Aspects of upscaling in simulation of flow in porous media

Abstract The ability to numerically simulate single phase and multiphase flow of fluids in porous media is extremely important in developing an understanding of the complex phenomena governing the flow. The flow is complicated by the presence of heterogeneities in the reservoir at many different length scales and by phenomena such as diffusion and dispersion. These effects must be effectively modeled by terms in coupled systems of nonlinear partial differential equations which form the basis of the simulator. The simulator must be able to model both single and multiphase flows and the transition regimes between the two in unsaturated flow applications. A discussion of some of the aspects of modeling unsaturated and multiphase flows in the presence of heterogeneities and channeling is presented along with directions for future work. © 1997 Elsevier Science Ltd. All rights reserved

[1]  J. Douglas,et al.  Prismatic mixed finite elements for second order elliptic problems , 1989 .

[2]  T. F. Russell,et al.  An Anistropic Coarse-Grid Dispersion Model of Heterogeneity and Viscous Fingering in Five-Spot Miscible Displacement That Matches Experiments and Fine-Grid Simulations , 1989 .

[3]  Philip John Binning,et al.  A mass conservative numerical solution for two‐phase flow in porous media with application to unsaturated flow , 1992 .

[4]  Richard E. Ewing,et al.  Efficient Use of Locally Refined Grids for Multiphase Reservoir Simulation , 1989 .

[5]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[6]  Richard E. Ewing,et al.  Local refinement techniques for elliptic problems on cell-centered grids , 1991, Numer. Linear Algebra Appl..

[7]  The Asymptotic Scaling Behavior of Mixing Induced by a Random Velocity Field , 1995 .

[8]  H. Haldorsen,et al.  Stochastic Modeling (includes associated papers 21255 and 21299 ) , 1990 .

[9]  S. P. Neuman,et al.  A quasi-linear theory of non-Fickian and Fickian subsurface dispersion , 1990 .

[10]  T. F. Russell,et al.  Time Stepping Along Characteristics with Incomplete Iteration for a Galerkin Approximation of Miscible Displacement in Porous Media , 1985 .

[11]  Todd Arbogast,et al.  Logically rectangular mixed methods for Darcy flow on general geometry , 1995 .

[12]  T. F. Russell,et al.  Finite element and finite difference methods for continuous flows in porous media. , 1800 .

[13]  M. Fortin,et al.  Mixed finite elements for second order elliptic problems in three variables , 1987 .

[14]  C. Axness,et al.  Three‐dimensional stochastic analysis of macrodispersion in aquifers , 1983 .

[15]  Richard E. Ewing,et al.  Mixed finite element approximation of phase velocities in compositional reservoir simulation , 1984 .

[16]  Richard E. Ewing,et al.  Analysis of mixed finite element methods on locally refined grids , 1992 .

[17]  M. Wheeler,et al.  A characteristics-mixed finite element method for advection-dominated transport problems , 1995 .

[18]  L. Durlofsky.  Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities , 1994 .

[19]  Yoram Rubin,et al.  Mapping permeability in heterogeneous aquifers using hydrologic and seismic data , 1992 .

[20]  T. F. Russell,et al.  An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation , 1990 .

[21]  M. Wheeler,et al.  Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences , 1997 .

[22]  J. V. Koebbe.  A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors , 1993 .

[23]  P. Vassilevski,et al.  Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis , 1991 .

[24]  Richard H. Gallagher,et al.  Finite elements in fluids , 1975 .

[25]  T. F. Russell,et al.  Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis , 1994 .

[26]  Magne S. Espedal,et al.  Heterogeneous Reservoir Models: Local Refinement and Effective Parameters , 1991 .

[27]  G. Dagan.  Flow and transport in porous formations , 1989 .

[28]  Richard E. Ewing,et al.  Eulerian-Lagrangian Localized Adjoint Methods for a Nonlinear Advection-Diffusion Equation , 1994 .

[29]  R. Ewing,et al.  Characteristics Petrov-Galerkin subdomain methods for two-phase immiscible flow , 1987 .

[30]  G. Dagan.  Stochastic Modeling of Groundwater Flow by Unconditional and Conditional Probabilities: The Inverse Problem , 1985 .

[31]  E. Sudicky.  A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process , 1986 .

[32]  Todd Arbogast,et al.  Efficient Mixed Methods for Groundwater Flow on Triangular or Tetrahedral Meshes , 1994 .

[33]  M. Celia,et al.  The effect of heterogeneity on the drainage capillary pressure‐saturation relation , 1992 .

[34]  Ismael Herrera,et al.  A new numerical approach for the advective-diffusive transport equation , 1989 .

[35]  John W. Barrett,et al.  Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems , 1984 .

[36]  Effective behavior of two-phase flow in heterogeneous reservoirs , 1987 .

[37]  W. B. Lindquist,et al.  Characterization of Mixing Length Growth for Flow in Heterogeneous Porous Media , 1991 .

