Jonathan Higdon

Fluid Mechanics and Computational Algorithms

Jonathan Higdon
Jonathan Higdon

Geophysical Fluid Mechanics

As rivers flow across open plains, they follow an ever-changing course that changes the landscape and composition of the land they traverse. We are developing advanced computational algorithms to study the dynamics of meandering rivers and to predict the geological evolution including the path of the river channels and the composition of sediment deposited by the evolving channel system.

Over time scales ranging from decades up to a hundred years or so, an initially straight stretch of river will develop an exaggerated meandering course as it flows downstream. As the meanders become more extreme, seasonal floods may cause the river to jump its banks, shortening the channel and leaving behind detached isolated sections forming oxbow lakes. At each stage of the meandering process, the eroding bank material as well as sediment carried by the river flow is deposited on the river bottom and on the newly formed inner bank of the river. While the initial time scale of the river migration is typically of order of a hundred years, the pattern of the river channel belt is preserved as successive generations of river evolution bury the original river plain beneath hundreds or thousands of meters of sedimentary deposits. On geologic time scales of 105 to 106 years, pressure and high temperatures convert the organic materials into petroleum and natural gas, while sand and other materials are converted into sedimentary rock.

In principle, one might start with a known geological river configuration and predict the future evolution of the geological formations based on sound physical models and known process conditions. In practice, this is not feasible owing to approximations in the physical models and uncertainty in geophysical process variables. Nonetheless, computational simulations provide a knowledge base of plausible scenarios which may be combined with limited geological observations to provide the best estimates of the detailed structure of geological strata.

This information is of critical interest in understanding environment changes associated with natural phenomena, and in guiding the energy industry to yield optimum resource management with minimal environmental impact.

Computational Algorithms for Petroleum Reservoir Simulation

The simulation of multiphase fluid flows in large scale porous media is an important long term strategic requirement for reservoir modeling and simulation in the petroleum industry. A great challenge in predicting the future production of oil and gas from a reservoir is to reduce the uncertainty associated with unknown details of the geological formation, unknown initial saturation of water and organic phases in the reservoir, approximations in the governing equations for microscale multiphase flow and imperfect solution of the final partial differential equations. By combining heuristics and modeling of geological formation processes with reliable reservoir simulations to predict reservoir output, one may formulate an optimization problem to solve the large scale inverse problem and improve estimates of unknown parameters. Solution of this problem provides more reliable predictions of future output and improved reservoir management.

The reservoir simulation approach in this project focuses on development of finite element algorithms for two and three phase oil reservoir simulations. The goal is to develop stable high order finite element methods for hyperbolic PDE’s with discontinuous data in the governing equations (e.g. permeability) or the dependent variables (e.g. saturation front). Algorithms include discontinuous Galerkin methods, spectral finite elements and extended finite element (XFEM) approaches. In the XFEM approach, supplemental interfacial discretizations are introduced to accommodate discontinuous data fields progressing across a fixed main grid. High order PDE solvers facilitate optimization algorithms employing hybrid approaches with deterministic (gradient based, descent, adjoint) and stochastic components.

Colloidal Gels: Microstructure, Dynamics and Rheology

Our group has developed novel simulations for the study of the microstructure, dynamics, and rheology of colloidal gels. A colloidal gel is a system of sub-micron particles suspended in a fluid with interparticle forces such that it behaves as a viscoelastic material – acting similar to a viscous fluid under some conditions and as an elastic solid under others. We have focused on systems with short-range attraction and long-range repulsion induced by polymer depletion forces and adsorbed surface charge respectively. Our simulation techniques approach the accuracy of refined Stokesian dynamics computations with nearly the speed of simple Brownian dynamics algorithms.

Gels exhibit non-equilibrium phase behavior in attractive forces induce interparticle bonds which act to frustrate the formation of ordered crystalline solids. Thus the system may exist as a fluid phase when attractive forces EA/kT are weak, as a crystalline solid as attractive forces reach the predicted equilibrium threshold and as a disordered gel as the attractive forces increase further. The addition of long-range repulsive forces further complicates the phase behavior as it eliminates crystalline structures at all attractive strengths and modifies the microstructure and rheology of the gel. We find that the mechanical properties of gels may be tuned by the adjustment of the attractive and repulsive forces. Stronger attractive forces yield stiff unyielding gels which are brittle and fracture under excessive stress. The addition of repulsive forces leads to more pliable gels which can be greatly deformed, rapidly rebuilding interparticle bonds and maintaining their elastic properties over a wide range. Repulsion forces can also increase the time needed for gelation a thousand-fold.

Gels formed under quiescent conditions form an interesting class of materials, however, particle suspensions formed under more complicated circumstances are also highly important in advanced chemical processes. Many novel materials are formed by moving a bidisperse mixture of two distinct particle types through a series of processing operations. These may include spray coating a thin film of suspension onto a substrate, thickening by sedimentation or drainage, further consolidating with rollers or presses, and finally drying operations to form the final solid material. During each of the processes, the suspension microstructure may evolve, particles may partially segregate and the material may develop internal stresses similar to those observed in quiescent gels subjected to rheological flow conditions. Management of the microstructure and microscopic stresses is critical in the manufacture of these exotic materials as suboptimal processing may lead to defects, cracking, or other undesirable material performance. The study of these phenomena under high throughput manufacturing operations is a primary focus of our ongoing efforts in large-scale computational simulations of complex fluids.

Faculty Profile