Bridging from Pore to Continuum: A Hybrid Mortar Domain Decomposition Framework for Subsurface Flow and Transport


Flow and transport in the subsurface occurs over a wide range of spatial scales (nanometer to kilometer). Modeling at the pore scale becomes imperative where scales are not separable. Since pore-scale models are computationally limited to small domain sizes, accurate field-scale modeling requires simulating parts of the reservoir at the pore scale and other parts at the continuum. The need for modeling large pore-scale domains for ascertaining macroscopic parameters is prevalent in the literature. We develop a hybrid mortar domain decomposition framework for parallel modeling (linear and nonlinear) flow and transport across scales and in large pore-scale domains. Novel and efficient mortar methods for coupling flow and transport are adapted and developed and comparisons are presented. Mortars are developed for pore-to-pore and pore-to-continuum interfaces and shown to be more suitable than the commonly used Lagrangian mortars for these interfaces. The methods are shown to produce accurate results and demonstrated to be much more efficient than solving the problem as a single domain even in the absence of parallelism (especially for transport). The coupling algorithms are also extended to include diffusive transport. The methods are further applied and validated for coupling pore-scale to continuum-scale subdomains. Finally, the methods are shown to be computationally efficient for nonlinear flow problems. The results of this work provide a promising step forward in closing the spatial gap between the pore scale and the continuum.


  1. mortar coupling
  2. hybrid modeling
  3. multiscale modeling
  4. domain decomposition
  5. pore-scale modeling
  6. flow and transport

