Data-driven discovery of the governing equations for transport in heterogeneous media by symbolic regression and stochastic optimization.

With advances in instrumentation and the tremendous increase in computational power, vast amounts of data are becoming available for many complex phenomena in macroscopically heterogeneous media, particularly those that involve flow and transport processes, which are problems of fundamental interest that occur in a wide variety of physical systems. The absence of a length scale beyond which such systems can be considered as homogeneous implies that the traditional volume or ensemble averaging of the equations of continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equations for flow and transport processes is an open question. We propose a data-driven approach that uses stochastic optimization and symbolic regression to discover the governing equations for flow and transport processes in heterogeneous media. The data could be experimental or obtained by microscopic simulation. As an example, we discover the governing equation for anomalous diffusion on the critical percolation cluster at the percolation threshold, which is in the form of a fractional partial differential equation, and agrees with what has been proposed previously.

Temporal flow variations interact with spatial physical heterogeneity to impact solute transport in managed river corridors.

The interactions between surface water and groundwater in river corridors lead to temporal fluctuations in subsurface water fluxes which have a critical role on solute transport dynamics. In this work, we develop a framework to analyze the relative impacts of different temporal frequencies of the flow field in a spatially heterogeneous aquifer on solute transport. Our analysis indicates that the advection-dispersion equation behaves as a low-pass filter by wiping out the effect of high-frequency velocity fluctuations on the first two spatial moments of the solute plume, namely its center of mass and spreading. The concepts discussed in the theoretical analysis are then applied to understand solute transport dynamics at the 300 Area of the Hanford site (USA) adjacent to the Columbia River. We examine the temporal behavior of the solute plume's spatial moments for different temporal frequencies utilizing geostatistical parameters estimated in the 300 Area. Due to the proximity to the Columbia river, groundwater fluxes at the Hanford site are highly dynamic resulting in a large range of characteristic temporal frequencies. Nonetheless, similar to the theoretical analysis, our results show that the effect of high-frequency fluctuations is filtered, with most of the solute transport dynamics being controlled by fluctuations characterized by a large characteristic period.

Resilience of groundwater systems in the presence of Bisphenol A under uncertainty.

Assessing the health risks associated with emerging contaminants in groundwater systems is a complex issue that has been receiving increased attention in indirect potable reuse applications. Among several emerging contaminants, our study focuses on developing a numerical model that aims to compute the transport characteristics of Bisphenol A (BPA) in a 3D spatially heterogeneous aquifer under uncertainty. Traditional approaches that characterize the health risk of BPA to humans rely on the monotonic dose-response (MDR) relationship with a regulatory dose limit. Recent public health studies indicate that BPA can cause endocrine-related health effects in specific low dose ranges, which requires the consideration of the non-monotonic dose-response (NMDR) model. This work investigates the impact of different BPA DR models (i.e., monotonic vs. non-monotonic) on the resilience of the aquifer against BPA contamination in the presence of hydrogeological heterogeneity. For the resilience estimation, a systematic stochastic methodology linking risk characterization to aquifer resilience is established. Our results show the importance of the interplay between the DR models and aquifer heterogeneity on controlling the uncertainty of the resilience loss RL (d) at a specified environmentally sensitive target. In the increased level of aquifer heterogeneity, the uncertainty bounds are higher for RL estimated through the NMDR model as opposed to the MDR model. Moreover, RL is controlled by η (-), the ratio of the volumetric flow rate at the source zone to the average flow rate at the background aquifer. In a risk management perspective, the consideration of the NMDR model needs to be emphasized due to its impact on the uncertainty of RL. A critical case is when the land use of a contamination site indicates a large number of the vulnerable population to endocrine-related health effects. In this case, η as an indicator of aquifer resilience can reduce the uncertainty of RL.

Climate change impact on residual contaminants under sustainable remediation.

This study investigates the potential impact of climate change on residual contaminants in vadose zones and groundwater. We assume that the effect of climate changes can be represented by perturbations in the natural recharge through the aquifer system. We perform numerical modeling of unsaturated/saturated flow and transport and consider different performance metrics: contaminant concentrations at observation wells and contaminant export at the site's boundary. We evaluate the effect of increasing and decreasing recharge as well as the impact of potential failure of surface capping structures employed to immobilize vadose zone contaminants. Our approach is demonstrated in a real case study by simulating transport of non-reactive radioactive tritium at the U.S. Department of Energy's Savannah River Site. Results show that recharge changes significantly affect well concentrations: after an initial slight dilution we identify a significant concentration increase at different observation wells some years after the recharge increase and/or the cap failure, as a consequence of contaminants' mobilization. This effect is generally emphasized and occurs earlier as the recharge increases. Under decreased aquifers' recharge the concentration could slightly increase for some years, due to a decrease of dilution, depending on the magnitude of the negative recharge shift. We identify trigger levels of recharge above which the concentration/export breakthrough curves and the time of exceedance of the Maximum Contaminant Level for tritium are remarkably affected. Moreover, we observe that the contaminant export at the control plane, identified as the risk pathway to the downgradient population, may only be minimally affected by shifts in the natural recharge regime, except for some extreme cases. We conclude that more frequent sampling and in-situ monitoring near the source zone should be adopted to better explain concentrations' anomalies under changing climatic conditions. Moreover, the maintenance of the cap is critical not only to sequester residual contaminants in the vadose zone, but also to reduce the uncertainty associated with future precipitation changes. Finally, realistic flow and transport simulations achieved through proper calibration processes, rather than conservative modeling, should be adopted to identify non-trivial trade-offs which enable better allocation of resources towards reducing uncertainty in decision making.

