Shu-Guang Li - Academia.edu (original) (raw)
Papers by Shu-Guang Li
Groundwater, 2019
In west-central Lower Peninsula of Michigan, population growth and expanded agricultural activiti... more In west-central Lower Peninsula of Michigan, population growth and expanded agricultural activities over recent decades have resulted in significant increases in distributed groundwater withdrawals. The growth of the extensive well network and anecdotes of water shortages (dry wells) have raised concerns over the region's groundwater sustainability. We developed an unsteady, three-dimensional (3D) groundwater This article is protected by copyright. All rights reserved. flow model to describe system dynamics over the last 50 years and evaluate long-term impacts of groundwater use. Simulating this large aquifer system was challenging; the site is characterized by strong, spatially distributed and statistically nonstationary heterogeneity, making it difficult to avoid overparameterization using traditional approaches for conceptualizing and calibrating a flow model. Moreover, traditional pumping and water level data were lacking and prohibitively expensive to collect given the large-scale and long-term nature of this study. An integrated, stochastic-deterministic approach was developed to characterize the system and calibrate the flow model through innovative use of highdensity water well datasets. This approached allowed 1) implementation of a 'zone-based', non-stationary stochastic approach to conceptualize complex spatial variability using a small set of geologic material types; 2) modeling the spatiotemporal evolution of many water well withdrawals across several decades using sector-based parameterization; and 3) critical analysis of long-term water level changes at different locations in the aquifer system for characterizing the system dynamics and calibrating the model. Results show the approach is reasonably successful in calibrating a complex model for a highly complex site in a way that honors complex distributed heterogeneity and stress configurations.
Groundwater, 2019
Managing non-point-source (NPS) pollution of groundwater systems is a significant challenge becau... more Managing non-point-source (NPS) pollution of groundwater systems is a significant challenge because of the heterogeneous nature of the subsurface, high costs of data collection, and the multitude of scales involved. In this study, we assessed a particularly complex NPS groundwater pollution problem in Michigan, namely, the salinization of shallow aquifer systems due to natural upwelling of deep brines. We applied a system-based approach to characterize, across multiple scales, the integrated groundwater quantity-quality dynamics associated with the brine upwelling process, assimilating a variety of modeling tools and data-including statewide water well datasets scarcely used for larger scientific analysis. Specifically, we combined 1) data-driven modeling of massive amounts of groundwater/geologic information across multiple spatial scales with 2) detailed analysis of groundwater salinity dynamics and process-based flow modeling at local scales. Statewide "hot-spots" were delineated and county-level severity rankings were developed based on dissolved chloride (Cl-) concentration percentiles. Within local hot spots, the relative impact of upwelling was determined to be controlled by: 1) streams-which act as 'natural pumps' that bring deeper (more mineralized) groundwater to the surface; 2) the occurrence of nearly impervious geologic material at the surface-which restricts freshwater dilution of deeper, saline groundwater; and 3) the space-time evolution of water well withdrawals-which slowly induces migration of saline groundwater from its natural course. This multi-scale, data-intensive approach significantly improved our understanding of the brine upwelling processes in Michigan, and has applicability elsewhere given the growing availability of statewide water well databases.
Hydrological Processes, 2016
Fens, which are among the most biodiverse of wetland types in the USA, typically occur in glacial... more Fens, which are among the most biodiverse of wetland types in the USA, typically occur in glacial landscapes characterized by geo‐morphologic variability at multiple spatial scales. As a result, the hydrologic systems that sustain fens are complex and not well understood. Traditional approaches for characterizing such systems use simplifying assumptions that cannot adequately capture the impact of variability in geology and topography. In this study, a hierarchical, multi‐scale groundwater modelling approach coupled with a geologic model is used to understand the hydrology of a fen in Michigan. This approach uses high‐resolution data to simulate the multi‐scale topographic and hydrologic framework and lithologic data from more than 8500 boreholes in a statewide water well database to capture the complex geology. A hierarchy of dynamically linked models is developed that simulates groundwater flow at all scales of interest and to delineate the areas that contribute groundwater to the fen. The results show the fen receiving groundwater from multiple sources: an adjacent wetland, local recharge, a nearby lake and a regional groundwater mound. Water from the regional mound flows to an intermediate source before reaching the fen, forming a ‘cascading’ connection, while other sources provide water through ‘direct’ connections. The regional mound is also the source of water to other fens, streams and lakes in this area, thus creating a large, interconnected hydrologic system that sustains the entire ecosystem. In order to sustainably manage such systems, conservation efforts must include both site‐based protection and management, as well as regional protection and management of groundwater source areas. Copyright © 2016 John Wiley & Sons, Ltd.
