusion operator on cubic meshes. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water breakthrough time. We compare two two-point flux approximations (TPFA), the proposed nonlinear TPFA and the conventional linear TPFA, and multipoint flux approximation (MPFA). The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the nonlinear TPFA demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For nonorthogonal grids or full anisotropic permeability tensors, the conventional linear TPFA provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the nonnegativity of the differential solution. 相似文献
2.
I. Aavatsmark G. T. Eigestad R. A. Klausen M. F. Wheeler I. Yotov 《Computational Geosciences》2007,11(4):333-345
This paper investigates different variants of the multipoint flux approximation (MPFA) O-method in 2D, which rely on a transformation
to an orthogonal reference space. This approach yields a system of equations with a symmetric matrix of coefficients. Different
methods appear, depending on where the transformed permeability is evaluated. Midpoint and corner-point evaluations are considered.
Relations to mixed finite element (MFE) methods with different velocity finite element spaces are further discussed. Convergence
of the MPFA methods is investigated numerically. For corner-point evaluation of the reference permeability, the same convergence
behavior as the O-method in the physical space is achieved when the grids are refined uniformly or when grid perturbations
of order h
2 are allowed. For h
2-perturbed grids, the convergence of the normal velocities is slower for the midpoint evaluation than for the corner-point
evaluation. However, for rough grids, i.e., grids with perturbations of order h, contrary to the physical space method, convergence cannot be claimed for any of the investigated reference space methods.
The relations to the MFE methods are used to explain the loss of convergence.
Wheeler was partially supported by NSF grant DMS 0411413 and the DOE grant DE-FGO2-04ER25617. Yotov was supported in part
by the DOE grant DE-FG02-04ER25618, the NSF grant DMS 0411694 and the J. Tinsley Oden Faculty Fellowship, The University of
Texas at Austin. 相似文献
3.
Subsurface flow models can exhibit strong full-tensor anisotropy due to either permeability or grid nonorthogonality effects.
Upscaling procedures, for example, generate full-tensor effects on the coarse scale even for cases in which the underlying
fine-scale permeability is isotropic. A multipoint flux approximation (MPFA) is often needed to accurately simulate flow for
such systems. In this paper, we present and apply a different approach, nonlinear two-point flux approximation (NTPFA), for
modeling systems with full-tensor effects. In NTPFA, transmissibility (which provides interblock connections) is determined
from reference global flux and pressure fields for a specific flow problem. These fields can be generated using either fully
resolved or approximate global simulations. The use of fully resolved simulations leads to an NTPFA method that corresponds
to global upscaling procedures, while the use of approximate simulations gives a method corresponding to recently developed
local–global techniques. For both approaches, NTPFA algorithms applicable to both single-scale full-tensor permeability systems
and two-scale systems are described. A unified framework is introduced, which enables single-scale and two-scale problems
to be viewed in a consistent manner. Extensive numerical results demonstrate that the global and local–global NTPFA techniques
provide accurate flow predictions over wide parameter ranges for both single-scale and two-scale systems, though the global
procedure is more accurate overall. The applicability of NTPFA to the simulation of two-phase flow in upscaled models is also
demonstrated. 相似文献
4.
This paper extends the multipoint flux-approximation (MPFA) control-volume method to quadrilateral grids for which the adjacent cells do not necessarily share corners. Examples are grids with faults and locally refined grids. This paper gives a derivation of the method for such grids. The difference between two-point flux-approximation (TPFA) results and MPFA results for faults and local grid refinements is demonstrated for synthetic problems. Further, the results are compared with results from uniform fine-grid simulations. The effect of repeated fault patterns as well as anisotropy is investigated. Large errors may be found for the TPFA method for flow through a series of faults in an anisotropic medium. Finally, a comparison is done for a reservoir field application. 相似文献
5.
