ICES REPORT 17-13 June 2017 Adaptive Numerical Homogenization for Non-Linear Multiphase Flow and Transport

One of the major objectives in the development of upscaling approaches is to reduce the computational costs associated with solving fine scale flow and transport problems in heterogeneous porous media. This is due to the availability of reservoir rock property data at fine spatial scales, such as facies distributions, obtained from geological models and field data from well logs. The data sets from each of these sources are themselves at different spatial scales which further adds to the computational challenge. The upscaling approach must not only accommodate these disparate data sets but also capture the flow physics accurately while maintaining computational efficiency. We present a novel upscaling approach which draws upon previous developments of two-scale homogenization [4] to obtain coarse scale properties in addition to dynamic mesh refinement using an enhanced velocity mixed finite element method (EV MFEM) [27]. A transient region is defined where changes in saturation/concentration are above a chosen threshold compared to a non-transient region where these are relatively small. Since most of the recovery technologies employed in oil and gas field operations involve flooding the subsurface porous medium; as in the case of water-flooding, chemical or gas enhanced oil recovery (EOR), these aforementioned transient regions are usually restricted to much smaller subdomains of the entire reservoir domain. The computational efficiency is achieved by using coarse scale parameters, from numerical homogenization, in the non-transient region. Furthermore, the solution accuracy is preserved by using fine scale information only in the transient region. The numerical results section shows that our adaptive homogenization approach closely captures fine scale flow and transport features while maintaining a computational speedup of approximately 4 times for a variety of permeability distributions extracted from the SPE 10 comparative upscaling project [13].

[1]  S. E. Buckley,et al.  Mechanism of Fluid Displacement in Sands , 1942 .

[2]  E. C. Childs.  Dynamics of fluids in Porous Media , 1973 .

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

[4]  Van Genuchten,et al.  A closed-form equation for predicting the hydraulic conductivity of unsaturated soils , 1980 .

[5]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

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

[7]  Homer F. Walker,et al.  Globally Convergent Inexact Newton Methods , 1994, SIAM J. Optim..

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

[9]  Mary F. Wheeler,et al.  A parallel, implicit, cell‐centered method for two‐phase flow with a preconditioned Newton–Krylov solver , 1997 .

[10]  Andrey L. Piatnitski,et al.  Averaging a transport equation with small diffusion and oscillating velocity , 2001 .

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

[12]  C. L. Farmer.  Upscaling: a review , 2002 .

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

[14]  Mary F. Wheeler,et al.  Enhanced Velocity Mixed Finite Element Methods for Flow in Multiblock Domains , 2002 .

[15]  L. J. Durlofsky,et al.  Upscaling of Geocellular Models for Flow Simulation , 2004 .

[16]  Alain Bourgeat,et al.  Effective Macrodiffusion in Solute Transport through Heterogeneous Porous Media , 2005, Multiscale Model. Simul..

[17]  Andro Mikelic,et al.  Rigorous Upscaling of the Reactive Flow through a Pore, under Dominant Peclet and Damkohler Numbers , 2006, SIAM J. Math. Anal..

[18]  Yalchin Efendiev,et al.  An Adaptive Multiscale Method for Simulation of Fluid Flow in Heterogeneous Porous Media , 2006, Multiscale Model. Simul..

[19]  Todd Arbogast,et al.  A Multiscale Mortar Mixed Finite Element Method , 2007, Multiscale Model. Simul..

[20]  R. H. Brooks,et al.  Hydraulic properties of porous media , 1963 .

[21]  Abraham K. John,et al.  Dispersion in large scale permeable media , 2008 .

[22]  Yalchin Efendiev,et al.  Multiscale Finite Element Methods: Theory and Applications , 2009 .

[23]  Mary F. Wheeler,et al.  Enhanced velocity mixed finite element methods for modeling coupled flow and transport on non-matching multiblock grids , 2011 .

[24]  T. Arbogast.  Mixed Multiscale Methods for Heterogeneous Elliptic Problems , 2012 .

[25]  Mojdeh Delshad,et al.  Comparison of ensemble filtering algorithms and null‐space Monte Carlo for parameter estimation and uncertainty quantification using CO2 sequestration data , 2013 .

[26]  Mary F. Wheeler,et al.  Compositional flow modeling using a multi-point flux mixed finite element method , 2014, Computational Geosciences.

[27]  Yalchin Efendiev,et al.  Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods , 2016, J. Comput. Phys..

[28]  Yerlan Amanbek,et al.  Adaptive numerical homogenization for upscaling single phase flow and transport , 2019, J. Comput. Phys..