Finite Element Modeling of Diffusion in Fractured Porous Media by Using Hierarchical Material Properties

Beskardes, G. D.; Weiss, C. J.; Kuhlman, K. L.; Chang, K. W.

doi:10.1029/2020WR029510

Finite Element Modeling of Diffusion in Fractured Porous Media by Using Hierarchical Material Properties

Journal Article · Wed Feb 09 23:00:00 EST 2022 · Water Resources Research

DOI:https://doi.org/10.1029/2020WR029510· OSTI ID:1844460

^[1]; Weiss, C. J. ^[2]; ^[3]; ^[4]

Geophysics Department Sandia National Laboratories Albuquerque NM USA
Geophysics Department Sandia National Laboratories Albuquerque NM USA, Department of Earth and Planetary Sciences Albuquerque University of New Mexico Albuquerque NM USA
Applied Systems Analysis &, Research Sandia National Laboratories Albuquerque NM USA
Geosciences Engineering Department Sandia National Laboratories Albuquerque NM USA

Abstract

Fractured porous media challenge modeling approaches due to high computational costs and excessive mesh refinement imposed by the extreme scale variability of fractures and the heterogeneity of the surrounding porous rock. To overcome such difficulties, we utilize the hierarchical finite element method ( Hi ‐FEM) that has been developed previously to simulate the electrical potential distribution in complex geologic environments. The method employs the hierarchical basis functions in classical finite element analysis to enable representation of material properties on each dimensional component of a given 3D unstructured finite element, thereby inherently allowing for interactions at the boundary between fracture and a host rock. In this study, we extend its application to transient fluid flow and heat conduction in the Laplace domain. Time‐domain flow solutions are obtained by numerical inverse Laplace transform. We evaluate the accuracy of the method using different flow models and demonstrate its robustness for large‐scale, rock mass models featuring complex fracture networks. Moreover, for the computation of nodal Darcian velocity fields in fractured porous media where the fractures are represented as 2D features, a new approach that employs the Yeh's Galerkin model for both volume and facet elements is proposed. Results show that Hi ‐FEM can produce accurate flow solutions for fractured porous media without any need of coupling or transfer mechanism while still being computationally economical and numerically robust, even for large‐scale simulations.

View Accepted Manuscript (Publisher)

Sponsoring Organization:: USDOE

OSTI ID:: 1844460

Journal Information:: Water Resources Research, Journal Name: Water Resources Research Journal Issue: 2 Vol. 58; ISSN 0043-1397

Publisher:: American Geophysical Union (AGU)Copyright Statement

Country of Publication:: United States

Language:: English

References (78)

