Gowri Srinivasan

Random walk particle tracking simulations of non-Fickian transport in heterogeneous media

Derivations of continuum nonlocal models of non-Fickian (anomalous) transport require assumptions that might limit their applicability. We present a particle-based algorithm, which obviates the need for many of these assumptions by allowing stochastic processes that represent spatial and temporal random increments to be correlated in space and time, be stationary or non-stationary, and to have arbitrary distributions. The approach treats a particle trajectory as a subordinated stochastic process that is described by a set of Langevin equations, which represent a continuous time random walk (CTRW). Convolution-based particle tracking (CBPT) is used to increase the computational efficiency and accuracy of these particle-based simulations. The combined CTRW-CBPT approach enables one to convert any particle tracking legacy code into a simulator capable of handling non-Fickian transport.

Lagrangian models of reactive transport in heterogeneous porous media with uncertain properties

We consider multi-component reactive transport in heterogeneous porous media with uncertain hydraulic and chemical properties. This parametric uncertainty is quantified by treating relevant flow and transport parameters as random fields, which renders the governing equations stochastic. We adopt a stochastic Lagrangian framework to replace a three-dimensional advection–reaction transport equation with a one-dimensional equation for solute travel times. We derive approximate expressions for breakthrough curves and their temporal moments. To illustrate our general theory, we consider advective transport of dissolved species undergoing an irreversible bimolecular reaction.

Predictions of first passage times in sparse discrete fracture networks using graph-based reductions

We present a graph-based methodology to reduce the computational cost of obtaining first passage times through sparse fracture networks. We derive graph representations of generic three-dimensional discrete fracture networks (DFNs) using the DFN topology and flow boundary conditions. Subgraphs corresponding to the union of the k shortest paths between the inflow and outflow boundaries are identified and transport on their equivalent subnetworks is compared to transport through the full network. The number of paths included in the subgraphs is based on the scaling behavior of the number of edges in the graph with the number of shortest paths. First passage times through the subnetworks are in good agreement with those obtained in the full network, both for individual realizations and in distribution. Accurate estimates of first passage times are obtained with an order of magnitude reduction of CPU time and mesh size using the proposed method.

Machine learning for graph-based representations of three-dimensional discrete fracture networks

Structural and topological information play a key role in modeling flow and transport through fractured rock in the subsurface. Discrete fracture network (DFN) computational suites such as dfnWorks (Hyman et al. Comput. Geosci. 84, 10–19 2015) are designed to simulate flow and transport in such porous media. Flow and transport calculations reveal that a small backbone of fractures exists, where most flow and transport occurs. Restricting the flowing fracture network to this backbone provides a significant reduction in the network’s effective size. However, the particle-tracking simulations needed to determine this reduction are computationally intensive. Such methods may be impractical for large systems or for robust uncertainty quantification of fracture networks, where thousands of forward simulations are needed to bound system behavior. In this paper, we develop an alternative network reduction approach to characterizing transport in DFNs, by combining graph theoretical and machine learning methods. We consider a graph representation where nodes signify fractures and edges denote their intersections. Using random forest and support vector machines, we rapidly identify a subnetwork that captures the flow patterns of the full DFN, based primarily on node centrality features in the graph. Our supervised learning techniques train on particle-tracking backbone paths found by dfnWorks, but run in negligible time compared to those simulations. We find that our predictions can reduce the network to approximately 20% of its original size, while still generating breakthrough curves consistent with those of the original network.

Convolution-based particle tracking method for transient flow

A convolution-based particle tracking (CBPT) method was recently developed for calculating solute concentrations (Robinson et al., Comput Geosci 14(4): 779–792, 2010). This method is highly efficient but limited to steady-state flow conditions. Here, we present an extension of this method to transient flow conditions. This extension requires a single-particle tracking process model run, with a pulse of particles introduced at a sequence of times for each source location. The number and interval of particle releases depends upon the transients in the flow. Numerical convolution of particle paths obtained at each release time and location with a time-varying source term is performed to yield the shape of the plume. Many factors controlling transport such as variation in source terms, radioactive decay, and in some cases linear processes such as sorption and diffusion into dead-end pores can be simulated in the convolution step for Monte Carlo-based analysis of transport uncertainty. We demonstrate the efficiency of the transient CBPT method, by showing that it requires fewer particles than traditional random walk particle tracking methods to achieve the same levels of accuracy, especially as the source term increases in duration or is uncertain. Since flow calculations under transient conditions are often very expensive, this is a computationally efficient yet accurate method.