MSC codes

  1. 65M55
  2. 65N
  3. 68Q22
  4. 76M25
  5. 35-04
  6. 35J15
  7. 35K10
  8. 35K57
  9. 35L65

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R. Al-Raoush, K. Thompson, and C. S. Willson, Comparison of network generation techniques for unconsolidated porous media, Soil Sci. Soc. Amer. J., 67 (2003), pp. 1687--1700.
T. Arbogast, L. C. Cowsar, M. F. Wheeler, and I. Yotov, Mixed finite element methods on nonmatching multiblock grids, SIAM J. Numer. Anal., 37 (2000), pp. 1295--1315.
T. Arbogast, G. Pencheva, M. F. Wheeler, and I. Yotov, A multiscale mortar mixed finite element method, Multiscale Model. Simul., 6 (2007), pp. 319--346.
M. T. Balhoff, K. E. Thompson, and M. Hjortsø, Coupling pore-scale networks to continuum-scale models of porous media, Comput. Geosci., 33 (2007), pp. 393--410.
M. T. Balhoff, S. G. Thomas, and M. F. Wheeler, Mortar coupling and upscaling of pore-scale models, Comput. Geosci., 12 (2008), pp. 15--27.
M. T. Balhoff, D. Sanchez-Rivera, A. Kwok, Y. Mehmani, and M. Prodanović, Numerical algorithms for network modeling of yield stress and other non-Newtonian fluids in porous media, Transp. Porous Media, 93 (2012), pp. 363--379.
I. Battiato, D. M. Tartakovsky, A. M. Tartakovsky, and T. Scheibe, On breakdown of macroscopic models of mixing-controlled heterogeneous reactions in porous media, Adv. Water Resources, 32 (2009), pp. 1664--1673.
I. Battiato and D. M. Tartakovsky, Applicability regimes for macroscopic models of reactive transport in porous media, J. Contaminant Hydrology, 120 (2011), pp. 18--26.
I. Battiato, D. M. Tartakovsky, A. M. Tartakovsky, and T. D. Scheibe, Hybrid models of reactive transport in porous and fractured media, Adv. Water Resources, 34 (2011), pp. 1140--1150.
L. E. Beckingham, C. A. Peters, W. Um, K. W. Jones, and W. B. Lindquist, $2$D and $3$D imaging resolution trade-offs in quantifying pore throats for prediction of permeability, Adv. Water Resources, 62 (2013), pp. 1--12.
C. Bernardi, Y. Maday, and A. T. Patera, A new nonconforming approach to domain decomposition: The mortar element method, in Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. XI (Paris, 1989--1991), H. Brezis and J.-L. Lions, eds., Longman Sci. Tech., Harlow, UK, (1994), pp. 13--51.
B. Bijeljic, A. H. Muggeridge, and M. J. Blunt, Pore-scale modeling of longitudinal dispersion, Water Resources Research, 40 (2004).
C. Bruderer and Y. Bernabé, Network modeling of dispersion: Transition from Taylor dispersion in homogeneous networks to mechanical dispersion in very heterogeneous ones, Water Resources Res., 37 (2001), pp. 897--908.
S. Chen, and G. D. Doolen, Lattice Boltzmann method for fluid flows, Ann. Rev. Fluid Mech., 30 (1998), pp. 329--364.
J. Chu, B. Engquist, M. Prodanović, and R. Tsai, A Multiscale Method Coupling Network and Continuum Models in Porous Media II: Single and Two Phase Flow, ICES Report 11-42, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, (2011).
J. Chu, B. Engquist, M. Prodanović, and R. Tsai, A multiscale method coupling network and continuum models in porous media I: Steady-state single phase flow, Multiscale Model. Simul., 10 (2012), pp. 515--549.
I. Fatt, The network model of porous media, I. Capillary pressure characteristics, Pet. Trans. AIME, 207 (1956), pp. 144--159.
B. Ganis, M. Juntunen, G. Pencheva, M. F. Wheeler, and I. Yotov, A Global Jacobian Method for Simultaneous Solution of Mortar and Subdomain Variables in Nonlinear Porous Media Flow, ICES Report 12-46, The Institute for Computational Engineering and Sciences, The University of Texas at Austin, (2012).
D. Kim, C. A. Peters, and W. B. Lindquist, Upscaling geochemical reaction rates accompanying acidic CO$_2$-saturated brine flow in sandstone aquifers, Water Resources Res., 47 (1), (2011).
L. Li, C. A. Peters, and M. A. Celia, Upscaling geochemical reaction rates using pore-scale network modeling, Adv. Water Resources, 29 (2006), pp. 1351--1370.
Y. Mehmani, T. Sun, M. T. Balhoff, P. Eichhubl, and S. L. Bryant, Multiblock pore-scale modeling and upscaling of reactive transport: Application to carbon sequestration, Transp. Porous Media. 95 (2012), pp. 305--326.
M. Peszyńska, M. F. Wheeler, and I. Yotov, Mortar upscaling for multiphase flow in porous media, Comput. Geosci., 6 (2002), pp. 73--100.
M. Peszyńska, Mortar adaptivity in mixed methods for flow in porous media, Int. J. Numer. Anal. Model., 2 (2005), pp. 241--282.
M. Prodanovic and S. L. Bryant, A level set method for determining critical curvatures for drainage and imbibition, J. Colloid Interface Sci., 304 (2006), pp. 442--458.
Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, (2003).
T. Sun, Y. Mehmani, J. Bhagmane, and M. T. Balhoff, Pore to continuum upscaling of permeability in heterogeneous porous media using mortars, Intern. J. Oil Gas Coal Technol., 5 (2012), pp. 249--266.
T. Sun, Y. Mehmani, and M. T. Balhoff, Hybrid multiscale modeling through direct substitution of pore-scale models into near-well reservoir simulators, Energy & Fuels, 26 (2012), pp. 5828--5836.
A. M. Tartakovsky and P. Meakin, Simulation of unsaturated flow in complex fractures using smoothed particle hydrodynamics, Vadose Zone J., 4 (2005), pp. 848--855.
A. M. Tartakovsky, D. M. Tartakovsky, T. D. Scheibe, and P. Meakin, Hybrid simulations of reaction-diffusion systems in porous media, SIAM J. Sci. Comput., 30 (2008), pp. 2799--2816.
M. F. Wheeler, T. Arbogast, S. Bryant, J. Eaton, Q. Lu, M. Peszyńska, and I. Yotov, A parallel multiblock/multidomain approach for reservoir simulation, in Proceedings of the 15th SPE Reservoir Simulation Symposium, 51884, 1999, pp. 51--61.

Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 667 - 693
ISSN (online): 1540-3467


Submitted: 10 June 2013
Accepted: 10 February 2014
Published online: 29 May 2014


  1. mortar coupling
  2. hybrid modeling
  3. multiscale modeling
  4. domain decomposition
  5. pore-scale modeling
  6. flow and transport

MSC codes

  1. 65M55
  2. 65N
  3. 68Q22
  4. 76M25
  5. 35-04
  6. 35J15
  7. 35K10
  8. 35K57
  9. 35L65



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