Fluid Deformation in Random Steady Three Dimensional Flow.

The deformation of elementary fluid volumes by velocity gradients is a key process for scalar mixing, chemical reactions and biological processes in flows. Whilst fluid deformation in unsteady, turbulent flow has gained much attention over the past half century, deformation in steady random flows with complex structure - such as flow through heterogeneous porous media - has received significantly less attention. In contrast to turbulent flow, the steady nature of these flows constrains fluid deformation to be anisotropic with respect to the fluid velocity, with significant implications for e.g. longitudinal and transverse mixing and dispersion. In this study we derive an ab initio coupled continuous time random walk (CTRW) model of fluid deformation in random steady three-dimensional flow that is based upon a streamline coordinate transform which renders the velocity gradient and fluid deformation tensors upper-triangular. We apply this coupled CTRW model to several model flows and find these exhibit a remarkably simple deformation structure in the streamline coordinate frame, facilitating solution of the stochastic deformation tensor components. These results show that the evolution of longitudinal and transverse fluid deformation for chaotic flows is governed by both the Lyapunov exponent and power-law exponent of the velocity PDF at small velocities, whereas algebraic deformation in non-chaotic flows arises from the intermittency of shear events following similar dynamics as that for steady two-dimensional flow.

Solute concentration at a well in non-Gaussian aquifers under constant and time-varying pumping schedule.

Our study is keyed to the analysis of the interplay between engineering factors (i.e., transient pumping rates versus less realistic but commonly analyzed uniform extraction rates) and the heterogeneous structure of the aquifer (as expressed by the probability distribution characterizing transmissivity) on contaminant transport. We explore the joint influence of diverse (a) groundwater pumping schedules (constant and variable in time) and (b) representations of the stochastic heterogeneous transmissivity (T) field on temporal histories of solute concentrations observed at an extraction well. The stochastic nature of T is rendered by modeling its natural logarithm, Y=lnT, through a typical Gaussian representation and the recently introduced Generalized sub-Gaussian (GSG) model. The latter has the unique property to embed scale-dependent non-Gaussian features of the main statistics of Y and its (spatial) increments, which have been documented in a variety of studies. We rely on numerical Monte Carlo simulations and compute the temporal evolution at the well of low order moments of the solute concentration (C), as well as statistics of the peak concentration (Cp), identified as the environmental performance metric of interest in this study. We show that the pumping schedule strongly affects the pattern of the temporal evolution of the first two statistical moments of C, regardless the nature (Gaussian or non-Gaussian) of the underlying Y field, whereas the latter quantitatively influences their magnitude. Our results show that uncertainty associated with C and Cp estimates is larger when operating under a transient extraction scheme than under the action of a uniform withdrawal schedule. The probability density function (PDF) of Cp displays a long positive tail in the presence of time-varying pumping schedule. All these aspects are magnified in the presence of non-Gaussian Y fields. Additionally, the PDF of Cp displays a bimodal shape for all types of pumping schemes analyzed, independent of the type of heterogeneity considered.

Uncertainty quantification of environmental performance metrics in heterogeneous aquifers with long-range correlations.

We investigate how the uncertainty stemming from disordered porous media that display long-range correlation in the hydraulic conductivity (K) field propagates to predictions of environmental performance metrics (EPMs). In this study, the EPMs are quantities that are of relevance to risk analysis and remediation, such as peak flux-averaged concentration, early and late arrival times among others. By using stochastic simulations, we quantify the uncertainty associated with the EPMs for a given disordered spatial structure of the K-field and identify the probability distribution function (PDF) model that best captures the statistics of the EPMs of interest. Results indicate that the probabilistic distribution of the EPMs considered in this study follows lognormal PDF. Finally, through the use of information theory, we reveal how the persistent/anti-persistent correlation structure of the K-field influences the EPMs and corresponding uncertainties.

Coupled continuous-time random walks for fluid stretching in two-dimensional heterogeneous media.