We develop a finite element discretization and multigrid solver for a Darcy–Stokes system of three-dimensional vuggy porous media, i.e., porous media with cavities. The finite element method uses low-order mixed finite elements in the Darcy
and Stokes domains and special transition elements near the Darcy–Stokes interface to allow for tangential discontinuities
implied by the Beavers–Joseph boundary condition. We design a multigrid method to solve the resulting saddle point linear
system. The intertwining of the Darcy and Stokes subdomains makes the resulting matrix highly ill-conditioned. The velocity
field is very irregular, and its discontinuous tangential component at the Darcy–Stokes interface makes it difficult to define
intergrid transfer operators. Our definition is based on mass conservation and the analysis of the orders of magnitude of
the solution. The coarser grid equations are defined using the Galerkin method. A new smoother of Uzawa type is developed
based on taking an optimal step in a good search direction. Our algorithm has a measured convergence factor independent of
the size of the system, at least when there are no disconnected vugs. We study the macroscopic effective permeability of a
vuggy medium, showing that the influence of vug orientation; shape; and, most importantly, interconnectivity determine the
macroscopic flow properties of the medium.
This work was supported by the U.S. National Science Foundation under grants DMS-0074310 and DMS-0417431. 相似文献
6.
In this article we present a series of tests to study how well suited the TPFA coefficient matrix is as a preconditioner for the MPFA discrete system of equations in an iterative solver, using a flux splitting method. These tests have been conducted for single-phase flow for a wide range of anisotropy, heterogeneity, and grid skewness (mainly parallelogram grids). We use the K-orthogonal part of the MPFA transmissibilities for a parallelogram grid to govern the TPFA transmissibilities. The convergence of the flux splitting method is for each test case measured by the spectral radius of the iteration matrix. 相似文献
7.
We present a high-order method for miscible displacement simulation in porous media. The method is based on discontinuous Galerkin discretization with weighted average stabilization technique and flux reconstruction post processing. The mathematical model is decoupled and solved sequentially. We apply domain decomposition and algebraic multigrid preconditioner for the linear system resulting from the high-order discretization. The accuracy and robustness of the method are demonstrated in the convergence study with analytical solutions and heterogeneous porous media, respectively. We also investigate the effect of grid orientation and anisotropic permeability using high-order discontinuous Galerkin method in contrast with cell-centered finite volume method. The study of the parallel implementation shows the scalability and efficiency of the method on parallel architecture. We also verify the simulation result on highly heterogeneous permeability field from the SPE10 model. 相似文献
8.
A numerical investigation of the desaturation process at the argillaceous Tournemire site has been carried out. This desaturation
is initialized by the contact of the saturated rock with the ambient air in excavated openings. The used hydraulical model
is based on the Richards’ approximation for unsaturated one phase flow coupled with the deformation of a porous medium with
anisotropic linearly elastic behavior. In relation to the extent of the desaturated zone around an excavated opening, the
intrinsic permeability and the relative permeability have been identified to be the most important model parameters. The mechanical
deformation process itself, the seasonal influences and the tunnel lining are less important for the formation of the desaturated
zone. The comparison with measured saturation values bares some difficulties but indicates the principle capacity of the applied
finite element codes to simulate the desaturation process. The consideration of seasonal changes in humidity in the ambient
air results in a constantly recurring desaturation–resaturation cycle in the near field of the openings. This seasonally influenced
zone amount 1–2 m and is nearly independent from time and from a variation of model parameters within a reasonable range.
The possibility of material weakening in this zone is of special interest, since claystone is a potential host rock for the
disposal of radioactive waste. 相似文献
9.
In this work, lowest-order Raviart–Thomas and Brezzi–Douglas–Marini mixed methods are considered for groundwater flow simulations.
Typically, mixed methods lead to a saddle-point problem, which is expensive to solve. Two approaches are numerically compared
here to allow an explicit velocity elimination: (1) the well-known hybrid formulation leading to a symmetric positive definite
system where the only unknowns are the Lagrange multipliers and (2) a more recent approach, inspired from the multipoint flux
approximation method, reducing low-order mixed methods to cell-centered finite difference schemes. Selected groundwater flow
scenarios are used for the comparison between hybrid and multipoint approaches. The simulations are performed in the bidimensional
case with a general triangular discretization because of its practical interest for hydrogeologists. 相似文献
10.
Eren Ucar Eirik Keilegavlen Inga Berre Jan Martin Nordbotten 《Computational Geosciences》2018,22(4):993-1007
Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-volume approach, termed the multi-point stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (face pairs in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation, considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate our proposed approach using both problems, for which analytical solutions are available, and more complex benchmark problems, including comparison with a finite-element discretization. 相似文献
11.