[38]  P. Vassilevski,et al.  Local refinement techniques for elliptic problems on cell-centered grids , 1991 .

[39]  Richard E. Ewing,et al.  Eulerian–Lagrangian localized adjoint methods for reactive transport with biodegradation , 1995 .

[40]  Yoram Rubin,et al.  Stochastic identification of recharge, transmissivity, and storativity in aquifer transient flow: A quasi‐steady approach , 1988 .

[41]  Leszek Demkowicz,et al.  An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables , 1986 .

[42]  B. Amaziane,et al.  Numerical Simulation and Homogenization of Two-Phase Flow in Heterogeneous Porous Media , 1991 .

[43]  Magne S. Espedal,et al.  Macrodispersion for two-phase, immiscible flow in porous media , 1994 .

[44]  G. Chavent,et al.  A new formulation of diphasic incompressible flows in porous media , 1976 .

[45]  G. Dagan.  Dispersion of a passive solute in non-ergodic transport by steady velocity fields in heterogeneous formations , 1991, Journal of Fluid Mechanics.

[46]  Magne S. Espedal,et al.  Multiphase Flow Simulation with Various Boundary Conditions , 1994 .

[47]  Louis J. Durlofsky,et al.  Representation of grid block permeability in coarse scale models of randomly heterogeneous porous media , 1992 .

[48]  Todd Arbogast,et al.  Logically Rectangular Mixed Methods for Groundwater Flow and Transport on General Geometry , 1994 .

[49]  J. Mandel,et al.  Balancing domain decomposition for mixed finite elements , 1995 .

[50]  Richard E. Ewing,et al.  Adaptive techniques for time-dependent problems , 1992 .

[51]  Richard E. Ewing,et al.  A preconditioning technique for the efficient solution of problems with local grid refinement , 1988 .

[52]  Larry C. Young,et al.  A study of spatial approximations for simulating fluid displacements in petroleum reservoirs , 1984 .

[53]  R. Ewing,et al.  Characteristic adaptive subdomain methods for reservoir flow problems , 1990 .

[54]  Scaling Laws For Macrodispersion James Glimm and , 2007 .

[55]  Richard E. Ewing,et al.  A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media , 1983 .

[56]  J. H. Cushman,et al.  Inverse methods for subsurface flow: A critical review of stochastic techniques , 1990 .

[57]  Richard E. Ewing,et al.  Vectorizable preconditioners for mixed finite element solution of second-order elliptic problems , 1992 .

[58]  Richard E. Ewing,et al.  Optimal‐order convergence rates for Eulerian‐Lagrangian localized adjoint methods for reactive transport and contamination in groundwater , 1995 .

[59]  Richard E. Ewing,et al.  Mixed finite element methods for accurate fluid velocities. , 1985 .

[60]  W. B. Lindquist,et al.  A theory of macrodispersion for the scale-up problem , 1993 .

[61]  E. G. Vomvoris,et al.  A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one‐dimensional simulations , 1983 .

[62]  Richard E. Ewing,et al.  Approximation of parabolic problems on grids locally refined in time and space , 1994 .

[63]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 1. Maximum Likelihood Method Incorporating Prior Information , 1986 .

[64]  Y. Rubin.  Stochastic modeling of macrodispersion in heterogeneous porous media , 1990 .

[65]  M. R. Todd,et al.  Development, testing and application of a numerical simulator for predicting miscible flood performance , 1972 .

[66]  J. Wang,et al.  Analysis of the Schwarz algorithm for mixed finite elements methods , 1992 .

[67]  W. Yeh.  Review of Parameter Identification Procedures in Groundwater Hydrology: The Inverse Problem , 1986 .

[68]  William G. Gray,et al.  Eulerian-Lagrangian localized adjoint methods with variable coefficients in multiple dimensions. , 1990 .

[69]  Richard E. Ewing,et al.  Well-Conditioned Iterative Schemes for Mixed Finite-Element Models of Porous-Media Flows , 1992, SIAM J. Sci. Comput..

[70]  T. F. Russell,et al.  Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics , 1984 .

[71]  Michael A. Celia,et al.  Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts: 3. Hydraulic c , 1992 .

[72]  I. Herrera.  Unified formulation of numerical methods. I. Green's formulas for operators in discontinuous fields , 1985 .

[73]  Richard E. Ewing,et al.  Eulerian‐Lagrangian localized adjoint method: The theoretical framework , 1993 .

[74]  Qiang Zhang,et al.  A multi-length-scale theory of the anomalous mixing-length growth for tracer flow in heterogeneous porous media , 1992 .

[75]  Gedeon Dagan,et al.  The significance of heterogeneity of evolving scales to transport in porous formations , 1994 .

[76]  A Stochastic Kriging Algorithm a Stochastic Kriging Algorithm , 1994 .