Search Menu
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information
Search Scholarly Publications

Title: Multiscale simulations for multi-continuum Richards equations

Abstract

In this paper, we study a multiscale method for simulating a dual-continuum unsaturated flow problem within complex heterogeneous fractured porous media. Mathematically, each of the dual continua is modeled by a multiscale Richards equation (for pressure head), and these equations are coupled to one another by transfer terms. On its own, Richards equation is already a nonlinear partial differential equation, and it is exceedingly difficult to solve numerically due to the extra nonlinear dependencies involving the soil water. To deal with multiple scales, our strategy is that starting from a microscopic scale, we upscale the coupled system of dual-continuum Richards equations via homogenization by the two-scale asymptotic expansion, to obtain a homogenized system, at an intermediate scale (level). Based on a hierarchical approach, the homogenization’s effective coefficients are computed through solving the arising cell problems. Furthermore, to tackle the nonlinearity, after time discretization, we use Picard iteration procedure for linearization of the homogenized Richards equations. At each Picard iteration, some degree of multiscale still remains from the intermediate level, so we utilize the generalized multiscale finite element method (GMsFEM) combining with a multi-continuum approach, to upscale the homogenized system to a macroscopic (coarse-grid) level. This scheme involves building uncoupled andmore » coupled multiscale basis functions, which are used not only to construct coarse-grid solution approximation with high accuracy but also (with the coupled multiscale basis) to capture the interactions among continua. These prospects and convergence are demonstrated by several numerical results for the proposed method.« less

Authors:
 [1]; ORCiD logo [2]; ORCiD logo [3]
+ Show Author Affiliations
  1. Univ. of Iowa, Iowa City, IA (United States)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
  3. Duy Tan Univ., Da Nang (Vietnam). Inst. of Research and Development
Publication Date:
Research Org.:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); Vietnam National Foundation for Science and Technology Development (NAFOSTED)
OSTI Identifier:
1810663
Alternate Identifier(s):
OSTI ID: 1815329
Report Number(s):
LLNL-JRNL-822603
Journal ID: ISSN 0377-0427; 1035173
Grant/Contract Number:  
AC52-07NA27344; 101.99-2019.326
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 397; Journal ID: ISSN 0377-0427
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; coupled system of nonlinear Richards equations; unsaturated multi-continuum flow problem; upscaling; hierarchical finite element; heterogeneous fractured porous media; generalized multiscale finite element method

Citation Formats

Park, Jun Richard, Cheung, Siu Wun, and Mai, Tina. Multiscale simulations for multi-continuum Richards equations. United States: N. p., 2021. Web. doi:10.1016/j.cam.2021.113648.
Copy to clipboard
Park, Jun Richard, Cheung, Siu Wun, & Mai, Tina. Multiscale simulations for multi-continuum Richards equations. United States. https://doi.org/10.1016/j.cam.2021.113648
Copy to clipboard
Park, Jun Richard, Cheung, Siu Wun, and Mai, Tina. Mon . "Multiscale simulations for multi-continuum Richards equations". United States. https://doi.org/10.1016/j.cam.2021.113648. https://www.osti.gov/servlets/purl/1810663.
Copy to clipboard
@article{osti_1810663,
title = {Multiscale simulations for multi-continuum Richards equations},
author = {Park, Jun Richard and Cheung, Siu Wun and Mai, Tina},
abstractNote = {In this paper, we study a multiscale method for simulating a dual-continuum unsaturated flow problem within complex heterogeneous fractured porous media. Mathematically, each of the dual continua is modeled by a multiscale Richards equation (for pressure head), and these equations are coupled to one another by transfer terms. On its own, Richards equation is already a nonlinear partial differential equation, and it is exceedingly difficult to solve numerically due to the extra nonlinear dependencies involving the soil water. To deal with multiple scales, our strategy is that starting from a microscopic scale, we upscale the coupled system of dual-continuum Richards equations via homogenization by the two-scale asymptotic expansion, to obtain a homogenized system, at an intermediate scale (level). Based on a hierarchical approach, the homogenization’s effective coefficients are computed through solving the arising cell problems. Furthermore, to tackle the nonlinearity, after time discretization, we use Picard iteration procedure for linearization of the homogenized Richards equations. At each Picard iteration, some degree of multiscale still remains from the intermediate level, so we utilize the generalized multiscale finite element method (GMsFEM) combining with a multi-continuum approach, to upscale the homogenized system to a macroscopic (coarse-grid) level. This scheme involves building uncoupled and coupled multiscale basis functions, which are used not only to construct coarse-grid solution approximation with high accuracy but also (with the coupled multiscale basis) to capture the interactions among continua. These prospects and convergence are demonstrated by several numerical results for the proposed method.},
doi = {10.1016/j.cam.2021.113648},
journal = {Journal of Computational and Applied Mathematics},
number = ,
volume = 397,
place = {United States},
year = {Mon May 17 00:00:00 EDT 2021},
month = {Mon May 17 00:00:00 EDT 2021}
}
Copy to clipboard
Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1016/j.cam.2021.113648
Copyright Statement
Other availability
Search WorldCat Search WorldCat to find libraries that may hold this journal
Save / Share:
Export Metadata
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.