We study the relation between flow structure and fluid deformation in steady flows through two-dimensional heterogeneous media, which are characterized by a broad spectrum of stretching behaviors, ranging from sub- to superlinear. We analyze these behaviors from first principles, which uncovers intermittent shear events to be at the origin of subexponential stretching. We derive explicit expressions for Lagrangian deformation and demonstrate that stretching obeys a coupled continuous-time random walk, which for broad distributions of flow velocities becomes a Lévy walk. The derived model provides a direct link between the flow and deformation statistics, and a natural way to quantify the impact of intermittent shear events on the stretching behavior.

Scaling forms of particle densities for Lévy walks and strong anomalous diffusion.

We study the scaling behavior of particle densities for Lévy walks whose transition length r is coupled with the transition time t as |r|∝t^{α} with an exponent α>0. The transition-time distribution behaves as ψ(t)∝t^{-1-ß} with ß>0. For 1<ß<2α and α≥1, particle displacements are characterized by a finite transition time and confinement to |r|q_{c}. These results give insight into the possible origins of strong anomalous diffusion and anomalous behaviors in disordered systems in general.

Dynamic interactions between hydrogeological and exposure parameters in daily dose prediction under uncertainty and temporal variability.

We study the time dependent interaction between hydrogeological and exposure parameters in daily dose predictions due to exposure of humans to groundwater contamination. Dose predictions are treated stochastically to account for an incomplete hydrogeological and geochemical field characterization, and an incomplete knowledge of the physiological response. We used a nested Monte Carlo framework to account for uncertainty and variability arising from both hydrogeological and exposure variables. Our interest is in the temporal dynamics of the total dose and their effects on parametric uncertainty reduction. We illustrate the approach to a HCH (lindane) pollution problem at the Ebro River, Spain. The temporal distribution of lindane in the river water can have a strong impact in the evaluation of risk. The total dose displays a non-linear effect on different population cohorts, indicating the need to account for population variability. We then expand the concept of Comparative Information Yield Curves developed earlier (see de Barros et al. [29]) to evaluate parametric uncertainty reduction under temporally variable exposure dose. Results show that the importance of parametric uncertainty reduction varies according to the temporal dynamics of the lindane plume. The approach could be used for any chemical to aid decision makers to better allocate resources towards reducing uncertainty.

Stochastic evaluation of mixing-controlled steady-state plume lengths in two-dimensional heterogeneous domains.

We study plumes originating from continuous sources that require a dissolved reaction partner for their degradation. The length of such plumes is typically controlled by transverse mixing. While analytical expressions have been derived for homogeneous flow fields, incomplete characterization of the hydraulic conductivity field causes uncertainty in predicting plume lengths in heterogeneous domains. In this context, we analyze the effects of three sources of uncertainty: (i) The uncertainty of the effective mixing rate along the plume fringes due to spatially varying flow focusing, (ii) the uncertainty of the volumetric discharge through (and thus total mass flux leaving) the source area, and (iii) different parameterizations of the Darcy-scale transverse dispersion coefficient. The first two are directly related to heterogeneity of hydraulic conductivity. In this paper, we derive semi-analytical expressions for the probability distribution of plume lengths at different levels of complexity. The results are compared to numerical Monte Carlo simulations. Uncertainties in mixing and in the source strength result in a statistical distribution of possible plume lengths. For unconditional random hydraulic conductivity fields, plume lengths may vary by more than one order of magnitude even for moderate degrees of heterogeneity. Our results show that the uncertainty of volumetric flux through the source is the most relevant contribution to the variance of the plume length. The choice of different parameterizations for the local dispersion coefficient leads to differences in the mean estimated plume length.

On the link between contaminant source release conditions and plume prediction uncertainty.

The initial width of contaminant plumes is known to have a key influence on expected plume development, dispersion and travel time statistics. In past studies, initial plume width has been perceived identical to the geometric width of a contaminant source or injection volume. A recent study on optimal sampling layouts (Nowak et al., 2009) showed that a significant portion of uncertainty in predicting plume migration stems from the uncertain total hydraulic flux through the source area. This result points towards a missing link between source geometry and plume statistics, which we denote as the effective source width. We define the effective source width by the ratio between the actual and expected hydraulic fluxes times the geometric source width. The actual hydraulic flux through the source area is given by individual realizations while the expected one represents the mean over the ensemble. It is a stochastic quantity that may strongly differ from the actual geometric source width for geometrically small sources, and becomes identical only at the limit of wide sources (approaching ergodicity). We derive its stochastic ensemble moments in order to explore the dependency on source scale. We show that, if the effective source width is known rather than the geometric width, predictions of plume development can greatly increase in predictive power. This is illustrated on plume statistics such as the distribution of plume length, average width, transverse dispersion, total mass flux and overall concentration variance. The analysis is limited to 2D depth-averaged systems, but implications hold for 3D cases.