The IV AMMCS International Conference
Waterloo, Ontario, Canada | August 20-25, 2017
AMMSCS 2017 Plenary Talk
Adaptive Enriched Galerkin Methods for Miscible Displacement in Porous Media
Mary Wheeler, Sanghyun Lee, Young-Ju Lee (University of Texas at Austin)
Miscible displacement of one fluid by another in a porous medium
has attracted considerable attention in subsurface modeling with
emphasis on enhanced oil recovery applications. Here flow
instabilities arising when a fluid with higher mobility
displaces another fluid with lower mobility is referred to as
viscous fingering. The latter has been the topic of major
physical and mathematical studies for over half a
century. Recently, viscous fingering has been applied for
proppant-filled hydraulic fracture propagation to efficiently
transport the proppant to the tip of fractures. The governing
mathematical system that represents the displacement of the
fluid mixtures consists of pressure, velocity, and
concentration.
Here we present a novel approach to the simulation of miscible displacement by employing an adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. EG provides locally and globally conservative fluxes, which is crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. We have shown theoretically and computationally that a robust preconditioner can be achieved if one adds pre- and post smoothings to a block preconditioner involving CG and jumps in the discontinuous piecewise constants. Dynamic adaptive mesh refinement is applied in treating geological material discontinuities.
An additional advantage of EG is that only those subdomains that require local conservation need to be enriched with a treatment of high order non-matching grids. Our high order EG transport system is coupled with an entropy viscosity residual stabilization method introduced in to avoid spurious oscillations near shocks. Instead of using limiters and non-oscillatory reconstructions, this method employs the local residual of an entropy equation to construct the numerical diffusion, which is added as a nonlinear dissipation to the numerical discretization of the system. The amount of numerical diffusion added is proportional to the computed entropy residual. This technique is independent of mesh and order of approximation and has been shown to be efficient and stable in solving many physical problems with CG. Finally we note that it is crucial to have dynamic mesh adaptivity in order to reduce computational costs for large-scale three dimensional applications; both for flow and transport. We employ the entropy residual for dynamic adaptive mesh refinement to capture the moving interface between the miscible fluids. It Our computational results indicate that the entropy residual can be used as a efficient posteriori error indicator.
Here we present a novel approach to the simulation of miscible displacement by employing an adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. EG provides locally and globally conservative fluxes, which is crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. We have shown theoretically and computationally that a robust preconditioner can be achieved if one adds pre- and post smoothings to a block preconditioner involving CG and jumps in the discontinuous piecewise constants. Dynamic adaptive mesh refinement is applied in treating geological material discontinuities.
An additional advantage of EG is that only those subdomains that require local conservation need to be enriched with a treatment of high order non-matching grids. Our high order EG transport system is coupled with an entropy viscosity residual stabilization method introduced in to avoid spurious oscillations near shocks. Instead of using limiters and non-oscillatory reconstructions, this method employs the local residual of an entropy equation to construct the numerical diffusion, which is added as a nonlinear dissipation to the numerical discretization of the system. The amount of numerical diffusion added is proportional to the computed entropy residual. This technique is independent of mesh and order of approximation and has been shown to be efficient and stable in solving many physical problems with CG. Finally we note that it is crucial to have dynamic mesh adaptivity in order to reduce computational costs for large-scale three dimensional applications; both for flow and transport. We employ the entropy residual for dynamic adaptive mesh refinement to capture the moving interface between the miscible fluids. It Our computational results indicate that the entropy residual can be used as a efficient posteriori error indicator.
Mary Fanett Wheeler is well-known researcher in computational science. She has been a member of the
faculty at The University of Texas at Austin since 1995 and holds the Ernest and Virginia Cockrell Chair in the departments of Aerospace Engineering and Engineering Mechanics, and Petroleum and Geosystems Engineering. She is also director of the Center for Subsurface Modeling (CSM) at the Institute for Computational Engineering and Sciences (ICES).
Before joining the faculty at UT Austin, Dr. Wheeler was the Noah Harding Professor in engineering at Rice University in Houston.
Dr. Wheeler’s research interests involve 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 porous media, contaminant transport in groundwater, and sequestration of carbon in geological formations. Dr. Wheeler has published more than 300 technical papers and edited seven books; she is currently an editor of five technical journals.
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 SIAM Activity Group in the Geosciences, and with Dr. Hans van Duijn, started the Journal on Computational Geosciences.
In 1998, Dr. Wheeler was elected to the National Academy of Engineering. In 2006, she received an honorary doctorate from Technische Universiteit 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. 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, most notably in geosciences. 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 contaminant transport, 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 named an SPE honorary member, the organization’s highest honor.
Dr. Wheeler’s research interests involve 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 porous media, contaminant transport in groundwater, and sequestration of carbon in geological formations. Dr. Wheeler has published more than 300 technical papers and edited seven books; she is currently an editor of five technical journals.
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 SIAM Activity Group in the Geosciences, and with Dr. Hans van Duijn, started the Journal on Computational Geosciences.
In 1998, Dr. Wheeler was elected to the National Academy of Engineering. In 2006, she received an honorary doctorate from Technische Universiteit 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. 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, most notably in geosciences. 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 contaminant transport, 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 named an SPE honorary member, the organization’s highest honor.