Jonathan J.L. Higdon

The geologic evolution of meandering rivers is a dynamic process controlled by geophysical fluid dynamics for three dimensional fluid flow in channels with erodible beds and channel walls.  The fluid flow equations combined with models for erosion and sediment transport provide a complex moving boundary problem wherein the bottom topology and bank geometry influence the fluid flow field which in turn affects the soil deposition and emerging channel geometry.  The goal of this study is to determine the instantaneous fluid flow field and to model the sediment transport and deposition over geological time scales.  The overall goal is to predict the stratigraphic profile of the emerging landforms and eventual rock formation over the long term geologic evolution.

Computational simulations of the depositional process for geologic landforms plays an important role in our understanding of the complex heterogeneities of geologic formations and its impact on natural and human driven processes affecting our environment.  In recent years, there has been increasing interest in using computer simulation to model the depositional process, combined with geophysical models for subsurface evolution, fracturing, fault generation and other natural processes to produce a library of realistic synthetic subsurface models.  These synthetic subsurface models may be employed in simulations of environmental processes including ground water transport and surface water runoff and transport of chemicals into natural river systems.   In industrial processes, synthetic subsurface models are valuable in conducting multiphase flow simulations for simulations of oil recovery in petroleum reservoirs.  Underlying all of these simulation efforts is the enormous uncertainty in the details of the subsurface geology – rock or soil type, permeability, porosity, etc.  It is impossible to accumulate sufficient experimental data over typical domains with horizontal scales of 10’s of kilometers and vertical scales of hundreds of meters.  By conducting simulations over a library of varying synthetic subsurface profiles, one may hope to assess the range of possible outcomes for industrial and natural transport processes.

We briefly review the overall governing equations for the fluid dynamics and transport systems.  We show how averaging processes leads to a reduced system of PDE’s with closures required for modelling turbulent Reynolds stresses and three dimensional momentum flux associated with axial swirling flows in curved channels with variable bottom topography.  The resulting PDE’s are of a strongly hyperbolic character which motivates our use of a hybridizable discontinuous Galerkin (HDG, Cockburn et al 2009) method with spectral element discretization.  We find significant similarities with the hyperbolic character of reservoir simulation equations and our approach for meandering rivers builds on previous efforts of Taneja and Higdon (2018) in that application.

Cockburn, J. Gopalakrishnan, R. Lazarov, Unified hybridization of discontinuous Galerkin,

mixed and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer.

Anal. 47, 1319–1365 (2009)

Taneja, A. and Higdon, J.  A fully-coupled discontinuous Galerkin spectral element method for two-phase flow in petroleum reservoirs.  J. Computat. Phys.  352, 341-372, (2018).

Jonathan Higdon is the Houston Professor of Chemical Engineering at the University of Illinois at Urbana-Champaign.  Higdon has been on the faculty at Illinois from 1980 -2019.  His research interests include high performance computation, complex fluids and soft matter, geophysical fluid mechanics, reservoir simulation.

• Time: 11.00 AM
• Location: 206 Furnas Hall
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