Random domain decomposition for flow in heterogeneous stratified aquifers

We study two-dimensional flow in a layered heterogeneous medium composed of two materials whose hydraulic properties and spatial distribution are known statistically but are otherwise uncertain. Our analysis relies on the composite media theory, which employs random domain decomposition in the context of groundwater flow moment equations to explicitly account for the separate effects of material and geometric uncertainty on ensemble moments of head and flux. Flow parallel and perpendicular to the layering in a two-material composite layered medium is considered. The hydraulic conductivity of each material is log-normally distributed with a much higher mean in one material than in the other. The hydraulic conductivities of points within different materials are uncorrelated. The location of the internal boundary between the two contrasting materials is random and normally distributed with given mean and variance. We solve the equations for (ensemble) moments of hydraulic head and flux and analyze the impact of unknown geometry of materials on statistical moments of head and flux. We compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudo-homogeneous random fields. This work was performed under the auspices of the US Department of Energy (DOE): DOE/BES (Bureau of Energy Sciences) Program in the Applied Mathematical Sciences contract KC-07–01–01 and Los Alamos National Laboratory under LDRD 98604. This work made use of STC shared experimental facilities supported by the National Science Foundation under Agreement No. EAR-9876800. This work was supported in part by the European Commission under Contract No. EVK1-CT-1999–00041 (W-SAHaRA).     

Conditioning mean steady state flow on hydraulic head and conductivity through geostatistical inversion

Nonlocal moment equations allow one to render deterministically optimum predictions of flow in randomly heterogeneous media and to assess predictive uncertainty conditional on measured values of medium properties. We present a geostatistical inverse algorithm for steady-state flow that makes it possible to further condition such predictions and assessments on measured values of hydraulic head (and/or flux). Our algorithm is based on recursive finite-element approximations of exact first and second conditional moment equations. Hydraulic conductivity is parameterized via universal kriging based on unknown values at pilot points and (optionally) measured values at other discrete locations. Optimum unbiased inverse estimates of natural log hydraulic conductivity, head and flux are obtained by minimizing a residual criterion using the Levenberg-Marquardt algorithm. We illustrate the method for superimposed mean uniform and convergent flows in a bounded two-dimensional domain. Our examples illustrate how conductivity and head data act separately or jointly to reduce parameter estimation errors and model predictive uncertainty.This work is supported in part by NSF/ITR Grant EAR-0110289. The first author was additionally supported by scholarships from CONACYT and Instituto de Investigaciones Electricas of Mexico. Additional support was provided by the European Commission under Contract EVK1-CT-1999-00041 (W-SAHaRA-Stochastic Analysis of Well Head Protection and Risk Assessment).     

Analytical solutions to steady state unsaturated flow in layered,randomly heterogeneous soils via Kirchhoff transformation

In this study, we derive analytical solutions of the first two moments (mean and variance) of pressure head for one-dimensional steady state unsaturated flow in a randomly heterogeneous layered soil column under random boundary conditions. We first linearize the steady state unsaturated flow equations by Kirchhoff transformation and solve the moments of the transformed variable up to second order in terms of σ_Y and σ_β, the standard deviations of log hydraulic conductivity Y=ln(K_s) and of the log pore size distribution parameter β=ln(α). In addition, we also give solutions for the mean and variance of the unsaturated hydraulic conductivity. The analytical solutions of moment equations are validated via Monte Carlo simulations.     

Numerical method of moments for solute transport in a nonstationary flow field conditioned on hydraulic conductivity and head measurements

The unconditional stochastic studies on groundwater flow and solute transport in a nonstationary conductivity field show that the standard deviations of the hydraulic head and solute flux are very large in comparison with their mean values (Zhang et al. in Water Resour Res 36:2107–2120, 2000; Wu et al. in J Hydrol 275:208–228, 2003; Hu et al. in Adv Water Resour 26:513–531, 2003). In this study, we develop a numerical method of moments conditioning on measurements of hydraulic conductivity and head to reduce the variances of the head and the solute flux. A Lagrangian perturbation method is applied to develop the framework for solute transport in a nonstationary flow field. Since analytically derived moments equations are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. Instead of using an unconditional conductivity field as an input to calculate groundwater velocity, we combine a geostatistical method and a method of moment for flow to conditionally simulate the distributions of head and velocity based on the measurements of hydraulic conductivity and head at some points. The developed theory is applied in several case studies to investigate the influences of the measurements of hydraulic conductivity and/or the hydraulic head on the variances of the predictive head and the solute flux in nonstationary flow fields. The study results show that the conditional calculation will significantly reduce the head variance. Since the hydraulic head measurement points are treated as the interior boundary (Dirichlet boundary) conditions, conditioning on both the hydraulic conductivity and the head measurements is much better than conditioning only on conductivity measurements for reduction of head variance. However, for solute flux, variance reduction by the conditional study is not so significant.     