In this paper, we present a semi-implicit method for the incompressible three-phase flow equations in two dimensions. In particular, a high-order discontinuous Galerkin spatial discretization is coupled with a backward Euler discretization in time. We consider a pressure-saturation formulation, decouple the pressure and saturation equations, and solve them sequentially while still keeping each equation implicit in its respective unknown. We present several numerical examples on both homogeneous and heterogeneous media, with varying permeability and porosity. Our results demonstrate the robustness of the scheme. In particular, no slope limiters are required and a relatively large time step may be taken. 相似文献
12.
Control volume methods are frequently used in porous media flow. This article gives an example on how one method, the Multipoint Flux Approximation method (MPFA), fails to satisfy the maximum principle for strong anisotropies or grid skewnesses, and develops conditions for when the monotonicity property holds for uniform parallelogram grids in homogeneous media. The conditions developed are applicable to any nine-point scheme in 2D or 27-point scheme in 3D, and is useful when the method produces a system matrix which is not an M-matrix. 相似文献
13.
Dmitry Eydinov Sigurd Ivar Aanonsen Jarle Haukås Ivar Aavatsmark 《Computational Geosciences》2008,12(2):209-225
A method for history matching of an in-house petroleum reservoir compositional simulator with multipoint flux approximation
is presented. This method is used for the estimation of unknown reservoir parameters, such as permeability and porosity, based
on production data and inverted seismic data. The limited-memory Broyden–Fletcher–Goldfarb–Shanno method is employed for minimization
of the objective function, which represents the difference between simulated and observed data. In this work, we present the
key features of the algorithm for calculations of the gradients of the objective function based on adjoint variables. The
test example shows that the method is applicable to cases with anisotropic permeability fields, multipoint flux approximation,
and arbitrary fluid compositions. 相似文献
14.
We propose a methodology, called multilevel local–global (MLLG) upscaling, for generating accurate upscaled models of permeabilities
or transmissibilities for flow simulation on adapted grids in heterogeneous subsurface formations. The method generates an
initial adapted grid based on the given fine-scale reservoir heterogeneity and potential flow paths. It then applies local–global
(LG) upscaling for permeability or transmissibility [7], along with adaptivity, in an iterative manner. In each iteration of MLLG, the grid can be adapted where needed to reduce
flow solver and upscaling errors. The adaptivity is controlled with a flow-based indicator. The iterative process is continued
until consistency between the global solve on the adapted grid and the local solves is obtained. While each application of
LG upscaling is also an iterative process, this inner iteration generally takes only one or two iterations to converge. Furthermore,
the number of outer iterations is bounded above, and hence, the computational costs of this approach are low. We design a
new flow-based weighting of transmissibility values in LG upscaling that significantly improves the accuracy of LG and MLLG
over traditional local transmissibility calculations. For highly heterogeneous (e.g., channelized) systems, the integration
of grid adaptivity and LG upscaling is shown to consistently provide more accurate coarse-scale models for global flow, relative
to reference fine-scale results, than do existing upscaling techniques applied to uniform grids of similar densities. Another
attractive property of the integration of upscaling and adaptivity is that process dependency is strongly reduced, that is,
the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic
boundary conditions used to compute the upscaled parameters. The method is demonstrated on Cartesian cell-based anisotropic
refinement (CCAR) grids, but it can be applied to other adaptation strategies for structured grids and extended to unstructured
grids. 相似文献
15.
We have used different techniques for permeability prediction using porosity core data from one well at the Maracaibo Lake,
Venezuela. One of these techniques is statistical and uses neuro-fuzzy concepts. Another has been developed by Pape et al.
(Geophysics 64(5):1447–1460, 1999), based on fractal theory and the Kozeny–Carman equations. We have also calculated permeability values using the empirical
model obtained in 1949 by Tixier and a simple linear regression between the logarithms of permeability and porosity. We have
used 100% of the permeability–porosity data to obtain the predictor equations in each case. The best fit, in terms of the
root mean-square error, was obtained with the statistical approach. The results obtained from the fractal model, the Tixier
equation or the linear approach do not improve the neuro-fuzzy results. We have also randomly taken 25% of the porosity data
to obtain the predictor equations. The increase of the input data density for the neuro-fuzzy approach improves the results,
as is expected for a statistical analysis. On the contrary, for the physical model based on the fractal theory, the decrease
in the data density could allow reaching the ideal theoretical Kozeny–Carman model, on which are based the fractal equations,
and hence, the permeability prediction using these expressions is improved. 相似文献
16.