Implementation of a finite-volume method for the determination of effective parameters in fissured porous media Caillabet, Yannick; Fabrie, Pierre; Landereau, Pascal Numerical Methods for Partial Differential Equations, Vol. 16, Issue 2 https://doi.org/10.1002/(sici)1098-2426(200003)16:2<237::aid-num6>3.0.co;2-w	journal	March 2000
Hydrologic issues associated with nuclear waste repositories: HYDROLOGIC ISSUES ASSOCIATED WITH NUCLEAR WASTE REPOSITORIES Tsang, Chin-Fu; Neretnieks, Ivars; Tsang, Yvonne Water Resources Research, Vol. 51, Issue 9 https://doi.org/10.1002/2015wr017641	journal	September 2015
Assessment of CO ₂ Storage Potential in Naturally Fractured Reservoirs With Dual‐Porosity Models March, Rafael; Doster, Florian; Geiger, Sebastian Water Resources Research, Vol. 54, Issue 3 https://doi.org/10.1002/2017wr022159	journal	March 2018
A Laplace domain finite element method (LDFEM) applied to diffusion and propagation problems in electrical engineering Cai, X. -D.; Costache, G. I. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 7, Issue 6 https://doi.org/10.1002/jnm.1660070606	journal	November 1994
A hybrid boundary element-finite element analysis procedure for fluid flow simulation in fractured rock masses Elsworth, D. International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 10, Issue 6 https://doi.org/10.1002/nag.1610100603	journal	November 1986
An Efficient Boundary Integral Formulation for Flow Through Fractured Porous Media Lough, M. F.; Lee, S. H.; Kamath, J. Journal of Computational Physics, Vol. 143, Issue 2 https://doi.org/10.1006/jcph.1998.5858	journal	July 1998
Permeability tensor and representative elementary volume of fractured rock masses Rong, Guan; Peng, Jun; Wang, Xiaojiang Hydrogeology Journal, Vol. 21, Issue 7 https://doi.org/10.1007/s10040-013-1040-x	journal	September 2013
Numerical simulation of fracture flow with a mixed-hybrid FEM stochastic discrete fracture network model Maryška, Jiří; Severýn, Otto; Vohralík, Martin Computational Geosciences, Vol. 8, Issue 3 https://doi.org/10.1007/s10596-005-0152-3	journal	January 2005
An embedded fracture modeling framework for simulation of hydraulic fracturing and shear stimulation Norbeck, Jack H.; McClure, Mark W.; Lo, Jonathan W. Computational Geosciences, Vol. 20, Issue 1 https://doi.org/10.1007/s10596-015-9543-2	journal	November 2015
A Lagrange multiplier method for a discrete fracture model for flow in porous media Köppel, Markus; Martin, Vincent; Jaffré, Jérôme Computational Geosciences, Vol. 23, Issue 2 https://doi.org/10.1007/s10596-018-9779-8	journal	September 2018
Review of inverse Laplace transform algorithms for Laplace-space numerical approaches Kuhlman, Kristopher L. Numerical Algorithms, Vol. 63, Issue 2 https://doi.org/10.1007/s11075-012-9625-3	journal	August 2012
Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata] Barenblatt, G. I.; Zheltov, Iu. P.; Kochina, I. N. Journal of Applied Mathematics and Mechanics, Vol. 24, Issue 5 https://doi.org/10.1016/0021-8928(60)90107-6	journal	January 1960
Free vibration analysis of a flat plate using the hierarchical finite element method Bardell, N. S. Journal of Sound and Vibration, Vol. 151, Issue 2 https://doi.org/10.1016/0022-460x(91)90855-e	journal	December 1991
A mixed-dimensional finite volume method for two-phase flow in fractured porous media Reichenberger, Volker; Jakobs, Hartmut; Bastian, Peter Advances in Water Resources, Vol. 29, Issue 7 https://doi.org/10.1016/j.advwatres.2005.09.001	journal	July 2006
An efficient numerical model for incompressible two-phase flow in fractured media Hoteit, Hussein; Firoozabadi, Abbas Advances in Water Resources, Vol. 31, Issue 6 https://doi.org/10.1016/j.advwatres.2008.02.004	journal	June 2008
Simulation of density-driven flow in fractured porous media Grillo, A.; Logashenko, D.; Stichel, S. Advances in Water Resources, Vol. 33, Issue 12 https://doi.org/10.1016/j.advwatres.2010.08.004	journal	December 2010