Interface dynamics in randomly heterogeneous porous media

We consider the dynamics of a fluid interface in heterogeneous porous media, whose hydraulic properties are uncertain. Modeling hydraulic conductivity as a random field of given statistics allows us to predict the interface dynamics and to estimate the corresponding predictive uncertainty by means of statistical moments. The novelty of our approach to obtaining the interface statistics consists of dynamically mapping the Cartesian coordinate system onto a coordinate system associated with the moving front. This transforms a difficult problem of deriving closure relationships for highly nonlinear stochastic flows with free surfaces into a relatively simple problem of deriving stochastic closures for linear flows in domains with fixed boundaries. We derive a set of deterministic equations for the statistical moments of the interfacial dynamics, which hold in one and two spatial dimensions, and analyze their solutions for one-dimensional flow.     

Stochastic analysis of spatial variability in unconfined groundwater flow

In this paper, spatial variability in steady one-dimensional unconfined groundwater flow in heterogeneous formations is investigated. An approach to deriving the variance of the hydraulic head is developed using the nonlinear filter theory. The nonlinear governing equation describing the one-dimensional unconfined groundwater flow is decomposed into three linear partial differential equations using the perturbation method. The linear and quadratic frequency response functions are obtained from the first- and second-order perturbation equations using the spectral method. Furthermore, under the assumption of the exponential covariance function of log hydraulic conductivity, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the linear system are derived. The results show that the variance derived herein is less than that of Gelhar (1977). The reason is that the log transmissivity is linearized in Gelhars work. In addition, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the quadratic system are derived as well. It is found that the correlation scale and the trend in mean of log hydraulic conductivity are important to the dimensionless variance ratio.     

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].     

Comparison of Ensemble Kalman Filter groundwater-data assimilation methods based on stochastic moment equations and Monte Carlo simulation

Traditional Ensemble Kalman Filter (EnKF) data assimilation requires computationally intensive Monte Carlo (MC) sampling, which suffers from filter inbreeding unless the number of simulations is large. Recently we proposed an alternative EnKF groundwater-data assimilation method that obviates the need for sampling and is free of inbreeding issues. In our new approach, theoretical ensemble moments are approximated directly by solving a system of corresponding stochastic groundwater flow equations. Like MC-based EnKF, our moment equations (ME) approach allows Bayesian updating of system states and parameters in real-time as new data become available. Here we compare the performances and accuracies of the two approaches on two-dimensional transient groundwater flow toward a well pumping water in a synthetic, randomly heterogeneous confined aquifer subject to prescribed head and flux boundary conditions.     

Stochastic perturbation analysis of groundwater flow. Spatially variable soils,semi-infinite domains and large fluctuations

As is well known, a complete stochastic solution of the stochastic differential equation governing saturated groundwater flow leads to an infinite hierarchy of equations in terms of higher-order moments. Perturbation techniques are commonly used to close this hierarchy, using power-series expansions. These methods are applied by truncating the series after a finite number of terms, and products of random gradients of conductivity and head potential are neglected. Uncertainty regarding the number or terms required to yield a sufficiently accurate result is a significant drawback with the application of power series-based perturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and can lead to physically unreasonable and inaccurate solutions of the stochastic flow equation. To support this argument, one-dimensional, steady-state, saturated groundwater flow is examined, for the case of a spatially distributed hydraulic conductivity field. An ordinary power-series perturbation method is used to approximate the mean head, using second-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improved results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such approximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the original stochastic flow problem to a closed set of equations for the mean and the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bounded domains and large fluctuations.     

3D inverse modelling of groundwater flow at a fractured site using a stochastic continuum model with multiple statistical populations

 3D groundwater flow at the fractured site of Asp? (Sweden) is simulated. The aim was to characterise the site as adequately as possible and to provide measures on the uncertainty of the estimates. A stochastic continuum model is used to simulate both groundwater flow in the major fracture planes and in the background. However, the positions of the major fracture planes are deterministically incorporated in the model and the statistical distribution of the hydraulic conductivity is modelled by the concept of multiple statistical populations; each fracture plane is an independent statistical population. Multiple equally likely realisations are built that are conditioned to geological information on the positions of the major fracture planes, hydraulic conductivity data, steady state head data and head responses to six different interference tests. The experimental information could be reproduced closely. The results of the conditioning are analysed in terms of ensemble averaged average fracture plane conductivities, the ensemble variance of average fracture plane conductivities and the statistical distribution of the hydraulic conductivity in the fracture planes. These results are evaluated after each conditioning stage. It is found that conditioning to hydraulic head data results in an increase of the hydraulic conductivity variance while the statistical distribution of log hydraulic conductivity, initially Gaussian, becomes more skewed for many of the fracture planes in most of the realisations.     