Works referenced in this record:

Derivation of the Double Porosity Model of Single Phase Flow via Homogenization Theory
journal, July 1990

  • Arbogast, Todd; Douglas, Jr., Jim; Hornung, Ulrich
  • SIAM Journal on Mathematical Analysis, Vol. 21, Issue 4
  • DOI: 10.1137/0521046

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains
journal, July 2016

Efficient uncertainty quantification techniques in inverse problems for Richards’ equation using coarse-scale simulation models
journal, March 2009

Multiscale model reduction for transport and flow problems in perforated domains
journal, March 2018

  • Chung, Eric T.; Leung, Wing Tat; Vasilyeva, Maria
  • Journal of Computational and Applied Mathematics, Vol. 330
  • DOI: 10.1016/j.cam.2017.08.017

Computational multiscale methods for linear poroelasticity with high contrast
journal, October 2019

Multiscale model reduction for shale gas transport in fractured media
journal, May 2016

  • Akkutlu, I. Y.; Efendiev, Yalchin; Vasilyeva, Maria
  • Computational Geosciences, Vol. 20, Issue 5
  • DOI: 10.1007/s10596-016-9571-6

Local discontinuous Galerkin approximations to Richards’ equation
journal, March 2007

Nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media
journal, August 2019

  • Vasilyeva, Maria; Chung, Eric T.; Cheung, Siu Wun
  • Journal of Computational and Applied Mathematics, Vol. 355
  • DOI: 10.1016/j.cam.2019.01.024

Numerical Simulation of Water-Oil Flow in Naturally Fractured Reservoirs
journal, December 1976

  • Kazemi, H.; Merrill, L. S.; Porterfield, K. L.
  • Society of Petroleum Engineers Journal, Vol. 16, Issue 06
  • DOI: 10.2118/5719-PA

Constraint energy minimizing generalized multiscale finite element method for dual continuum model
journal, January 2020

  • Cheung, Siu Wun; Chung, Eric T.; Efendiev, Yalchin
  • Communications in Mathematical Sciences, Vol. 18, Issue 3
  • DOI: 10.4310/CMS.2020.v18.n3.a4

Residual-driven online generalized multiscale finite element methods
journal, December 2015

  • Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat
  • Journal of Computational Physics, Vol. 302
  • DOI: 10.1016/j.jcp.2015.07.068

A Three-Level Multi-Continua Upscaling Method for Flow Problems in Fractured Porous Media
journal, June 2019

Hydraulic conductivity estimation in partially saturated soils using the adjoint method
journal, December 2006

  • Santos, J.; Efendiev, Y.; Guarracino, L.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 196, Issue 1-3
  • DOI: 10.1016/j.cma.2006.03.004

Nonlocal multicontinua with representative volume elements. Bridging separable and non-separable scales
journal, April 2021

  • Chung, Eric T.; Efendiev, Yalchin; Leung, Wing T.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 377
  • DOI: 10.1016/j.cma.2021.113687