Light dynamics in nonlinear trimers and twisted multicore fibers

Novel photonic structures such as multi-core fibers and graphene based arrays present unique opportunities to manipulate and control the propagation of light. Here we discuss nonlinear dynamics for structures with a few (2 to 6) elements for which linear and nonlinear properties can be tuned. Specifically we show how nonlinearity, coupling, and parity-time PT symmetric gain/loss relate to existence, stability and in general, dynamical properties of nonlinear optical modes. The main emphasis of our presentation will be on systems with few degrees of freedom, most notably couplers, trimers and generalizations thereof to systems with 6 nodes.

Quantifying Topological Uncertainty in Fractured Systems using Graph Theory and Machine Learning

Fractured systems are ubiquitous in natural and engineered applications as diverse as hydraulic fracturing, underground nuclear test detection, corrosive damage in materials and brittle failure of metals and ceramics. Microstructural information (fracture size, orientation, etc.) plays a key role in governing the dominant physics for these systems but can only be known statistically. Current models either ignore or idealize microscale information at these larger scales because we lack a framework that efficiently utilizes it in its entirety to predict macroscale behavior in brittle materials. We propose a method that integrates computational physics, machine learning and graph theory to make a paradigm shift from computationally intensive high-fidelity models to coarse-scale graphs without loss of critical structural information. We exploit the underlying discrete structure of fracture networks in systems considering flow through fractures and fracture propagation. We demonstrate that compact graph representations require significantly fewer degrees of freedom (dof) to capture micro-fracture information and further accelerate these models with Machine Learning. Our method has been shown to improve accuracy of predictions with up to four orders of magnitude speedup.

Learning on Graphs for Predictions of Fracture Propagation, Flow and Transport

Microstructural information plays a key role in governing the dominant physics for various applications involving fracture networks. Resolving the interactions of thousands of interconnected sub-micron scale fractures is computationally intensive, and is intractable with current technologies. Coarsening of the domain and simplification of the physics are two commonly used workarounds, but these methods often eliminate features critical to accurately predicting macroscale behavior. Additionally, traditional Uncertainty Quantification (UQ) methods which account for parametric and model uncertainties have been shown to be inadequate in failure predictions that do not include these subgrid scale effects. We propose to overcome this hurdle by exploiting the fact that fracture networks have an underlying discrete structure that can be compactly represented and propagated via graphs. We have outlined two separate approaches for two separate applications -- prediction of flow in the subsurface and brittle failure at the macroscale. In the first approach, we expect to discover accurate graph representations of previously neglected microscale physics. An alternate approach would be using machine learning algorithms to mimic the detailed physics at the microscale. The resulting workflow using either approach will be memory/computationally efficient by at least one to two orders of magnitude over existing methods.

Robust system size reduction of discrete fracture networks: a multi-fidelity method that preserves transport characteristics

We propose a multi-fidelity system reduction technique that uses weighted graphs paired with three-dimensional discrete fracture network (DFN) modelling for efficient simulation of subsurface flow and transport in fractured media. DFN models are used to simulate flow and transport in subsurface fractured rock with low-permeability. One method to alleviate the heavy computational overhead associated with these simulations is to reduce the size of the DFN using a graph representation of it to identify the primary flow sub-network and only simulate flow and transport thereon. The first of these methods used unweighted graphs constructed solely on DFN topology and could be used for accurate predictions of first-passage times. However, these techniques perform poorly when predicting later stages of the mass breakthrough. We utilize a weighted-graph representation of the DFN where edge weights are based on hydrological parameters in the DFN that allows us to exploit the kinematic quantities derivable a posteriori from the flow solution obtained on the graph representation of the DFN to perform system reduction and predict the later stages of the breakthrough curve with high fidelity. We also propose and demonstrate the use of an adaptive pruning algorithm with error control that produces a pruned DFN sub-network whose predicted mass breakthrough agrees with the original DFN within a user-specified tolerance. The method allows for the level of accuracy to be a user-controlled parameter.

Existence, Stability and Dynamics of Discrete Solitary Waves in a Binary Waveguide Array

Recent work has explored binary waveguide arrays in the long-wavelength, near-continuum limit, here we examine the opposite limit, namely the vicinity of the so-called anti-continuum limit. We provide a systematic discussion of states involving one, two and three excited waveguides, and provide comparisons that illustrate how the stability of these states differ from the monoatomic limit of a single type of waveguide. We do so by developing a general theory which systematically tracks down the key eigenvalues of the linearized system. When we find the states to be unstable, we explore their dynamical evolution through direct numerical simulations. The latter typically illustrate, for the parameter values considered herein, the persistence of localized dynamics and the emergence for the duration of our simulations of robust quasi-periodic states for two excited sites. As the number of excited nodes increase, the unstable dynamics feature less regular oscillations of the solutions amplitude.