Diagrammatic solutions for hydraulic head moments in 1-D and 2-D bounded domains

We present a diagrammatic method for solving stochastic 1-D and 2-D steady-state flow equations in bounded domains. The diagrammatic method results in explicit solutions for the moments of the hydraulic head. This avoids certain numerical constraints encountered in realization-based methods. The diagrammatic technique also allows for the consideration of finite domains or large fluctuations, and is not restricted by distributional assumptions. The results of the method for 1-D and 2-D finite domains are compared with those obtained through a realization-based approach. Mean and variance of head are well reproduced for all log-conductivity variances inputted, including those larger than one. The diagrammatic results also compare favorably to hydraulic head moments derived by standard analytic methods requiring a linearized form of the flow equation.     

Diagrammatic solutions for hydraulic head moments in 1-D and 2-D bounded domains

We present a diagrammatic method for solving stochastic 1-D and 2-D steady-state flow equations in bounded domains. The diagrammatic method results in explicit solutions for the moments of the hydraulic head. This avoids certain numerical constraints encountered in realization-based methods. The diagrammatic technique also allows for the consideration of finite domains or large fluctuations, and is not restricted by distributional assumptions. The results of the method for 1-D and 2-D finite domains are compared with those obtained through a realization-based approach. Mean and variance of head are well reproduced for all log-conductivity variances inputted, including those larger than one. The diagrammatic results also compare favorably to hydraulic head moments derived by standard analytic methods requiring a linearized form of the flow equation.     

Using data assimilation method to calibrate a heterogeneous conductivity field conditioning on transient flow test data

A data assimilation method is developed to calibrate a heterogeneous hydraulic conductivity field conditioning on transient pumping test data. The ensemble Kalman filter (EnKF) approach is used to update model parameters such as hydraulic conductivity and model variables such as hydraulic head using available data. A synthetical two-dimensional flow case is used to assess the capability of the EnKF method to calibrate a heterogeneous conductivity field by assimilating transient flow data from observation wells under different hydraulic boundary conditions. The study results indicate that the EnKF method will significantly improve the estimation of the hydraulic conductivity field by assimilating continuous hydraulic head measurements and the hydraulic boundary condition will significantly affect the simulation results. For our cases, after a few data assimilation steps, the assimilated conductivity field with four Neumann boundaries matches the real field well while the assimilated conductivity field with mixed Dirichlet and Neumann boundaries does not. We found in our cases that the ensemble size should be 300 or larger for the numerical simulation. The number and the locations of the observation wells will significantly affect the hydraulic conductivity field calibration.     

Stochastic analysis of immiscible displacement of the fluids with arbitrary viscosities and its dependence on support scale of hydrological data

Stochastic analysis is commonly used to address uncertainty in the modeling of flow and transport in porous media. In the stochastic approach, the properties of porous media are treated as random functions with statistics obtained from field measurements. Several studies indicate that hydrological properties depend on the scale of measurements or support scales, but most stochastic analysis does not address the effects of support scale on stochastic predictions of subsurface processes. In this work we propose a new approach to study the scale dependence of stochastic predictions. We present a stochastic analysis of immiscible fluid–fluid displacement in randomly heterogeneous porous media. While existing solutions are applicable only to systems in which the viscosity of one phase is negligible compare with the viscosity of the other (water–air systems for example), our solutions can be applied to the immiscible displacement of fluids having arbitrarily viscosities such as NAPL–water and water–oil. Treating intrinsic permeability as a random field with statistics dependant on the permeability support scale (scale of measurements) we obtained, for one-dimensional systems, analytical solutions for the first moments characterizing unbiased predictions (estimates) of system variables, such as the pressure and fluid–fluid interface position, and we also obtained second moments, which characterize the uncertainties associated with such predictions. Next we obtained empirically scale dependent exponential correlation function of the intrinsic permeability that allowed us to study solutions of stochastic equations as a function of the support scale. We found that the first and second moments converge to asymptotic values as the support scale decreases. In our examples, the statistical moments reached asymptotic values for support scale that were approximately 1/10000 of the flow domain size. We show that analytical moment solutions compare well with the results of Monte Carlo simulations for moderately heterogeneous porous media, and that they can be used to study the effects of heterogeneity on the dynamics and stability of immiscible flow.     

