Modelling of Multiscale Structures in Flow Simulations for Petroleum Reservoirs

Flow in petroleum reservoirs occurs on a wide variety of physical scales. This poses a continuing challenge to modelling and simulation of reservoirs since fine-scale effects often have a profound impact on flow patterns on larger scales. Resolving all pertinent scales and their interaction is therefore imperative to give reliable qualitative and quantitative simulation results. To overcome the problem of multiple scales it is customary to use some kind of upscaling or homogenisation procedure, in which the reservoir properties are represented by some kind of averaged properties and the flow is solved on a coarse grid. Unfortunately, most upscaling techniques give reliable results only for a limited range of flow scenarios. Increased demands for reservoir simulation studies have therefore led researchers to develop more rigorous multiscale methods that incorporate subscale effects more directly.

[1]  Zhiming Chen,et al.  A mixed multiscale finite element method for elliptic problems with oscillating coefficients , 2003, Math. Comput..

[2]  Yalchin Efendiev,et al.  Modeling of subgrid effects in coarse‐scale simulations of transport in heterogeneous porous media , 2000 .

[3]  Knut-Andreas Lie,et al.  An Introduction to the Numerics of Flow in Porous Media using Matlab , 2007, Geometric Modelling, Numerical Simulation, and Optimization.

[4]  Yalchin Efendiev,et al.  Accurate multiscale finite element methods for two-phase flow simulations , 2006, J. Comput. Phys..

[5]  Michael Andrew Christie,et al.  A New Upscaling Approach for Highly Heterogeneous Reservoirs , 2005 .

[6]  Louis J. Durlofsky,et al.  Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques , 1998 .

[7]  G. W. Thomas,et al.  An Extension of Pseudofunction Concepts , 1983 .

[8]  Clayton V. Deutsch,et al.  Power Averaging for Block Effective Permeability , 1986 .

[9]  S. Begg,et al.  Assigning Effective Values to Simulator Gridblock Parameters for Heterogeneous Reservoirs , 1989 .

[10]  W. G. Price,et al.  Renormalization calculations of immiscible flow , 1993 .

[11]  Larry W. Lake,et al.  Reservoir Characterization II , 1991 .

[12]  Khalid Aziz,et al.  Evaluation of Dynamic Pseudo Functions for Reservoir Simulation , 1996 .

[13]  R. A. Behrens,et al.  SCALING LAWS IN RESERVOIR SIMULATION AND THEIR USE IN A HYBRID FINITE DIFFERENCE/STREAMTUBE APPROACH TO SIMULATING THE EFFECTS OF PERMEABILITY HETEROGENEITY , 1991 .

[14]  E. Smith,et al.  THE INFLUENCE OF SMALL-SCALE HETEROGENEITY ON AVERAGE RELATIVE PERMEABILITY , 1991 .

[15]  Louis J. Durlofsky,et al.  Use of Higher Moments for the Description of Upscaled, Process Independent Relative Permeabilities , 1997 .

[16]  Michael Andrew Christie,et al.  Analytical Calculation of Coarse-Grid Corrections for Use in Pseudofunctions , 1998 .

[17]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .

[18]  Khalid Aziz,et al.  Automatic grid generation for modeling reservoir heterogeneities , 1992 .

[19]  Y. Efendiev The Multiscale Finite Element Method (MsFEM) and Its Applications , 1999 .

[20]  Jørg E. Aarnes,et al.  On the Use of a Mixed Multiscale Finite Element Method for GreaterFlexibility and Increased Speed or Improved Accuracy in Reservoir Simulation , 2004, Multiscale Model. Simul..

[21]  L. Holden,et al.  Global Upscaling of Permeability in Heterogeneous Reservoirs; The Output Least Squares (OLS) Method , 2000 .

[22]  Kenneth Stuart Sorbie,et al.  The Scaleup of Two-Phase Flow in Porous Media Using Phase Permeability Tensors , 1996 .

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

[24]  P. King The use of renormalization for calculating effective permeability , 1989 .

[25]  J. Gómez-Hernández,et al.  Upscaling hydraulic conductivities in heterogeneous media: An overview , 1996 .

[26]  Yalchin Efendiev,et al.  Numerical modeling of subgrid heterogeneity in two phase flow simulations , 2002 .