Application and Validation of an Upscaling Method for Unsaturated Water Flow Processes in Heterogeneous Soils
journal, July 2015

A study on iterative methods for solving Richards’ equation
journal, March 2016

A general mass-conservative numerical solution for the unsaturated flow equation
journal, July 1990

  • Celia, Michael A.; Bouloutas, Efthimios T.; Zarba, Rebecca L.
  • Water Resources Research, Vol. 26, Issue 7
  • DOI: 10.1029/WR026i007p01483

Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy–Forchheimer flow in the fracture
journal, August 2014

  • Knabner, Peter; Roberts, Jean E.
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 48, Issue 5
  • DOI: 10.1051/m2an/2014003

Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
journal, September 2016

Modelling fluid flow in fractured-porous rock masses by finite-element techniques
journal, April 1984

  • Baca, R. G.; Arnett, R. C.; Langford, D. W.
  • International Journal for Numerical Methods in Fluids, Vol. 4, Issue 4
  • DOI: 10.1002/fld.1650040404

An adaptive multiscale finite element method for unsaturated flow problems in heterogeneous porous media
journal, July 2009

Generalized multiscale finite element method for a strain-limiting nonlinear elasticity model
journal, October 2019

  • Fu, Shubin; Chung, Eric; Mai, Tina
  • Journal of Computational and Applied Mathematics, Vol. 359
  • DOI: 10.1016/j.cam.2019.03.047

Homogenization of a multiscale multi-continuum system
journal, June 2020

Space-Time Nonlinear Upscaling Framework Using Nonlocal Multicontinuum Approach
journal, January 2019

  • Leung, Wing T.; Chung, Eric T.; Efendiev, Yalchin
  • International Journal for Multiscale Computational Engineering, Vol. 17, Issue 5
  • DOI: 10.1615/IntJMultCompEng.2019031829

Central Discontinuous Galerkin Methods on Overlapping Cells with a Nonoscillatory Hierarchical Reconstruction
journal, January 2007

  • Liu, Yingjie; Shu, Chi-Wang; Tadmor, Eitan
  • SIAM Journal on Numerical Analysis, Vol. 45, Issue 6
  • DOI: 10.1137/060666974

Multiscale finite element methods for high-contrast problems using local spectral basis functions
journal, February 2011

  • Efendiev, Yalchin; Galvis, Juan; Wu, Xiao-Hui
  • Journal of Computational Physics, Vol. 230, Issue 4
  • DOI: 10.1016/j.jcp.2010.09.026

Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]
journal, January 1960

  • Barenblatt, G. I.; Zheltov, Iu. P.; Kochina, I. N.
  • Journal of Applied Mathematics and Mechanics, Vol. 24, Issue 5
  • DOI: 10.1016/0021-8928(60)90107-6

Convergence of the CEM-GMsFEM for Stokes flows in heterogeneous perforated domains
journal, June 2021

  • Chung, Eric; Hu, Jiuhua; Pun, Sai-Mang
  • Journal of Computational and Applied Mathematics, Vol. 389
  • DOI: 10.1016/j.cam.2020.113327

Non-local multi-continua upscaling for flows in heterogeneous fractured media
journal, November 2018

  • Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat
  • Journal of Computational Physics, Vol. 372
  • DOI: 10.1016/j.jcp.2018.05.038

Constraint energy minimizing generalized multiscale finite element method for nonlinear poroelasticity and elasticity
journal, September 2020

A hierarchical finite element Monte Carlo method for stochastic two-scale elliptic equations
journal, October 2017

Generalized Multiscale Finite Element method for multicontinua unsaturated flow problems in fractured porous media
journal, May 2020

  • Spiridonov, Denis; Vasilyeva, Maria; Chung, Eric T.
  • Journal of Computational and Applied Mathematics, Vol. 370
  • DOI: 10.1016/j.cam.2019.112594

Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media
journal, May 1991

  • Durlofsky, Louis J.
  • Water Resources Research, Vol. 27, Issue 5
  • DOI: 10.1029/91WR00107

Learning macroscopic parameters in nonlinear multiscale simulations using nonlocal multicontinua upscaling techniques
journal, July 2020