PLOS ONE, 2015
The sources of water and corresponding delivery mechanisms to groundwater-fed fens are not well u... more The sources of water and corresponding delivery mechanisms to groundwater-fed fens are not well understood due to the multi-scale geo-morphologic variability of the glacial landscape in which they occur. This lack of understanding limits the ability to effectively conserve these systems and the ecosystem services they provide, including biodiversity and water provisioning. While fens tend to occur in clusters around regional groundwater mounds, Ives Road Fen in southern Michigan is an example of a geographically-isolated fen. In this paper, we apply a multi-scale groundwater modeling approach to understand the groundwater sources for Ives Road fen. We apply Transition Probability geo-statistics on more than 3000 well logs from a statewide water well database to characterize the complex geology using conditional simulations. We subsequently implement a 3-dimensional reverse particle tracking to delineate groundwater contribution areas to the fen. The fen receives water from multiple sources: local recharge, regional recharge from an extensive till plain, a regional groundwater mound, and a nearby pond. The regional sources deliver water through a tortuous, 3-dimensional "pipeline" consisting of a confined aquifer lying beneath an extensive clay layer. Water in this pipeline reaches the fen by upwelling through openings in the clay layer. The pipeline connects the geographically-isolated fen to the same regional mound that provides water to other fen clusters in southern Michigan. The major implication of these findings is that fen conservation efforts must be expanded from focusing on individual fens and their immediate surroundings, to studying the much larger and interconnected hydrologic network that sustains multiple fens.
Journal of Hydrologic Engineering, 2015
ABSTRACT
Water Resources Research, 1992
Solute transport studies frequently rely on numerical solutions of the classical advection-diffus... more Solute transport studies frequently rely on numerical solutions of the classical advection-diffusion equation. Unfortunately, solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates. Recently developed variants of these techniques such as the finite analytic method (Chen and Li, 1979; Chen and Chen, 1984) and the optimal test function method (Celia et al., 1989a, b, c) perform well for steady state problems. Extensions of these methods to the transient case have, however, not been successful, primarily because of inadequate approximations of the temporal derivative. The new numerical method proposed in this paper avoids this difficulty by taking the Laplace transform of the transient equation. The transformed expression behaves like a steady state advection-diffusion equation with a first-order decay term. This expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse Laplace transform algorithm. The result is an accurate and robust transient solution which performs well over a very wide range of Peclet numbers. We demonstrate this approach by applying the finite analytic method to a Laplace transformed one-dimensional model problem. A comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection-dominated problems. A good space approximation combined with a poor temporal approximation (or vice versa) does not give satisfactory results. The method we propose provides a balanced space-time approximation which works very well for one-dimensional problems. Extensions to multiple dimensions are conceptually straightforward and briefly discussed. O t O x O x •xx ' with the following initial and boundary conditions c(x, O)=f(x); c(0, t)= go(t); c(l, t)= gl(t), (2) where c is the dependent variable (e.g., solute concentration); x is the spatial coordinate (0-< x-< l); t is time (t > 0); u is a steady state velocity; K is a first-order decay coefficient; s(x, t) is a source-sink term;f(x), go(t) and t7t(t) are, respectively, initial and boundary functions; and I is the length of the solution domain. Although analytical solutions of this equation are available for certain special cases, many problems of practical interest (e.g., transport through a heterogeneous velocity field) must be solved numerically. Numerical solutions of the advectiondiffusion equation can be surprisingly difficult, even though it is linear and has a simple mathematical form. Solutions based on conventional finite difference or finite element discretizations are, for example, almost always plagued with either spurious oscillations or excessive numerical diffusion when the advective term becomes dominant. Most traditional solution algorithms are forced to make compromises between these two types of undesirable behavior. The structure of the advection-diffusion equation suggests the use of "operator splitting" techniques which treat the
Water Resources Research, 2004
We present in this paper a critical review of recent research on nonuniform mean flows in heterog... more We present in this paper a critical review of recent research on nonuniform mean flows in heterogeneous porous media, examine why existing stochastic methods are computationally so difficult to implement, and introduce a new and efficient alternative. Specifically, we reformulate the nonstationary spectral method of Li and McLaughlin (1991, 1995) and present a new way for its numerical implementation, combining the best advantages of efficient analytical solutions and flexible numerical techniques. The result is a substantially improved stochastic technique that allows modeling efficiently the nonlinear scale effects for moderately heterogeneous media in the presence of general nonstationarity. In particular, the reformulated approach allows computing the nonlocal and nonstationary mean ''closure'' flux using a coarse grid without having to resolve numerically the small-scale heterogeneous dynamics. The methodological innovation significantly increases the size and expands the range of groundwater problems that can be analyzed with stochastic methods. The effectiveness of the new spectral approach is illustrated with two concrete examples and a systematic comparison with existing stochastic methods.
