Christopher E. Kees

Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow

Abstract Richards’ equation (RE) is commonly used to model flow in variably saturated porous media. However, its solution continues to be difficult for many conditions of practical interest. Among the various time discretizations applied to RE, the method of lines (MOL) has been used successfully to introduce robust, accurate, and efficient temporal approximations. At the same time, a mixed-hybrid finite element method combined with an adaptive, higher order time discretization has shown benefits over traditional, lower order temporal approximations for modeling single-phase groundwater flow in heterogeneous porous media. Here, we extend earlier work for single-phase flow and consider two mixed finite element methods that have been used previously to solve RE using lower order time discretizations with either fixed time steps or empirically based adaption. We formulate the two spatial discretizations within a MOL context for the pressure head form of RE as well as a fully mass-conservative version. We conduct several numerical experiments for both spatial discretizations with each formulation, and we compare the higher order, adaptive time discretization to a first-order approximation with formal error control and adaptive time step selection. Based on the numerical results, we evaluate the performance of the methods for robustness and efficiency.

Efficient steady-state solution techniques for variably saturated groundwater flow

Abstract We consider the simulation of steady-state variably saturated groundwater flow using Richards’ equation (RE). The difficulties associated with solving RE numerically are well known. Most discretization approaches for RE lead to nonlinear systems that are large and difficult to solve. The solution of nonlinear systems for steady-state problems can be particularly challenging, since a good initial guess for the steady-state solution is often hard to obtain, and the resulting linear systems may be poorly scaled. Common approaches like Picard iteration or variations of Newton’s method have their advantages but perform poorly with standard globalization techniques under certain conditions. Pseudo-transient continuation has been used in computational fluid dynamics for some time to obtain steady-state solutions for problems in which Newton’s method with standard line-search strategies fails. Here, we examine the use of pseudo-transient continuation as well as Newton’s method combined with standard globalization techniques for steady-state problems in heterogeneous domains. We investigate the methods’ performance with direct and preconditioned Krylov iterative linear solvers. We then make recommendations for robust and efficient approaches to obtain steady-state solutions for RE under a range of conditions.

Finite element methods for variable density flow and solute transport

Saltwater intrusion into coastal freshwater aquifers is an ongoing problem that will continue to impact coastal freshwater resources as coastal populations increase. To effectively model saltwater intrusion, the impacts of increased salt content on fluid density must be accounted for to properly model saltwater/freshwater transition zones and sharp interfaces. We present a model for variable density fluid flow and solute transport where a conforming finite element method discretization with a locally conservative velocity post-processing method is used for the flow model and the transport equation is discretized using a variational multiscale stabilized conforming finite element method. This formulation provides a consistent velocity and performs well even in advection-dominated problems that can occur in saltwater intrusion modeling. The physical model is presented as well as the formulation of the numerical model and solution methods. The model is tested against several 2-D and 3-D numerical and experimental benchmark problems, and the results are presented to verify the code.

A hydraulic capture application for optimal remediation design

The goal of a hydraulic capture model for remediation purposes is to design a well field so that the direction of groundwater flow is altered, thereby halting or reversing the migration of a contaminant plume. Management strategies typically require a well design that will contain or shrink a plume at minimum cost. Objective functions and constraints can be nonlinear, non-convex, non-differentiable, or even discontinuous. The solution uses optimization algorithms with groundwater flow and possibly transport simulators. The formulation of the objective function dictates possible optimization algorithms that can be used. For example, a gradient based method is likely to fail on a discontinuous objective function or gradient information may not be available. Computational efficiency as well as accuracy is desirable and often influences the choice of solution method. In this paper we present three hydraulic capture models. Our motivation is a hydraulic capture application proposed in the literature for benchmarking purposes. We present numerical results for the three models using the implicit filtering algorithm.

Versatile Two-Level Schwarz Preconditioners for Multiphase Flow

Numerically modeling groundwater flow on finely discretized two- and three-dimensional domains requires solution algorithms appropriate for distributed memory multiprocessor architectures. Multilevel and domain decomposition algorithms are appropriate for preconditioning or solving linear systems in parallel and have, therefore, been applied to linear models for saturated groundwater flow. These algorithms have also been incorporated into more complex nonlinear multiphase flow models in the context of a linearization procedure such as Newtons method. In this work, we study a class of parallel preconditioners based on two-level Schwarz domain decomposition applied in a nonlinear two-phase flow numerical model. The restriction and interpolation operators are based on an aggregation approach that has a straightforward implementation for a variety of applications arising in subsurface modeling: structured and unstructured discretizations, finite elements and finite differences, and multicomponent model equations. We present model formulations, results from numerical experiments, and a comparison of a standard one-level Schwarz method to three two-level aggregation-based methods.

Two-phase flow modeling of the influence of wave shapes and bed slope on nearshore hydrodynamics

ABSTRACT Bakhtyar, R., Razmi, A.M., Barry, D.A., Yeganeh-bakhtiary, A. Kees, C.E., and C.T. Miller, 2013. Two-Phase Flow Modeling of the Influence of Wave Shapes and Bed Slope on Nearshore Hydrodynamics. An Eulerian two-phase flow model (air-water) was used to simulate nearshore hydrodynamic processes driven by wave motion. The flow field was computed with the Reynolds-Averaged Navier-Stokes equations in conjunction with the Volume-Of-Fluid method and the RNG turbulence-closure scheme. To study the effects of different wave shapes on surf-swash zone hydrodynamics, a set of numerical experiments was carried out. Predictions of three wave theories (Airy, 2nd-order Stokes and 5th-order Stokes) were compared, with a focus on the turbulence and flow fields. Model performance was assessed by comparing numerical results with laboratory experimental observations. Relationships between the water depth, undertow, TKE and wave characteristics are presented. The results indicate that the characteristics of turbulence and flow, for example the position of wave breaking and magnitude of TKE, are affected by different wave types. Numerical simulations showed that only high-order Stokes wave theory predicts the nonlinearity required for predicting hydrodynamic characteristics in agreement with existing understanding of nearshore processes. Numerical simulations were run for different hydrodynamic conditions, but with a focus on different bed slopes. The transformation of incoming waves as they reach shallow water occurs closer to the shoreline for steeper profiles. Consistently, the peaks in TKE and wave set-up are shifted onshore for steeper slopes. The numerical results showed that TKE and undertow velocity are smaller on dissipative beaches than on intermediate beaches.