Iterative multiscale finite-volume method
journal, October 2008

  • Hajibeygi, Hadi; Bonfigli, Giuseppe; Hesse, Marc Andre
  • Journal of Computational Physics, Vol. 227, Issue 19
  • DOI: 10.1016/j.jcp.2008.06.013

Constraint Energy Minimizing Generalized Multiscale Finite Element Method
journal, September 2018

  • Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat
  • Computer Methods in Applied Mechanics and Engineering, Vol. 339
  • DOI: 10.1016/j.cma.2018.04.010

A Comparison of Numerical Simulation Models For One-Dimensional Infiltration
journal, March 1977

Global–local nonlinear model reduction for flows in heterogeneous porous media
journal, August 2015

  • Alotaibi, Manal; Calo, Victor M.; Efendiev, Yalchin
  • Computer Methods in Applied Mechanics and Engineering, Vol. 292
  • DOI: 10.1016/j.cma.2014.10.034

Multiscale model reduction for shale gas transport in a coupled discrete fracture and dual-continuum porous media
journal, December 2017

  • Akkutlu, I. Yucel; Efendiev, Yalchin; Vasilyeva, Maria
  • Journal of Natural Gas Science and Engineering, Vol. 48
  • DOI: 10.1016/j.jngse.2017.02.040

An exponential root-water-uptake model with water stress compensation
journal, October 2001

Coupling of multiscale and multi-continuum approaches
journal, March 2017

  • Chung, Eric T.; Efendiev, Yalchin; Leung, Tat
  • GEM - International Journal on Geomathematics, Vol. 8, Issue 1
  • DOI: 10.1007/s13137-017-0093-8

The Behavior of Naturally Fractured Reservoirs
journal, September 1963

  • Warren, J. E.; Root, P. J.
  • Society of Petroleum Engineers Journal, Vol. 3, Issue 03
  • DOI: 10.2118/426-PA

Constrained energy minimization based upscaling for coupled flow and mechanics
journal, January 2019

  • Vasilyeva, Maria; Chung, Eric T.; Efendiev, Yalchin
  • Journal of Computational Physics, Vol. 376
  • DOI: 10.1016/j.jcp.2018.09.054

Efficient 3D inversions using the Richards equation
journal, July 2018

Numerical Solution of Richards' Equation: A Review of Advances and Challenges
journal, October 2017

  • Farthing, Matthew W.; Ogden, Fred L.
  • Soil Science Society of America Journal, Vol. 81, Issue 6
  • DOI: 10.2136/sssaj2017.02.0058

A numerical method for singularly perturbed weakly coupled system of two second order ordinary differential equations with discontinuous source term
journal, May 2007

  • Tamilselvan, A.; Ramanujam, N.; Shanthi, V.
  • Journal of Computational and Applied Mathematics, Vol. 202, Issue 2
  • DOI: 10.1016/j.cam.2006.02.025

Multiscale model reduction for shale gas transport in poroelastic fractured media
journal, January 2018

  • Akkutlu, I. Yucel; Efendiev, Yalchin; Vasilyeva, Maria
  • Journal of Computational Physics, Vol. 353
  • DOI: 10.1016/j.jcp.2017.10.023

A mass conservative numerical solution for two-phase flow in porous media with application to unsaturated flow
journal, October 1992

  • Celia, Michael A.; Binning, Philip
  • Water Resources Research, Vol. 28, Issue 10
  • DOI: 10.1029/92WR01488

Generalized Multiscale Finite Element Method for the poroelasticity problem in multicontinuum media
journal, August 2020

  • Tyrylgin, Aleksei; Vasilyeva, Maria; Spiridonov, Denis
  • Journal of Computational and Applied Mathematics, Vol. 374
  • DOI: 10.1016/j.cam.2020.112783

Generalized multiscale finite element methods for problems in perforated heterogeneous domains
journal, June 2015

Hierarchical multiscale finite element method for multi-continuum media
journal, May 2020

  • Park, Jun Sur Richard; Hoang, Viet Ha
  • Journal of Computational and Applied Mathematics, Vol. 369
  • DOI: 10.1016/j.cam.2019.112588