FracKfinder: A MATLAB Toolbox for Computing Three-Dimensional Hydraulic Conductivity Tensors for Fractured Porous Media

Fractures in porous media have been documented extensively. However, they are often omitted from groundwater flow and mass transport models due to a lack of data on fracture hydraulic properties and the computational burden of simulating fractures explicitly in large model domains. We present a MATLAB toolbox, FracKfinder, that automates HydroGeoSphere (HGS), a variably saturated, control volume finite-element model, to simulate an ensemble of discrete fracture network (DFN) flow experiments on a single cubic model mesh containing a stochastically generated fracture network. Because DFN simulations in HGS can simulate flow in both a porous media and a fracture domain, this toolbox computes tensors for both the matrix and fractures of a porous medium. Each model in the ensemble represents a different orientation of the hydraulic gradient, thus minimizing the likelihood that a single hydraulic gradient orientation will dominate the tensor computation. Linear regression on matrices containing the computed three-dimensional hydraulic conductivity (K) values from each rotation of the hydraulic gradient is used to compute the K tensors. This approach shows that the hydraulic behavior of fracture networks can be simulated where fracture hydraulic data are limited. Simulation of a bromide tracer experiment using K tensors computed with FracKfinder in HGS demonstrates good agreement with a previous large-column, laboratory study. The toolbox provides a potential pathway to upscale groundwater flow and mass transport processes in fractured media to larger scales.     

Flow in unsaturated random porous media,nonlinear numerical analysis and comparison to analytical stochastic models

This work presents a rigorous numerical validation of analytical stochastic models of steady state unsaturated flow in heterogeneous porous media. It also provides a crucial link between stochastic theory based on simplifying assumptions and empirical field and simulation evidence of variably saturated flow in actual or realistic hypothetical heterogeneous porous media. Statistical properties of unsaturated hydraulic conductivity, soil water tension, and soil water flux in heterogeneous soils are investigated through high resolution Monte Carlo simulations of a wide range of steady state flow problems in a quasi-unbounded domain. In agreement with assumptions in analytical stochastic models of unsaturated flow, hydraulic conductivity and soil water tension are found to be lognormally and normally distributed, respectively. In contrast, simulations indicate that in moderate to strong variable conductivity fields, longitudinal flux is highly skewed. Transverse flux distributions are leptokurtic. the moments of the probability distributions obtained from Monte Carlo simulations are compared to modified first-order analytical models. Under moderate to strong heterogeneous soil flux conditions (σ²_y≥1), analytical solutions overestimate variability in soil water tension by up to 40% as soil heterogeneity increases, and underestimate variability of both flux components by up to a factor 5. Theoretically predicted model (cross-)covariance agree well with the numerical sample (cross-)covarianaces. Statistical moments are shown to be consistent with observed physical characteristics of unsaturated flow in heterogeneous soils.©1998 Elsevier Science Limited. All rights reserved     

An efficient tool for accelerating the numerical solution of the stochastic subsurface flow problem using neural networks

 An efficient numerical solution for the two-dimensional groundwater flow problem using artificial neural networks (ANNs) is presented. Under stationary velocity conditions with unidirectional mean flow, the conductivity realizations and the head gradients, obtained by a traditional finite difference solution to the flow equation, are given as input-output pairs to train a neural network. The ANN is trained successfully and a certain level of recognition of the relationship between input conductivity patterns and output head gradients is achieved. The trained network produced velocity realizations that are physically plausible without solving the flow equation for each of the conductivity realizations. This is achieved in a small fraction of the time necessary for solving the flow equations. The prediction accuracy of the ANN reaches 97.5% for the longitudinal head gradient and 94.7% for the transverse gradient. Head-gradient and velocity statistics in terms of the first two moments are obtained with a very high accuracy. The cross covariances between head gradients and the fluctuating log-conductivity (log-K) and between velocity and log-K obtained with the ANN approach match very closely those obtained by a traditional numerical solution. The same is true for the velocity components auto-covariances. The results are also extended to transport simulations with very good accuracy. Spatial moments (up to the fourth) of mean-concentration plumes obtained using ANNs are in very good agreement with the traditional Monte Carlo simulations. Furthermore, the concentration second moment (concentration variance) is very close between the two approaches. Considering the fact that higher moments of concentration need more computational effort in numerical simulations, the advantage of the presented approach in saving long computational times is evident. Another advantage of the ANNs approach is the ability to generalize a trained network to conductivity distributions different from those used in training. However, the accuracy of the approach in cases with higher conductivity variances is being investigated.