A general gridding, discretization, and coarsening methodology for modeling flow in porous formations with discrete geological features Karimi-Fard, M.; Durlofsky, L. J. Advances in Water Resources, Vol. 96 https://doi.org/10.1016/j.advwatres.2016.07.019	journal	October 2016
dfnWorks: A discrete fracture network framework for modeling subsurface flow and transport Hyman, Jeffrey D.; Karra, Satish; Makedonska, Nataliia Computers & Geosciences, Vol. 84 https://doi.org/10.1016/j.cageo.2015.08.001	journal	November 2015
The virtual element method for discrete fracture network simulations Benedetto, Matías Fernando; Berrone, Stefano; Pieraccini, Sandra Computer Methods in Applied Mechanics and Engineering, Vol. 280 https://doi.org/10.1016/j.cma.2014.07.016	journal	October 2014
Finite element mesh for complex flow simulation Mustapha, Hussein Finite Elements in Analysis and Design, Vol. 47, Issue 4 https://doi.org/10.1016/j.finel.2010.12.003	journal	April 2011
A Graph-theoretic Pipe Network Method for water flow simulation in a porous medium: GPNM Xu, Z. H.; Ma, G. W.; Li, S. C. International Journal of Heat and Fluid Flow, Vol. 45 https://doi.org/10.1016/j.ijheatfluidflow.2013.11.003	journal	February 2014
Estimating the hydraulic properties of the fracture network in a sandstone aquifer Hitchmough, Anna M.; Riley, Michael S.; Herbert, Alan W. Journal of Contaminant Hydrology, Vol. 93, Issue 1-4 https://doi.org/10.1016/j.jconhyd.2007.01.012	journal	August 2007
Robust streamline tracing for the simulation of porous media flow on general triangular and quadrilateral grids Matringe, Sébastien F.; Juanes, Ruben; Tchelepi, Hamdi A. Journal of Computational Physics, Vol. 219, Issue 2 https://doi.org/10.1016/j.jcp.2006.07.004	journal	December 2006
A hierarchical fracture model for the iterative multiscale finite volume method Hajibeygi, Hadi; Karvounis, Dimitris; Jenny, Patrick Journal of Computational Physics, Vol. 230, Issue 24 https://doi.org/10.1016/j.jcp.2011.08.021	journal	October 2011
Three-dimensional control-volume distributed multi-point flux approximation coupled with a lower-dimensional surface fracture model Ahmed, Raheel; Edwards, Michael G.; Lamine, Sadok Journal of Computational Physics, Vol. 303 https://doi.org/10.1016/j.jcp.2015.10.001	journal	December 2015
Solution of two key issues in arbitrary three-dimensional discrete fracture network flow models Zhang, Qi-Hua; Yin, Jian-Min Journal of Hydrology, Vol. 514 https://doi.org/10.1016/j.jhydrol.2014.04.027	journal	June 2014
Finite element generation of arbitrary 3-D fracture networks for flow analysis in complicated discrete fracture networks Zhang, Qi-Hua Journal of Hydrology, Vol. 529 https://doi.org/10.1016/j.jhydrol.2015.08.065	journal	October 2015
A semi-analytical model for the flow behavior of naturally fractured formations with multi-scale fracture networks Jia, Pin; Cheng, Linsong; Huang, Shijun Journal of Hydrology, Vol. 537 https://doi.org/10.1016/j.jhydrol.2016.03.022	journal	June 2016
Fractured porous medium flow analysis using numerical manifold method with independent covers Zhang, Qi-Hua; Lin, Shao-Zhong; Xie, Zhi-Qiang Journal of Hydrology, Vol. 542 https://doi.org/10.1016/j.jhydrol.2016.09.054	journal	November 2016
Unified pipe network method for simulation of water flow in fractured porous rock Ren, Feng; Ma, Guowei; Wang, Yang Journal of Hydrology, Vol. 547 https://doi.org/10.1016/j.jhydrol.2017.01.044	journal	April 2017
A domain decomposed finite element method for solving Darcian velocity in heterogeneous porous media Xie, Yifan; Wu, Jichun; Xue, Yuqun Journal of Hydrology, Vol. 554 https://doi.org/10.1016/j.jhydrol.2017.08.040	journal	November 2017
Numerical Manifold Method Modeling of Coupled Processes in Fractured Geological Media at Multiple Scales Hu, Mengsu; Rutqvist, Jonny Journal of Rock Mechanics and Geotechnical Engineering https://doi.org/10.1016/j.jrmge.2020.03.002	journal	May 2020