[27]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[28]  Stein Krogstad,et al.  Multiscale mixed/mimetic methods on corner-point grids , 2008 .

[29]  Kenneth Stuart Sorbie,et al.  The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow , 2001 .

[30]  H. L. Stone,et al.  Rigorous Black Oil Pseudo Functions , 1991 .

[31]  Ilio Galligani,et al.  Mathematical Aspects of Finite Element Methods , 1977 .

[32]  Ole Jakob Arntzen,et al.  Higher-Order Terms in the Capillary Limit Approximation of Viscous-Capillary Flow , 2004 .

[33]  Stein Krogstad,et al.  A Hierarchical Multiscale Method for Two-Phase Flow Based upon Mixed Finite Elements and Nonuniform Coarse Grids , 2006, Multiscale Model. Simul..

[34]  Louis J. Durlofsky,et al.  Adaptive Local–Global Upscaling for General Flow Scenarios in Heterogeneous Formations , 2006 .

[35]  P. Renard,et al.  Calculating equivalent permeability: a review , 1997 .

[36]  Aslak Tveito,et al.  An upscaling method for one‐phase flow in heterogeneous reservoirs. A weighted output least squares (WOLS) approach , 1998 .

[37]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[38]  T. Hou,et al.  Analysis of upscaling absolute permeability , 2002 .

[39]  J. W. Barker,et al.  A critical review of the use of pseudo-relative permeabilities for upscaling , 1997 .

[40]  Michael J. King,et al.  Application of Novel Upscaling Approaches to the Magnus and Andrew Reservoirs , 1998 .

[41]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[42]  L. Durlofsky,et al.  A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations , 2003 .

[43]  Yalchin Efendiev,et al.  On homogenization of nonlinear hyperbolic equations , 2005 .

[44]  H. Tchelepi,et al.  Multi-scale finite-volume method for elliptic problems in subsurface flow simulation , 2003 .

[45]  J. O. Aasen,et al.  Steady-State Upscaling , 1998 .

[46]  G. R. Jerauld,et al.  Impacts of Scale-up on Fluid Flow from Plug to Gridblock Scale in Reservoir Rock , 1996 .

[47]  Knut-Andreas Lie,et al.  Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels , 2005 .

[48]  Michael Andrew Christie,et al.  Tenth SPE Comparative Solution Project: a comparison of upscaling techniques , 2001 .

[49]  Knut-Andreas Lie,et al.  A comparison of multiscale methods for elliptic problems in porous media flow , 2008 .

[50]  Patrick Jenny,et al.  Adaptive Multiscale Finite-Volume Method for Multiphase Flow and Transport in Porous Media , 2005, Multiscale Model. Simul..

[51]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .

[52]  Yalchin Efendiev,et al.  A Generalized Convection-Diffusion Model for Subgrid Transport in Porous Media , 2003, Multiscale Model. Simul..

[53]  L. Durlofsky Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media , 1991 .

[54]  Kenneth Stuart Sorbie,et al.  Development and Application of a New Two-Phase Scaleup Method Based on Tensor Permeabilities , 1994 .

[55]  C. C. Mattax,et al.  The Modeling of a Three-Dimensional Reservoir with a Two-Dimensional Reservoir Simulator-The Use of Dynamic Pseudo Functions , 1973 .

[56]  Knut-Andreas Lie,et al.  Toward Reservoir Simulation on Geological Grid Models , 2004 .

[57]  J. W. Barker,et al.  An Analysis of Dynamic Pseudo Relative Permeability Methods , 1996 .

[58]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[59]  Michael Andrew Christie,et al.  Upscaling for reservoir simulation , 1996 .

[60]  Stig Bakke,et al.  Extending Predictive Capabilities to Network Models , 1998 .

[61]  U. Hornung Homogenization and porous media , 1996 .

[62]  Martin J. Blunt,et al.  Nested gridding and streamline-based simulation for fast reservoir performance prediction , 1999 .

[63]  J. W. Barker,et al.  Transport Coefficients for Compositional Simulation With Coarse Grids in Heterogeneous Media , 1994 .

[64]  L. Durlofsky,et al.  A nonuniform coarsening approach for the scale-up of displacement processes in heterogeneous porous media , 1997 .

[65]  J. R. Kyte,et al.  New Pseudo Functions To Control Numerical Dispersion , 1975 .