Prof. Mary F. Wheeler
Departments of Aerospace and Engineering Mechanics and
Petroleum and Geosystems Engineering
The University of Texas at Austin

Modeling Thermo-Hydro-Mechanical Processes Using Enriched Galerkin and Phase Field

ABSTRACT: A recent advancement in geothermal energy extraction is the Enhanced Geothermal System (EGS). Here a geothermal reservoir in hot, low permeability rock is created by injecting high pressure fluids to fracture the rock to allowing water to circulate and absorb heat. This process creates a heat exchange where hot water is pumped to the surface to generate steam which then turns a turbine to produce electricity. This technology expands 24/7 power generation. EGS reservoirs are dynamic systems that change in response to control pressures. Challenges include availability of water, installing fiber optic cables downhole in geothermal wells, gathering and analyzing real time data on temperature and performance. In addition, temperature sensitive issues, such as use of proppants, solubilities and viscosities, early thermal breakthroughs, and well spacing and layout need to be addressed. Thus, there is the need to model the coupled interactions between temperature, fluid flow, and mechanical deformation, which occur in a wide range of these natural and engineered systems.

To optimize geothermal energy production from EGS reservoirs, it is crucial to comprehend the coupled thermo-hydro-mechanical (THM) processes that are fundamental to various EGS systems; One widely employed mathematical model for describing THM processes is based on Biot’s non-isothermal consolidation theory, known as the thermo-poroelasticity model. This model extends Biot’s poroelasticity model, which characterizes the interaction between a deformable porous medium and fluid flow within it under isothermal conditions. The governing system of partial differential equations (PDEs) of the thermo-poroelasticity model consists of a system of coupled heat transfer, mass conservation, and momentum conservation. These equations are fully coupled through linear and nonlinear coupling terms.

In this presentation, we introduce a novel diffraction based THM model for fracture propagation using a phase-field fracture (PFF) approach. The key innovation of the THM-PFF model lies in its integrated treatment of four solution variables—displacements, phase-field, pressure, and temperature—each governed by a combination of conservation of momentum (mechanics problem), a variational inequality (constrained minimization problem), mass conservation (pressure problem), and energy conservation (temperature problem). This leads to a new formulation of a coupled variational inequality system.

A major advancement is the development of an extended fixed-stress algorithm, where displacements, phase-field, pressures, and temperatures are solved in a staggered sequence. An important aspect of this work is the global coupling of pressures and temperatures across the domain using diffraction systems, with diffraction coefficients defined by material parameters weighted by the diffusive phase-field variable. To ensure robust local mass conservation, we employ enriched Galerkin finite elements (EG) for both pressure and temperature diffraction equations. By enriching the continuous Galerkin basis functions with discontinuous piecewise constants, EG accurately represents solution and parameter discontinuities while preserving local mass and energy conservation—crucial aspects for THM problems and realistic behavior. Moreover, the use of a predictor–corrector local mesh adaptivity scheme is employed, allowing the model to handle small phase-field length-scale parameters while maintaining high numerical accuracy and reasonable computational cost. Computational examples of these algorithmic developments will be presented.

BIOGRAPHY: Dr. Mary Fanett Wheeler is an expert in computational science. She has been a member of the faculty at The University of Texas at Austin (UT Austin) since 1995 and held the Ernest and Virginia Cockrell Chair in the departments of Aerospace Engineering and Engineering Mechanics, and Petroleum and Geosystems Engineering. She also served as director of the Center for Subsurface Modeling at the Oden Institute for Computational Engineering and Sciences. Before joining the faculty at UT Austin, Dr. Wheeler was the Noah Harding Professor in engineering at Rice University in Houston, Texas, and was the first tenured female associate and full professor in engineering at Rice University. In August 2024, she became Professor Emeritus at UT Austin. Dr. Wheeler’s research has focused on computer simulations to model the behavior of fluids in geological formations. Her particular research interests include numerical solution of partial differential systems with application to the modeling of subsurface flows and parallel computation. Applications of her research include multiphase flow and geomechanics in fractured reservoirs, contaminant transport in groundwater, sequestration of carbon in geological formations. Dr. Wheeler has published more than 450 technical papers and edited 7 books; she is currently an editor of five technical journals. It should be noted that Dr. Wheeler co-authored the first papers on modeling flow and transport in porous media using discontinuous Galerkin (DG) and/or mixed finite element methods, as well as co-authored two papers (one with Tom Russell and one with Alan Weiser) demonstrating the first proofs on convergence of cell-centered finite differences on nonuniform mesh–standard approach employed in reservoir simulation. Dr. Wheeler is a member of the Society of Industrial and Applied Mathematics and the Society of Petroleum Engineers. She is a Fellow of the International Association for Computational Mechanics and is a certified Professional Engineer in the State of Texas. She was co-organizer of the Society for Industrial and Applied Mathematics (SIAM) Activity Group in the Geosciences, and alongside Dr. Hans van Duijn, started the journal Computational Geosciences. Dr. Wheeler has served on numerous National Science Foundation, Department of Energy, and Department of Defense committees. For 7 years, she was the university lead in the Department of Defense’s User Productivity Enhancement and Technology Transfer Program in environmental quality modeling. Dr. Wheeler has served on the Board of Governors for Argonne National Laboratory and advisory committees for Argonne, Oak Ridge and Pacific National Laboratories. In 1998, Dr. Wheeler was elected to the National Academy of Engineering and in 2000 was selected to be Distinguished Alumna at Rice University. In 2006, she received an honorary doctorate from University of Technology, Eindhoven, in the Netherlands. In 2008, she received an honorary doctorate from the Colorado School of Mines. In 2009, Dr. Wheeler was honored with the SIAM Geosciences Career Prize, as well as her third IBM Faculty Award. That same year, she was awarded the Theodore von Kármán prize at the SIAM national meeting, recognizing her seminal research in numerical methods for partial differential equations, her leadership in the field of scientific computation and service to the scientific community, and for her pioneering work in the application of computational methods to the engineering sciences. In 2010, she was elected to the American Academy of Arts and Sciences. In 2011, she received a Humboldt award. In February 2013, Dr. Wheeler was selected to receive the Lifetime Achievement Award of the International Society for Porous Media, InterPore. The award is given in recognition of her achievements in the area of subsurface flow and in recognition of her great contribution in increasing the visibility, credibility and prestige of porous media research. In May 2013, Dr. Wheeler received the John von Neumann Medal award from the Unites States Association for Computational Mechanics (USACM). It is the highest award given by USACM to honor individuals who have made outstanding, sustained contributions in the field of computational mechanics over substantial portions of their professional careers. In 2014 she was awarded Honorary Member by the Society of Petroleum Engineers, the highest award bestowed by this society. In 2017 she was the recipient of the University of Texas Book Award and in 2020 the Billy & Claude R. Hocott Distinguished Centennial Engineering Research Award.