Three-dimensional numerical simulation of single-phase transient compressible flows and well-tests in fractured formations Mourzenko, V. V.; Bogdanov, I. I.; Thovert, J. -F. Mathematics and Computers in Simulation, Vol. 81, Issue 10 https://doi.org/10.1016/j.matcom.2010.12.014	journal	June 2011
An efficient embedded discrete fracture model based on mimetic finite difference method Yan, Xia; Huang, Zhaoqin; Yao, Jun Journal of Petroleum Science and Engineering, Vol. 145 https://doi.org/10.1016/j.petrol.2016.03.013	journal	September 2016
A Graph-theoretic Pipe Network Method for water flow simulation in discrete fracture networks: GPNM Li, S. C.; Xu, Z. H.; Ma, G. W. Tunnelling and Underground Space Technology, Vol. 42 https://doi.org/10.1016/j.tust.2014.03.012	journal	May 2014
Accurate calculation of specific discharge in heterogeneous porous media Zhou, Quanlin; Bensabat, Jacob; Bear, Jacob Water Resources Research, Vol. 37, Issue 12 https://doi.org/10.1029/1998WR900105	journal	December 2001
Connectivity of joint networks with power law length distributions Renshaw, Carl E. Water Resources Research, Vol. 35, Issue 9 https://doi.org/10.1029/1999wr900170	journal	September 1999
Hierarchical modeling of flow in naturally fractured formations with multiple length scales Lee, S. H.; Lough, M. F.; Jensen, C. L. Water Resources Research, Vol. 37, Issue 3 https://doi.org/10.1029/2000wr900340	journal	March 2001
Effective permeability of fractured porous media in steady state flow: PERMEABILITY OF FRACTURED POROUS MEDIA Bogdanov, I. I.; Mourzenko, V. V.; Thovert, J. -F. Water Resources Research, Vol. 39, Issue 1 https://doi.org/10.1029/2001wr000756	journal	January 2003
Transport of conservative solutes in simulated fracture networks: 1. Synthetic data generation: TRANSPORT IN SIMULATED FRACTURE NETWORKS, 1 Reeves, Donald M.; Benson, David A.; Meerschaert, Mark M. Water Resources Research, Vol. 44, Issue 5 https://doi.org/10.1029/2007wr006069	journal	May 2008
The Laplace Transform Galerkin Technique for large-scale simulation of mass transport in discretely fractured porous formations Sudicky, E. A.; McLaren, R. G. Water Resources Research, Vol. 28, Issue 2 https://doi.org/10.1029/91wr02560	journal	February 1992
Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities Durlofsky, Louis J. Water Resources Research, Vol. 30, Issue 4 https://doi.org/10.1029/94wr00061	journal	April 1994
On the computation of Darcian velocity and mass balance in the finite element modeling of groundwater flow Yeh, Gour-Tsyh Water Resources Research, Vol. 17, Issue 5 https://doi.org/10.1029/wr017i005p01529	journal	October 1981
Porous media equivalents for networks of discontinuous fractures Long, J. C. S.; Remer, J. S.; Wilson, C. R. Water Resources Research, Vol. 18, Issue 3 https://doi.org/10.1029/wr018i003p00645	journal	June 1982
A model to evaluate the transient hydraulic response of three-dimensional sparsely fractured rock masses Elsworth, Derek Water Resources Research, Vol. 22, Issue 13 https://doi.org/10.1029/wr022i013p01809	journal	December 1986
Channel model of flow through fractured media Tsang, Y. W.; Tsang, C. F. Water Resources Research, Vol. 23, Issue 3 https://doi.org/10.1029/wr023i003p00467	journal	March 1987
Conditional simulations of fluid flow in three-dimensional networks of discrete fractures Andersson, Johan; Dverstorp, Björn Water Resources Research, Vol. 23, Issue 10 https://doi.org/10.1029/wr023i010p01876	journal	October 1987
Modeling fracture flow with a stochastic discrete fracture network: calibration and validation: 1. The flow model Cacas, M. C.; Ledoux, E.; de Marsily, G. Water Resources Research, Vol. 26, Issue 3 https://doi.org/10.1029/wr026i003p00479	journal	March 1990
A mixed finite element method for Darcy flow in fractured porous media with non-matching grids D’Angelo, Carlo; Scotti, Anna ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 46, Issue 2 https://doi.org/10.1051/m2an/2011148	journal	December 2011
Modelling dc responses of 3-D complex fracture networks Beskardes, G. D.; Weiss, C. J. Geophysical Journal International, Vol. 214, Issue 3 https://doi.org/10.1093/gji/ggy234	journal	June 2018
Permeability of three-dimensional fracture networks Koudina, N.; Gonzalez Garcia, R.; Thovert, J. -F. Physical Review E, Vol. 57, Issue 4 https://doi.org/10.1103/physreve.57.4466	journal	April 1998
Percolation parameter and percolation-threshold estimates for three-dimensional random ellipses with widely scattered distributions of eccentricity and size de Dreuzy, J. -R.; Davy, P.; Bour, O. Physical Review E, Vol. 62, Issue 5 https://doi.org/10.1103/physreve.62.5948	journal	November 2000
Geometry, percolation and transport properties of fracture networks derived from line data Sisavath, S.; Mourzenko, V.; Genthon, P. Geophysical Journal International, Vol. 157, Issue 2 https://doi.org/10.1111/j.1365-246x.2004.02185.x	journal	May 2004
Hybrid finite element?finite volume discretization of complex geologic structures and a new simulation workflow demonstrated on fractured rocks Paluszny, A.; MatthÄI, S. K.; Hohmeyer, M. Geofluids, Vol. 7, Issue 2 https://doi.org/10.1111/j.1468-8123.2007.00180.x	journal	May 2007
Flow Simulation in Three-Dimensional Discrete Fracture Networks Erhel, Jocelyne; de Dreuzy, Jean-Raynald; Poirriez, Baptiste SIAM Journal on Scientific Computing, Vol. 31, Issue 4 https://doi.org/10.1137/080729244	journal	January 2009
An Improved Method for Numerical Inversion of Laplace Transforms de Hoog, F. R.; Knight, J. H.; Stokes, A. N. SIAM Journal on Scientific and Statistical Computing, Vol. 3, Issue 3 https://doi.org/10.1137/0903022	journal	September 1982
An Implementation of the QMR Method Based on Coupled Two-Term Recurrences Freund, Roland W.; Nachtigal, Noël M. SIAM Journal on Scientific Computing, Vol. 15, Issue 2 https://doi.org/10.1137/0915022	journal	March 1994
Conforming Delaunay Triangulation of Stochastically Generated Three Dimensional Discrete Fracture Networks: A Feature Rejection Algorithm for Meshing Strategy Hyman, Jeffrey D.; Gable, Carl W.; Painter, Scott L. SIAM Journal on Scientific Computing, Vol. 36, Issue 4 https://doi.org/10.1137/130942541	journal	January 2014
Modeling Fractures and Barriers as Interfaces for Flow in Porous Media Martin, Vincent; Jaffré, Jérôme; Roberts, Jean E. SIAM Journal on Scientific Computing, Vol. 26, Issue 5 https://doi.org/10.1137/s1064827503429363	journal	January 2005
Basic Principles of Virtual Element Methods BeirÃO Da Veiga, L.; Brezzi, F.; Cangiani, A. Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 01 https://doi.org/10.1142/s0218202512500492	journal	November 2012
Advances in the study of naturally fractured hydrocarbon reservoirs: a broad integrated interdisciplinary applied topic Spence, Guy H.; Couples, Gary D.; Bevan, Tim G. Geological Society, London, Special Publications, Vol. 374, Issue 1 https://doi.org/10.1144/sp374.19	journal	January 2014
Predictability of properties of a fractured geothermal reservoir: the opportunities and limitations of an outcrop analogue study Bauer, Johanna F.; Krumbholz, Michael; Meier, Silke Geothermal Energy, Vol. 5, Issue 1 https://doi.org/10.1186/s40517-017-0081-0	journal	November 2017
Electromagnetic induction in a fully 3‐D anisotropic earth Weiss, Chester J.; Newman, Gregory A. GEOPHYSICS, Vol. 67, Issue 4 https://doi.org/10.1190/1.1500371	journal	July 2002
Finite-element analysis for model parameters distributed on a hierarchy of geometric simplices Weiss, Chester J. GEOPHYSICS, Vol. 82, Issue 4 https://doi.org/10.1190/geo2017-0058.1	journal	May 2017
A parametric analysis of oilfield design factors affecting the detectability and characterization of electrically conductive hydrofracks Didem Beskardes, G.; Weiss, Chester J. GEOPHYSICS, Vol. 85, Issue 4 https://doi.org/10.1190/geo2019-0297.1	journal	June 2020
A Multicomponent Isothermal System for Efficient Reservoir Simulation Blaskovich, F. T.; Cain, G. M.; Sonier, Fernand Middle East Oil Technical Conference and Exhibition, All Days https://doi.org/10.2118/11480-MS	conference	March 1983
A New Approach for Simulating Complex Fractured Reservoirs Hill, A. C.; Thomas, G. W. Middle East Oil Technical Conference and Exhibition https://doi.org/10.2118/13537-MS	conference	April 2013
A Multiple-Porosity Method for Simulation of Naturally Fractured Petroleum Reservoirs Wu, Yu-Shu; Pruess, Karsten SPE Reservoir Engineering, Vol. 3, Issue 01 https://doi.org/10.2118/15129-PA	journal	February 1988
Development of an Efficient Embedded Discrete Fracture Model for 3D Compositional Reservoir Simulation in Fractured Reservoirs Moinfar, Ali; Varavei, Abdoljalil; Sepehrnoori, Kamy SPE Journal, Vol. 19, Issue 02 https://doi.org/10.2118/154246-PA	journal	April 2014
The Behavior of Naturally Fractured Reservoirs Warren, J. E.; Root, P. J. Society of Petroleum Engineers Journal, Vol. 3, Issue 03 https://doi.org/10.2118/426-pa	journal	September 1963
An Efficient Discrete-Fracture Model Applicable for General-Purpose Reservoir Simulators Karimi-Fard, M.; Durlofsky, L. J.; Aziz, K. SPE Journal, Vol. 9, Issue 02 https://doi.org/10.2118/88812-pa	journal	June 2004
Compositional Modeling by the Combined Discontinuous Galerkin and Mixed Methods Hoteit, Hussein; Firoozabadi, Abbas SPE Journal, Vol. 11, Issue 01 https://doi.org/10.2118/90276-pa	journal	March 2006
CUBIT Mesh Generation Environment. Volume 1: Users Manual Blacker, Ted D.; Bohnhoff, William J.; Edwards, Tony L. https://doi.org/10.2172/10176386	report	May 1994
Hierarchal finite elements and their application to structural natural vibration problems Zhu, D. -C. 23rd Structures, Structural Dynamics and Materials Conference https://doi.org/10.2514/6.1982-706	conference	May 1982
Dual Virtual Element Methods for Discrete Fracture Matrix models Fumagalli, Alessio; Keilegavlen, Eirik Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles, Vol. 74 https://doi.org/10.2516/ogst/2019008	journal	January 2019
Identification of Fracture Properties in Shale Oil Reservoirs by a Well Testing Model: A Case Study Jun, Pu; Qin, Xuejie; Chen, Zhiming Day 1 Mon, November 02, 2020 https://doi.org/10.4043/30236-MS	conference	October 2020
A Two-Phase Flow Simulation of Discrete-Fractured Media using Mimetic Finite Difference Method Huang, Zhaoqin; Yan, Xia; Yao, Jun Communications in Computational Physics, Vol. 16, Issue 3 https://doi.org/10.4208/cicp.050413.170314a	journal	September 2014
Laplace-Transform Finite Element Solution of Nonlocal and Localized Stochastic Moment Equations of Transport Morales-Casique, Eric; Neuman, Shlomo P. Communications in Computational Physics https://doi.org/10.4208/cicp.2009.v6.p131	journal	January 2009

Similar Records

The Laplace transform Galerkin technique for large-scale simulation of mass transport in discretely fractured porous formations

Journal Article · Fri Jan 31 23:00:00 EST 1992 · Water Resources Research; (United States) · OSTI ID:5150131

Hydro-mechanical coupled modeling of hydraulic fracturing using the mesh fragmentation technique

Journal Article · Fri May 08 00:00:00 EDT 2020 · Computers and Geotechnics · OSTI ID:1801294

A Combined Experimental and Computational Approach for the Design of Mold Topography that Leads to Desired Ingot Surface and Microstructure in Aluminum Casting.

Journal Article · Thu May 27 00:00:00 EDT 2004 · International Journal for Numerical Methods in Engineering · OSTI ID:850515

Finite Element Modeling of Diffusion in Fractured Porous Media by Using Hierarchical Material Properties

Citation Formats

References (78)

Similar Records

Related Subjects