A novel mathematical framework for modeling folds in structural geology is presented. All the main fold classes from the classical
literature: parallel folds, similar folds, and other fold types with convergent and divergent dip isogons are modeled in 3D
space by linear and non-linear first-order partial differential equations. The equations are derived from a static Hamilton–Jacobi
equation in the context of isotropic and anisotropic front propagation. The proposed Hamilton–Jacobi framework represents
folded geological volumes in an Eulerian context as a time of arrival field relative to a reference layer. Metric properties
such as distances, gradients (dip and strike), curvature, and their spatial variations can then be easily derived and represented
as three-dimensional continua covering the whole geological volume. The model also serves as a basis for distributing properties
in folded geological volumes. 相似文献
17.
The Henry semi-analytical solution is developed for stratified aquifers with exponential and Gaussian permeability–depth relationships and small dispersion. The semi-analytical solution is sought by expanding the stream function and the concentration in an infinite Fourier series truncated at given orders. Due to the heterogeneity, the semi-analytical solution contains additional terms and quickly becomes impractical because of the high truncation orders required to yield stable results. An appropriate evaluation of the most expensive term, involving fairly complex summations, is proposed to render the computation of the semi-analytical solution affordable. Three configurations with the same transmissivity as that for the homogeneous Henry problem are investigated. The first configuration considers an exponential decay of the permeability (k) with depth. The second and third configurations consider an exponential increase and a Gaussian high-low-high distribution of k with depth, respectively. The three configurations are also investigated numerically using an accurate numerical code based on the method of lines and advanced spatial discretization schemes. An excellent agreement is obtained between the semi-analytical and the numerical solutions for all test cases. The results show that the amount of saltwater intrusion is strongly dependent on the heterogeneity. Further, the heterogeneous Henry problem is found to be more suitable for benchmarking density-driven numerical codes because the numerical solution is more sensitive to the grid size than for the homogeneous problem. 相似文献
18.
James Glimm Shuling Hou Hongjoong Kim Yoon-ha Lee David H. Sharp Kenny Ye Qisu Zou 《Computational Geosciences》2001,5(3):173-197
We consider numerical solutions of the Darcy and Buckley–Leverett equations for flow in porous media. These solutions depend on a realization of a random field that describes the reservoir permeability. The main content of this paper is to formulate and analyze a probability model for the numerical coarse grid solution error. We explore the extent to which the coarse grid oil production rate is sufficient to predict future oil production rates. We find that very early oil production data is sufficient to reduce the prediction error in oil production by about 30%, relative to the prior probability prediction. 相似文献
19.
A key ingredient in simulation of flow in porous media is accurate determination of the velocities that drive the flow. Large‐scale irregularities of the geology (faults, fractures, and layers) suggest the use of irregular grids in simulation. This paper presents a control‐volume mixed finite element method that provides a simple, systematic, easily implemented procedure for obtaining accurate velocity approximations on irregular (i.e., distorted logically rectangular) block‐centered quadrilateral grids. The control‐volume formulation of Darcy’s law can be viewed as a discretization into element‐sized “tanks” with imposed pressures at the ends, giving a local discrete Darcy law analogous to the block‐by‐block conservation in the usual mixed discretization of the mass‐conservation equation. Numerical results in two dimensions show second‐order convergence in the velocity, even with discontinuous anisotropic permeability on an irregular grid. The method extends readily to three dimensions. 相似文献
20.
We consider adaptive discontinuous Galerkin (DG) methods for solving reactive transport problems in porous media. To guide
anisotropic and dynamic mesh adaptation, a posteriori error estimators based on solving local problems are established. These error estimators are efficient to compute and effective
to capture local phenomena, and they apply to all the four primal DG schemes, namely, symmetric interior penalty Galerkin,
nonsymmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and the Oden–Babuška-Baumann version of DG.
Numerical results are provided to illustrate the effectiveness of the proposed error estimators. 相似文献