Water Resources Research, 1991
Stochastic analyses of groundwater flow and transport are frequently based on partial differentia... more Stochastic analyses of groundwater flow and transport are frequently based on partial differential equations which have random coefficients or forcing terms. Analytical methods for solving these equations rely on restrictive assumptions which may not hold in some practical applications. Numerically oriented alternatives are computationally demanding and generally not able to deal with large three-dimensional problems. In this paper we describe a hybrid solution approach which combines classical Fourier transform concepts with numerical solution techniques. Our approach is based on a nonstationary generalization of the spectral representation theorem commonly used in time series analysis. The generalized spectral representation is expressed in terms of an unknown transfer function which depends on space, time, and wave number. The transfer function is found by solving a linearized deterministic partial differential equation which has the same form as the original stochastic flow or transport equation. This approach can accomodate boundary conditions, spatially variable mean gradients, measurement conditioning, and other sources of nonstationarity which cannot be included in classical spectral methods. Here we introduce the nonstationary spectral method and show how it can be used to derive unconditional statistics of interest in groundwater flow and transport applications. use the general term "stationarity" to refer to the widesense case. Stationarity assumptions are advantageous both methodologically and conceptually. They enable us to use a variety of analytical techniques, such as Fourier transform theory, to solve stochastic flow and transport problems (see, for example,
Journal of Hydrology, 2010
ABSTRACT This study presents a hybrid spectral method (HSM) to estimate flow uncertainty in large... more ABSTRACT This study presents a hybrid spectral method (HSM) to estimate flow uncertainty in large-scale highly nonstationary groundwater systems. Taking advantages of spectral theories in solving unmodeled small-scale variability in hydraulic conductivity, the proposed HSM integrates analytical and numerical spectral solutions in the calculation procedures to estimate flow uncertainty. More specifically, the HSM involves two major computational steps after the mean flow equation is solved. The first step is to apply an analytical-based approximate spectral method (ASM) to predict nonstationary flow variances for entire modeling area. The perturbation-based numerical method, nonstationary spectral method (NSM), is then employed in the second step to correct the regional solution in local areas, where the variance dynamics is considered to be highly nonstationary (e.g., around inner boundaries or strong sources/sinks). The boundary conditions for the localized numerical solutions are based on the ASM closed form solutions at boundary nodes. Since the regional closed form solution is instantaneous and the more expensive perturbation-based numerical analysis is only applied locally around the strong stresses, the proposed HSM can be very efficient, making it possible to model strongly nonstationary variance dynamics with complex flow situations in large-scale groundwater systems. In this study the analytical-based ASM solutions was first assessed to quantify the solution accuracy under transient and inner boundary flow conditions. This study then illustrated the HSM accuracy and effectiveness with two synthetic examples. The HSM solutions were systematically compared with the corresponding numerical solutions of NSM and Monte Carlo simulation (MCS), and the analytical-based solutions of ASM. The simulation results have revealed that the HSM is computationally efficient and can provide accurate variance estimations for highly nonstationary large-scale groundwater flow problems.