Multiscale simulations for upscaled multi-continuum flows
journal, August 2020

  • Park, Jun Sur Richard; Cheung, Siu Wun; Mai, Tina
  • Journal of Computational and Applied Mathematics, Vol. 374
  • DOI: 10.1016/j.cam.2020.112782

An open source massively parallel solver for Richards equation: Mechanistic modelling of water fluxes at the watershed scale
journal, December 2014

An Efficient Hierarchical Multiscale Finite Element Method for Stokes Equations in Slowly Varying Media
journal, January 2013

  • Brown, Donald L.; Efendiev, Yalchin; Hoang, Viet Ha
  • Multiscale Modeling & Simulation, Vol. 11, Issue 1
  • DOI: 10.1137/110858525

Homogenization of nonlinear reaction-diffusion equation with a large reaction term
journal, February 2010

Generalized multiscale finite elements for simulation of elastic-wave propagation in fractured media
journal, January 2018

  • Cho, Yongchae; Gibson, Richard L.; Vasilyeva, Maria
  • GEOPHYSICS, Vol. 83, Issue 1
  • DOI: 10.1190/geo2017-0076.1

Simulating coupled surface–subsurface flows with ParFlow v3.5.0: capabilities, applications, and ongoing development of an open-source, massively parallel, integrated hydrologic model
journal, January 2020

  • Kuffour, Benjamin N. O.; Engdahl, Nicholas B.; Woodward, Carol S.
  • Geoscientific Model Development, Vol. 13, Issue 3
  • DOI: 10.5194/gmd-13-1373-2020

Generalized multiscale finite element methods (GMsFEM)
journal, October 2013

A General Convergence Result for a Functional Related to the Theory of Homogenization
journal, May 1989

  • Nguetseng, Gabriel
  • SIAM Journal on Mathematical Analysis, Vol. 20, Issue 3
  • DOI: 10.1137/0520043

Fast online generalized multiscale finite element method using constraint energy minimization
journal, February 2018

  • Chung, Eric T.; Efendiev, Yalchin; Leung, Wing Tat
  • Journal of Computational Physics, Vol. 355
  • DOI: 10.1016/j.jcp.2017.11.022

Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media
journal, January 2005

  • Chen, Zhiming; Deng, Weibing; Ye, Huang
  • Communications in Mathematical Sciences, Vol. 3, Issue 4
  • DOI: 10.4310/CMS.2005.v3.n4.a2

Generalized multiscale multicontinuum model for fractured vuggy carbonate reservoirs
journal, March 2020

  • Wang, Min; Cheung, Siu Wun; Chung, Eric T.
  • Journal of Computational and Applied Mathematics, Vol. 366
  • DOI: 10.1016/j.cam.2019.112370

Homogenization and Two-Scale Convergence
journal, November 1992

  • Allaire, Grégoire
  • SIAM Journal on Mathematical Analysis, Vol. 23, Issue 6
  • DOI: 10.1137/0523084

Capillary Conduction of Liquids Through Porous Mediums
journal, November 1931

Homogenization of a Phase Field Model for Binary Mixtures
journal, January 2005

Multiscale Finite Element Methods for Nonlinear Problems and Their Applications
journal, January 2004

  • Efendiev, Y.; Ginting, V.; Hou, T. Y.
  • Communications in Mathematical Sciences, Vol. 2, Issue 4
  • DOI: 10.4310/CMS.2004.v2.n4.a2

On fluid reserves and the production of superheated steam from fractured, vapor-dominated geothermal reservoirs
journal, January 1982

A new Duality Approach to Elasticity
journal, January 2012

  • Ciarlet, Philippe G.; Geymonat, Giuseppe; Krasucki, FranÇOise
  • Mathematical Models and Methods in Applied Sciences, Vol. 22, Issue 01
  • DOI: 10.1142/s0218202512005861

A study on iterative methods for solving Richards` equation
text, January 2015

    Sort by:
    [ × clear filter / sort ]

    Similar Records in DOE PAGES and OSTI.GOV collections: