Modelling of hydro-mechanical processes in heterogeneous fracture intersections using a fictitious domain method with variational transfer operators

Fluid flow in rough fractures and the coupling with the mechanical behaviour of the fractures pose great difficulties for numerical modeling approaches due to complex fracture surface topographies, the non-linearity of hydro-mechanical processes and their tightly coupled nature. To this end, we have adapted a fictitious domain method to enable the simulation of hydro-mechanical processes in fracture intersections. The main characteristic of the method is the immersion of the fracture, modelled as a linear elastic solid, in the surrounding computational fluid domain, modelled with the incompressible Navier-Stokes equations. The fluid and the solid problems are coupled with variational transfer operators. Variational transfer operators are also used to solve contact within the fracture using a dual mortar approach and to generate problem-specific fluid meshes. With respect to our applications, the key features of the method are the usage of different finite element discretizations for the solid and the fluid problem and the automatically generated representation of the fluid-solid boundary. We demonstrate that the presented methodology resolves small-scale roughness on the fracture surface, while capturing fluid flow field changes during mechanical loading. Starting with 2D/3D benchmark simulations of intersected fractures, we end with an intersected fracture composed of complex fracture surface topographies, which are in contact under increasing loads. The contributions of this article are as follows: (1) the application of the fictitious domain method to study flow in fractures with intersections, (2) a mortar-based contact solver for the solid problem, (3) generation of problem-specific grids using the geometry information from the variational transfer operators.

[1]  M. G. C. Nestola,et al.  Three-band decomposition analysis in multiscale FSI models of abdominal aortic aneurysms , 2016 .

[2]  Laurence C. Hull,et al.  Streamline routing through fracture junctions , 1986 .

[3]  Stephen R. Brown,et al.  Fluid flow through rock joints: The effect of surface roughness , 1987 .

[4]  Gordon R. J. Cooper.  Forward modelling of magnetic data , 1997 .

[5]  C. Zangerla,et al.  Normal stiffness of fractures in granitic rock : A compilation of laboratory and in-situ experiments , 2008 .

[6]  Peter Bayer,et al.  Permeability Evolution in Natural Fractures Subject to Cyclic Loading and Gouge Formation , 2016, Rock Mechanics and Rock Engineering.

[7]  Y. Tsang,et al.  The Effect of Tortuosity on Fluid Flow Through a Single Fracture , 1984 .

[8]  Brian Berkowitz,et al.  Mass transfer at fracture intersections: An evaluation of mixing models , 1994 .

[9]  Paolo Cignoni,et al.  MeshLab: an Open-Source Mesh Processing Tool , 2008, Eurographics Italian Chapter Conference.

[10]  Rolf Krause,et al.  3D non-conforming mesh model for flow in fractured porous media using Lagrange multipliers , 2019, Comput. Geosci..

[11]  Pierre M. Adler,et al.  Solute transport at fracture intersections , 2002 .

[12]  Brian Berkowitz,et al.  Effects of junction transfer characteristics on transport in fracture networks , 2001 .

[13]  Georg Kosakowski,et al.  Flow pattern variability in natural fracture intersections , 1999 .

[14]  Dominik Obrist,et al.  An immersed boundary method for fluid-structure interaction based on variational transfer , 2019, J. Comput. Phys..

[15]  Yujing Jiang,et al.  Estimating the relation between surface roughness and mechanical properties of rock joints , 2006 .

[16]  Rolf Krause,et al.  Variational Parallel Information Transfer between Unstructured Grids in Geophysics-Applications and Solutions Methods , 2018 .

[17]  M. A. Hyman,et al.  Non-iterative numerical solution of boundary-value problems , 1952 .

[18]  Chad W. Higgins,et al.  Albedo effect on radiative errors in air temperature measurements , 2009 .

[19]  Benjamin S. Kirk,et al.  Library for Parallel Adaptive Mesh Refinement / Coarsening Simulations , 2006 .

[20]  Rolf Krause,et al.  Simulation of hydro-mechanically coupled processes in rough rock fractures using an immersed boundary method and variational transfer operators , 2018, Computational Geosciences.

[21]  D. Tarboton,et al.  A sediment transport model for incision of gullies on steep topography , 2003 .

[22]  Philipp Blum,et al.  Numerical Simulations and Validation of Contact Mechanics in a Granodiorite Fracture , 2018, Rock Mechanics and Rock Engineering.

[23]  M. Saar,et al.  The seismo-hydromechanical behavior during deep geothermal reservoir stimulations: open questions tackled in a decameter-scale in situ stimulation experiment , 2017 .

[24]  Chin-Fu Tsang,et al.  Channel model of flow through fractured media , 1987 .

[25]  Géraldine Pichot,et al.  Influence of fracture scale heterogeneity on the flow properties of three-dimensional discrete fracture networks (DFN) , 2012 .

[26]  R. Glowinski,et al.  A distributed Lagrange multiplier/fictitious domain method for particulate flows , 1999 .

[27]  Daniel Vogler,et al.  Hydro-mechanically coupled processes in heterogeneous fractures: experiments and numerical simulations , 2016 .

[28]  Stephen R. Brown,et al.  Fluid flow and mixing in rough-walled fracture intersections , 2006 .

[29]  Roland N. Horne,et al.  Correlations between formation properties and induced seismicity during high pressure injection into granitic rock , 2014 .

[30]  Roland N. Horne,et al.  An investigation of stimulation mechanisms in Enhanced Geothermal Systems , 2014 .

[31]  Georg Kosakowski,et al.  Transport behavior in three‐dimensional fracture intersections , 2003 .

[32]  Paul W. J. Glover,et al.  Synthetic rough fractures in rocks , 1998 .

[33]  Rolf Krause,et al.  A Parallel Approach to the Variational Transfer of Discrete Fields between Arbitrarily Distributed Unstructured Finite Element Meshes , 2016, SIAM J. Sci. Comput..

[34]  Peter Betsch,et al.  A mortar approach for Fluid–Structure interaction problems: Immersed strategies for deformable and rigid bodies , 2014 .

[35]  Gudmundur S. Bodvarsson,et al.  Lubrication theory analysis of the permeability of rough-walled fractures , 1991 .

[36]  Thomas Richter,et al.  Fluid-structure Interactions , 2017 .

[37]  Derek Gaston,et al.  MOOSE: A parallel computational framework for coupled systems of nonlinear equations , 2009 .

[38]  Zhaosheng Yu.  A DLM/FD method for fluid/flexible-body interactions , 2005 .

[39]  Nobuo Hirano,et al.  Determination of aperture structure and fluid flow in a rock fracture by high‐resolution numerical modeling on the basis of a flow‐through experiment under confining pressure , 2008 .

[40]  Laura J. Pyrak-Nolte,et al.  Single fractures under normal stress: The relation between fracture specific stiffness and fluid flow , 2000 .

[41]  Anders Rasmuson,et al.  Radionuclide Transport in Fast Channels in Crystalline Rock , 1986 .

[42]  A. Quarteroni,et al.  Computational comparison of aortic root stresses in presence of stentless and stented aortic valve bio-prostheses , 2017, Computer methods in biomechanics and biomedical engineering.

[43]  Stephen R. Brown,et al.  Experimental mixing variability in intersecting natural fractures , 2001 .

[44]  Jonny Rutqvist,et al.  The role of hydromechanical coupling in fractured rock engineering , 2003 .

[45]  Noriaki Watanabe,et al.  Direct measurement of contact area and stress dependence of anisotropic flow through rock fracture with heterogeneous aperture distribution , 2009 .

[46]  E. D. Skouras,et al.  Mixing within fracture intersections during colloidal suspension flow , 2009 .

[47]  P. Bayer,et al.  Experiments and Simulations of Fully Hydro‐Mechanically Coupled Response of Rough Fractures Exposed to High‐Pressure Fluid Injection , 2018 .

[48]  G. Hou,et al.  Numerical Methods for Fluid-Structure Interaction — A Review , 2012 .

[49]  D. S. Sayres,et al.  Process-evaluation of tropospheric humidity simulated by general circulation models using water vapor isotopologues: 1. Comparison between models and observations , 2012 .

[50]  Sojwal Manoorkar,et al.  Particle transport in laboratory models of bifurcating fractures , 2016 .

[51]  Barbara I. Wohlmuth,et al.  A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier , 2000, SIAM J. Numer. Anal..

[52]  Stuart D. C. Walsh,et al.  Thermal Drawdown-Induced Flow Channeling in Fractured Geothermal Reservoirs , 2016, Rock Mechanics and Rock Engineering.

[53]  Vladimir Cvetkovic,et al.  Modeling of flow and mixing in 3D rough-walled rock fracture intersections , 2017 .

[54]  Matthew G. Knepley,et al.  PETSc Users Manual: Revision 3.11 , 2019 .

[55]  Rolf Krause,et al.  Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization , 2009 .

[56]  Martin P. Seybold,et al.  Estimating fluid flow rates through fracture networks using combinatorial optimization , 2018, Advances in Water Resources.

[57]  Rolf Krause,et al.  Presentation and comparison of selected algorithms for dynamic contact based on the Newmark scheme , 2012 .

[58]  Anozie Ebigbo,et al.  Inclusion-Based Effective Medium Models for the Permeability of a 3D Fractured Rock Mass , 2015, Transport in Porous Media.

[59]  Giovanni Grasselli,et al.  Characterization of the effect of normal load on the discontinuity morphology in direct shear specimens using X-ray micro-CT , 2015 .

[60]  John W. Peterson,et al.  Overview of the incompressible Navier-Stokes simulation capabilities in the MOOSE framework , 2017, Adv. Eng. Softw..

[61]  M. Saar,et al.  Solution of contact problems between rough body surfaces with non matching meshes using a parallel mortar method , 2018, 1811.02914.

[62]  N. Barton,et al.  FUNDAMENTALS OF ROCK JOINT DEFORMATION , 1983 .

[63]  John L. Wilson,et al.  A lattice‐gas and lattice Boltzmann study of mixing at continuous fracture Junctions: Importance of boundary conditions , 1997 .

[64]  Erik Eberhardt,et al.  Normal stiffness of fractures in granitic rock: A compilation of laboratory and in-situ experiments , 2008 .

[65]  S. Walsh,et al.  Comparison of Surface Properties in Natural and Artificially Generated Fractures in a Crystalline Rock , 2017, Rock Mechanics and Rock Engineering.

[66]  William J. Collins,et al.  Tropospheric ozone and El Niño–Southern Oscillation: Influence of atmospheric dynamics, biomass burning emissions, and future climate change , 2006 .

[67]  N. Barton,et al.  Strength, deformation and conductivity coupling of rock joints , 1985 .

[68]  Rolf Krause,et al.  Monotone Multigrid Methods on Nonmatching Grids for Nonlinear Multibody Contact Problems , 2003, SIAM J. Sci. Comput..

[69]  Kiyotoshi Sakaguchi,et al.  Estimation of closure of a fracture under normal stress based on aperture data , 2008 .

[70]  L. Heltai,et al.  A finite element approach to the immersed boundary method , 2003 .

[71]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[72]  Gudmundur S. Bodvarsson,et al.  Hydraulic conductivity of rock fractures , 1996 .