Journal of Hydraulic Engineering, 1997
A stochastic theory is developed for longitudinal dispersion in natural streams. Irregular variat... more A stochastic theory is developed for longitudinal dispersion in natural streams. Irregular variations in river width and bed elevation are conveniently represented as one-dimensional random fields. Longitudinal solute migration is described by a one-dimensional stochastic solute transport equation. When boundary variations are small and statistically homogeneous, the stochastic transport equation is solved in closed-form using a stochastic spectral technique. The results show that large scale longitudinal transport can be represented as a gradient dispersion process described by an effective longitudinal dispersion coefficient. The effective coefficient reflects longitudinal mixing due to flow variation both within the river cross section and along the flow and can be considerably greater than that of corresponding uniform channels. The discrepancy between uniform channels and natural rivers increases as the variances of river width and bed elevation increase, especially when the mean flow Froude number is high.
Advances in Water Resources, 2004
Despite the intensive research over the past two decades in the field of stochastic subsurface hy... more Despite the intensive research over the past two decades in the field of stochastic subsurface hydrology, a substantial gap remains between theory and application. The most popular stochastic theories are still based on closed-form solutions that apply, strictly speaking, only for statistically uniform flows. In this paper, we present an efficient, nonstationary spectral approach for modeling complex stochastic flows in moderately heterogeneous media. Specifically, we reformulate the governing stochastic equations and introduce a new transfer function to characterize the propagation of system uncertainty. The new transfer function plays a similar role as the commonly used GreenÕs functions in classical stochastic perturbation methods but is more amenable to numerical solution. The compact transfer function can be used to construct efficiently various spatial statistics of interest, such as covariances, cross-covariances, variances, and mean closure fluxes. We demonstrate the advantages of the proposed approach by applying it to a number of nonstationary problems, including a large, complex problem that is difficult to solve by traditional methods. In particular, we focus in this paper on demonstrating the new approachÕs ability to compute efficiently covariances and cross-covariances critical for measurement conditioning, monitoring network analyses, and stochastic transport modeling in the presence of complex mean flow nonstationarities (caused, e.g., by complex trends in aquifer properties, boundary conditions, and sources and sinks). This paper is an extension of our recent work that illustrated the basic approach for modeling nonlocal and nonstationary scale effects and uncertainty propagation in relatively simple situations.
Advances in Water Resources, 2003
Stochastic theories of subsurface flow and transport have changed the way we think about heteroge... more Stochastic theories of subsurface flow and transport have changed the way we think about heterogeneity but have not had much impact on practical groundwater modeling. Most numerical models still provide no information on prediction uncertainty. This gap between theory and practice is due largely to the excessive computational demands of available numerical methods for solving stochastic problems. The two primary alternatives, Monte Carlo and classical perturbation methods, both require solving large numbers of equations on fine grid spacings that are smaller than the log conductivity correlation length, no matter how smooth the mean flow and variance dynamics of interest are. This results in computational times that are many orders-of-magnitude greater than those required for conventional deterministic simulations. In addition, classical perturbation methods need much more real memory than comparable deterministic simulations. In this paper, we present an innovative approach for stochastic groundwater modeling-one that finally allows taking advantage of the scale disparity between the mean distributions and small-scale process. The new approach, which is called the nonstationary spectral method, implicitly separates the problem into rapidly varying random (stationary) part and a slowly varying deterministic (nonstationary) part and only the latter needs to be solved numerically. A general analysis and a specific numerical example both demonstrate that the nonstationary spectral method is much more efficient than the available alternatives. The methodological innovation dramatically increases the size and range of the problems that can be solved using stochastic methods and represents a major step forward in our effort to make stochastic uncertainty analysis a routine part of groundwater modeling.
Advances in Water Resources, 2004
Perturbation methods are of interest to hydrologists because they provide a way to incorporate up... more Perturbation methods are of interest to hydrologists because they provide a way to incorporate upscaling and accuracy assessment capabilities into practical groundwater models. In particular, these methods may be used to obtain approximate solutions for the ensemble moments of solute concentration and for related effective properties in randomly heterogeneous porous media. Although the many perturbation methods proposed in the groundwater literature seem to be rather different they are actually closely related. For example, we show here through a systematic analysis that the second-order asymptotic expansion approach [Flow and Transport in Porous Formations, Springer-Verlag, Berlin, 1989], random Green's function methods [Dispersion in Heterogeneous Geological Formations, Kluwer Academic Publishers, the Netherlands, 2001], and Taylor series methods [Water Resour. Res. 32(1) (1996) 85; Water Resour. Res. 28(12) (1992) 3269; Water Resour. Res. 21(3) (1985) 359] all give identical expressions for the ensemble mean, for random fluctuations about this mean, and for related closure covariances. The Eulerian truncation approach [Water Resour. Res. 14(2) (1978) 263; Water Resour. Res. 19(1) (1983) 161; Phys. Fluids 31 (1988) 965; Water Resour. Res. 27(7) (1991) 1598] gives expressions similar to the other methods except that its closure covariances depend on the unknown ensemble mean rather than the known deterministic solution (the solution obtained when random velocities are replaced by their means). This reflects the fact that Eulerian truncation works with perturbations taken about the mean while the other methods work with perturbations taken about the deterministic solution. The ensemble mean accounts for the large-scale effects of small-scale variability while the deterministic solution does not. Eulerian truncation solutions can be more difficult to compute than other alternatives but they can also be more accurate. This is demonstrated for the case of Gaussian velocities with a general truncation analysis and with a specific numerical example. The results provided here suggest that Eulerian truncation is superior to alternative perturbation methods of comparable computational complexity.
Journal of Hydrology, 2015
In this paper, we present a real-world demonstration of a generalized hierarchical approach for m... more In this paper, we present a real-world demonstration of a generalized hierarchical approach for modeling complex groundwater systems, the hierarchical patch dynamics paradigm (HPDP). In particular, we illustrate how the HPDP approach enables flexible and efficient simulation of a complex contaminant capture system at one of the largest groundwater pump-and-treat remediation operations in Michigan. The groundwater flow system at the site exhibits a multi-scale pattern that is difficult to simulate using standard modeling tools because of the complex interaction between ambient hydrologic stresses and on-site remediation operations. The hierarchical modeling system was calibrated to water level measurements collected from 208 monitoring wells located both on-site and in its immediate proximity and flux measurements from 6 trenches on-site. Systematic hierarchical simulations, including forward and reverse particle tracking as well as integrated water budget analyses, were performed to study the ongoing remediation. The hierarchical modeling results show that some contamination leaked off-site because of small-scale inefficiencies in the design of the remediation system. Thus, the HPDP approach provides an opportunity to analyze complex hydrological field environments in a pragmatic, time-efficient manner. Published by Elsevier B.V. 1.1. Local grid refinement Local grid refinement is the process of subdividing relatively large-sized cells in a numerical model into cells of smaller spatial
Groundwater, 2019
In west-central Lower Peninsula of Michigan, population growth and expanded agricultural activiti... more In west-central Lower Peninsula of Michigan, population growth and expanded agricultural activities over recent decades have resulted in significant increases in distributed groundwater withdrawals. The growth of the extensive well network and anecdotes of water shortages (dry wells) have raised concerns over the region's groundwater sustainability. We developed an unsteady, three-dimensional (3D) groundwater This article is protected by copyright. All rights reserved. flow model to describe system dynamics over the last 50 years and evaluate long-term impacts of groundwater use. Simulating this large aquifer system was challenging; the site is characterized by strong, spatially distributed and statistically nonstationary heterogeneity, making it difficult to avoid overparameterization using traditional approaches for conceptualizing and calibrating a flow model. Moreover, traditional pumping and water level data were lacking and prohibitively expensive to collect given the large-scale and long-term nature of this study. An integrated, stochastic-deterministic approach was developed to characterize the system and calibrate the flow model through innovative use of highdensity water well datasets. This approached allowed 1) implementation of a 'zone-based', non-stationary stochastic approach to conceptualize complex spatial variability using a small set of geologic material types; 2) modeling the spatiotemporal evolution of many water well withdrawals across several decades using sector-based parameterization; and 3) critical analysis of long-term water level changes at different locations in the aquifer system for characterizing the system dynamics and calibrating the model. Results show the approach is reasonably successful in calibrating a complex model for a highly complex site in a way that honors complex distributed heterogeneity and stress configurations.
Groundwater, 2019
Managing non-point-source (NPS) pollution of groundwater systems is a significant challenge becau... more Managing non-point-source (NPS) pollution of groundwater systems is a significant challenge because of the heterogeneous nature of the subsurface, high costs of data collection, and the multitude of scales involved. In this study, we assessed a particularly complex NPS groundwater pollution problem in Michigan, namely, the salinization of shallow aquifer systems due to natural upwelling of deep brines. We applied a system-based approach to characterize, across multiple scales, the integrated groundwater quantity-quality dynamics associated with the brine upwelling process, assimilating a variety of modeling tools and data-including statewide water well datasets scarcely used for larger scientific analysis. Specifically, we combined 1) data-driven modeling of massive amounts of groundwater/geologic information across multiple spatial scales with 2) detailed analysis of groundwater salinity dynamics and process-based flow modeling at local scales. Statewide "hot-spots" were delineated and county-level severity rankings were developed based on dissolved chloride (Cl-) concentration percentiles. Within local hot spots, the relative impact of upwelling was determined to be controlled by: 1) streams-which act as 'natural pumps' that bring deeper (more mineralized) groundwater to the surface; 2) the occurrence of nearly impervious geologic material at the surface-which restricts freshwater dilution of deeper, saline groundwater; and 3) the space-time evolution of water well withdrawals-which slowly induces migration of saline groundwater from its natural course. This multi-scale, data-intensive approach significantly improved our understanding of the brine upwelling processes in Michigan, and has applicability elsewhere given the growing availability of statewide water well databases.
Hydrological Processes, 2016
Fens, which are among the most biodiverse of wetland types in the USA, typically occur in glacial... more Fens, which are among the most biodiverse of wetland types in the USA, typically occur in glacial landscapes characterized by geo‐morphologic variability at multiple spatial scales. As a result, the hydrologic systems that sustain fens are complex and not well understood. Traditional approaches for characterizing such systems use simplifying assumptions that cannot adequately capture the impact of variability in geology and topography. In this study, a hierarchical, multi‐scale groundwater modelling approach coupled with a geologic model is used to understand the hydrology of a fen in Michigan. This approach uses high‐resolution data to simulate the multi‐scale topographic and hydrologic framework and lithologic data from more than 8500 boreholes in a statewide water well database to capture the complex geology. A hierarchy of dynamically linked models is developed that simulates groundwater flow at all scales of interest and to delineate the areas that contribute groundwater to the fen. The results show the fen receiving groundwater from multiple sources: an adjacent wetland, local recharge, a nearby lake and a regional groundwater mound. Water from the regional mound flows to an intermediate source before reaching the fen, forming a ‘cascading’ connection, while other sources provide water through ‘direct’ connections. The regional mound is also the source of water to other fens, streams and lakes in this area, thus creating a large, interconnected hydrologic system that sustains the entire ecosystem. In order to sustainably manage such systems, conservation efforts must include both site‐based protection and management, as well as regional protection and management of groundwater source areas. Copyright © 2016 John Wiley & Sons, Ltd.
PLOS ONE, 2015
The sources of water and corresponding delivery mechanisms to groundwater-fed fens are not well u... more The sources of water and corresponding delivery mechanisms to groundwater-fed fens are not well understood due to the multi-scale geo-morphologic variability of the glacial landscape in which they occur. This lack of understanding limits the ability to effectively conserve these systems and the ecosystem services they provide, including biodiversity and water provisioning. While fens tend to occur in clusters around regional groundwater mounds, Ives Road Fen in southern Michigan is an example of a geographically-isolated fen. In this paper, we apply a multi-scale groundwater modeling approach to understand the groundwater sources for Ives Road fen. We apply Transition Probability geo-statistics on more than 3000 well logs from a statewide water well database to characterize the complex geology using conditional simulations. We subsequently implement a 3-dimensional reverse particle tracking to delineate groundwater contribution areas to the fen. The fen receives water from multiple sources: local recharge, regional recharge from an extensive till plain, a regional groundwater mound, and a nearby pond. The regional sources deliver water through a tortuous, 3-dimensional "pipeline" consisting of a confined aquifer lying beneath an extensive clay layer. Water in this pipeline reaches the fen by upwelling through openings in the clay layer. The pipeline connects the geographically-isolated fen to the same regional mound that provides water to other fen clusters in southern Michigan. The major implication of these findings is that fen conservation efforts must be expanded from focusing on individual fens and their immediate surroundings, to studying the much larger and interconnected hydrologic network that sustains multiple fens.
Journal of Hydrologic Engineering, 2015
ABSTRACT
Water Resources Research, 1992
Solute transport studies frequently rely on numerical solutions of the classical advection-diffus... more Solute transport studies frequently rely on numerical solutions of the classical advection-diffusion equation. Unfortunately, solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates. Recently developed variants of these techniques such as the finite analytic method (Chen and Li, 1979; Chen and Chen, 1984) and the optimal test function method (Celia et al., 1989a, b, c) perform well for steady state problems. Extensions of these methods to the transient case have, however, not been successful, primarily because of inadequate approximations of the temporal derivative. The new numerical method proposed in this paper avoids this difficulty by taking the Laplace transform of the transient equation. The transformed expression behaves like a steady state advection-diffusion equation with a first-order decay term. This expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse Laplace transform algorithm. The result is an accurate and robust transient solution which performs well over a very wide range of Peclet numbers. We demonstrate this approach by applying the finite analytic method to a Laplace transformed one-dimensional model problem. A comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection-dominated problems. A good space approximation combined with a poor temporal approximation (or vice versa) does not give satisfactory results. The method we propose provides a balanced space-time approximation which works very well for one-dimensional problems. Extensions to multiple dimensions are conceptually straightforward and briefly discussed. O t O x O x •xx ' with the following initial and boundary conditions c(x, O)=f(x); c(0, t)= go(t); c(l, t)= gl(t), (2) where c is the dependent variable (e.g., solute concentration); x is the spatial coordinate (0-< x-< l); t is time (t > 0); u is a steady state velocity; K is a first-order decay coefficient; s(x, t) is a source-sink term;f(x), go(t) and t7t(t) are, respectively, initial and boundary functions; and I is the length of the solution domain. Although analytical solutions of this equation are available for certain special cases, many problems of practical interest (e.g., transport through a heterogeneous velocity field) must be solved numerically. Numerical solutions of the advectiondiffusion equation can be surprisingly difficult, even though it is linear and has a simple mathematical form. Solutions based on conventional finite difference or finite element discretizations are, for example, almost always plagued with either spurious oscillations or excessive numerical diffusion when the advective term becomes dominant. Most traditional solution algorithms are forced to make compromises between these two types of undesirable behavior. The structure of the advection-diffusion equation suggests the use of "operator splitting" techniques which treat the
Water Resources Research, 2004
We present in this paper a critical review of recent research on nonuniform mean flows in heterog... more We present in this paper a critical review of recent research on nonuniform mean flows in heterogeneous porous media, examine why existing stochastic methods are computationally so difficult to implement, and introduce a new and efficient alternative. Specifically, we reformulate the nonstationary spectral method of Li and McLaughlin (1991, 1995) and present a new way for its numerical implementation, combining the best advantages of efficient analytical solutions and flexible numerical techniques. The result is a substantially improved stochastic technique that allows modeling efficiently the nonlinear scale effects for moderately heterogeneous media in the presence of general nonstationarity. In particular, the reformulated approach allows computing the nonlocal and nonstationary mean ''closure'' flux using a coarse grid without having to resolve numerically the small-scale heterogeneous dynamics. The methodological innovation significantly increases the size and expands the range of groundwater problems that can be analyzed with stochastic methods. The effectiveness of the new spectral approach is illustrated with two concrete examples and a systematic comparison with existing stochastic methods.