Thermodynamically Constrained Averaging Theory: Principles, Model Hierarchies, and Deviation Kinetic Energy Extensions

The thermodynamically constrained averaging theory (TCAT) is a comprehensive theory used to formulate hierarchies of multiphase, multiscale models that are closed based upon the second law of thermodynamics. The rate of entropy production is posed in terms of the product of fluxes and forces of dissipative processes. The attractive features of TCAT include consistency across disparate length scales; thermodynamic consistency across scales; the inclusion of interfaces and common curves as well as phases; the development of kinematic equations to provide closure relations for geometric extent measures; and a structured approach to model building. The elements of the TCAT approach are shown; the ways in which each of these attractive features emerge from the TCAT approach are illustrated; and a review of the hierarchies of models that have been formulated is provided. Because the TCAT approach is mathematically involved, we illustrate how this approach can be applied by leveraging existing components of the theory that can be applied to a wide range of applications. This can result in a substantial reduction in formulation effort compared to a complete derivation while yielding identical results. Lastly, we note the previous neglect of the deviation kinetic energy, which is not important in slow porous media flows, formulate the required equations to extend the theory, and comment on applications for which the new components would be especially useful. This work should serve to make TCAT more accessible for applications, thereby enabling higher fidelity models for applications such as turbulent multiphase flows.

An ELLAM approximation for advective-dispersive transport with nonlinear sorption

We consider an Eulerian-Lagrangian localized adjoint method (ELLAM) applied to nonlinear model equations governing solute transport and sorption in porous media. Solute transport in the aqueous phase is modeled by standard advection and hydrodynamic dispersion processes, while sorption is modeled with a nonlinear local equilibrium model. We present our implementation of finite volume ELLAM (FVELLAM) and finite element (FE-ELLAM) discretizations to the reactive transport model and evaluate their performance for several test problems containing selfsharpening fronts. Preprint submitted to Elsevier Science 28 February 2005 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE FEB 2005 2. REPORT TYPE 3. DATES COVERED 00-00-2005 to 00-00-2005 4. TITLE AND SUBTITLE An ELLAM approximation for advective-dispersive transport with nonlinear sorption 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) University of North Carolina,Center for Integrated Study of the Environment,Department of Environmental Sciences and Engineering,Chapel Hill,NC,27599-7431 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES The original document contains color images. 14. ABSTRACT see report 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT 18. NUMBER OF PAGES 58 19a. NAME OF RESPONSIBLE PERSON a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

Dual-scale Galerkin methods for Darcy flow

Abstract The discontinuous Galerkin (DG) method has found widespread application in elliptic problems with rough coefficients, of which the Darcy flow equations are a prototypical example. One of the long-standing issues of DG approximations is the overall computational cost, and many different strategies have been proposed, such as the variational multiscale DG method, the hybridizable DG method, the multiscale DG method, the embedded DG method, and the Enriched Galerkin method. In this work, we propose a mixed dual-scale Galerkin method, in which the degrees-of-freedom of a less computationally expensive coarse-scale approximation are linked to the degrees-of-freedom of a base DG approximation. We show that the proposed approach has always similar or improved accuracy with respect to the base DG method, with a considerable reduction in computational cost. For the specific definition of the coarse-scale space, we consider Raviart–Thomas finite elements for the mass flux and piecewise-linear continuous finite elements for the pressure. We provide a complete analysis of stability and convergence of the proposed method, in addition to a study on its conservation and consistency properties. We also present a battery of numerical tests to verify the results of the analysis, and evaluate a number of possible variations, such as using piecewise-linear continuous finite elements for the coarse-scale mass fluxes.

PROJECTION-BASED MODEL REDUCTION FOR FINITE ELEMENT APPROXIMATION OF SHALLOW WATER FLOWS

Abstract. The shallow water equations (SWE) are used to model a wide range of environmental flows from dam breaks and riverine hydrodynamics to hurricane storm surge and atmospheric processes. Despite significant gains in numerical model efficiency stemming from algorithmic and hardware improvements, accurate shallow water modeling can still be very computationally intensive. The resulting computational expense remains as a barrier to the inclusion of fully resolved two-dimensional shallow water models in many applications, particularly when the analysis involves optimal design, parameter inversion, risk assessment, and/or uncertainty quantification. Here, we consider projection-based model reduction as a way to alleviate the computational burden associated with high-fidelity shallow-water approximations in ensemble forecast and sampling methodologies. In order to develop a robust approach that can resolve advectiondominated problems with shocks as well as more smoothly varying riverine and estuarine flows, we consider techniques using both Galerkin and Petrov-Galerkin projection on global bases provided by Proper Orthogonal Decomposition (POD). To achieve realistic speedup, we consider alternative methods for the reduction of the non-polynomial nonlinearities that arise in typical finite element formulations. We evaluate the schemes’ performance by considering their accuracy and robustness for test problems in one and two space dimensions.