Water Resources Research, 1991
Stochastic analyses of groundwater flow and transport are frequently based on partial differentia... more Stochastic analyses of groundwater flow and transport are frequently based on partial differential equations which have random coefficients or forcing terms. Analytical methods for solving these equations rely on restrictive assumptions which may not hold in some practical applications. Numerically oriented alternatives are computationally demanding and generally not able to deal with large three-dimensional problems. In this paper we describe a hybrid solution approach which combines classical Fourier transform concepts with numerical solution techniques. Our approach is based on a nonstationary generalization of the spectral representation theorem commonly used in time series analysis. The generalized spectral representation is expressed in terms of an unknown transfer function which depends on space, time, and wave number. The transfer function is found by solving a linearized deterministic partial differential equation which has the same form as the original stochastic flow or transport equation. This approach can accomodate boundary conditions, spatially variable mean gradients, measurement conditioning, and other sources of nonstationarity which cannot be included in classical spectral methods. Here we introduce the nonstationary spectral method and show how it can be used to derive unconditional statistics of interest in groundwater flow and transport applications. use the general term "stationarity" to refer to the widesense case. Stationarity assumptions are advantageous both methodologically and conceptually. They enable us to use a variety of analytical techniques, such as Fourier transform theory, to solve stochastic flow and transport problems (see, for example,
Journal of Hydrology, 2010
ABSTRACT This study presents a hybrid spectral method (HSM) to estimate flow uncertainty in large... more ABSTRACT This study presents a hybrid spectral method (HSM) to estimate flow uncertainty in large-scale highly nonstationary groundwater systems. Taking advantages of spectral theories in solving unmodeled small-scale variability in hydraulic conductivity, the proposed HSM integrates analytical and numerical spectral solutions in the calculation procedures to estimate flow uncertainty. More specifically, the HSM involves two major computational steps after the mean flow equation is solved. The first step is to apply an analytical-based approximate spectral method (ASM) to predict nonstationary flow variances for entire modeling area. The perturbation-based numerical method, nonstationary spectral method (NSM), is then employed in the second step to correct the regional solution in local areas, where the variance dynamics is considered to be highly nonstationary (e.g., around inner boundaries or strong sources/sinks). The boundary conditions for the localized numerical solutions are based on the ASM closed form solutions at boundary nodes. Since the regional closed form solution is instantaneous and the more expensive perturbation-based numerical analysis is only applied locally around the strong stresses, the proposed HSM can be very efficient, making it possible to model strongly nonstationary variance dynamics with complex flow situations in large-scale groundwater systems. In this study the analytical-based ASM solutions was first assessed to quantify the solution accuracy under transient and inner boundary flow conditions. This study then illustrated the HSM accuracy and effectiveness with two synthetic examples. The HSM solutions were systematically compared with the corresponding numerical solutions of NSM and Monte Carlo simulation (MCS), and the analytical-based solutions of ASM. The simulation results have revealed that the HSM is computationally efficient and can provide accurate variance estimations for highly nonstationary large-scale groundwater flow problems.
Journal of Hydraulic Engineering, 1997
A stochastic theory is developed for longitudinal dispersion in natural streams. Irregular variat... more A stochastic theory is developed for longitudinal dispersion in natural streams. Irregular variations in river width and bed elevation are conveniently represented as one-dimensional random fields. Longitudinal solute migration is described by a one-dimensional stochastic solute transport equation. When boundary variations are small and statistically homogeneous, the stochastic transport equation is solved in closed-form using a stochastic spectral technique. The results show that large scale longitudinal transport can be represented as a gradient dispersion process described by an effective longitudinal dispersion coefficient. The effective coefficient reflects longitudinal mixing due to flow variation both within the river cross section and along the flow and can be considerably greater than that of corresponding uniform channels. The discrepancy between uniform channels and natural rivers increases as the variances of river width and bed elevation increase, especially when the mean flow Froude number is high.
Advances in Water Resources, 2004
Despite the intensive research over the past two decades in the field of stochastic subsurface hy... more Despite the intensive research over the past two decades in the field of stochastic subsurface hydrology, a substantial gap remains between theory and application. The most popular stochastic theories are still based on closed-form solutions that apply, strictly speaking, only for statistically uniform flows. In this paper, we present an efficient, nonstationary spectral approach for modeling complex stochastic flows in moderately heterogeneous media. Specifically, we reformulate the governing stochastic equations and introduce a new transfer function to characterize the propagation of system uncertainty. The new transfer function plays a similar role as the commonly used GreenÕs functions in classical stochastic perturbation methods but is more amenable to numerical solution. The compact transfer function can be used to construct efficiently various spatial statistics of interest, such as covariances, cross-covariances, variances, and mean closure fluxes. We demonstrate the advantages of the proposed approach by applying it to a number of nonstationary problems, including a large, complex problem that is difficult to solve by traditional methods. In particular, we focus in this paper on demonstrating the new approachÕs ability to compute efficiently covariances and cross-covariances critical for measurement conditioning, monitoring network analyses, and stochastic transport modeling in the presence of complex mean flow nonstationarities (caused, e.g., by complex trends in aquifer properties, boundary conditions, and sources and sinks). This paper is an extension of our recent work that illustrated the basic approach for modeling nonlocal and nonstationary scale effects and uncertainty propagation in relatively simple situations.
Advances in Water Resources, 2003
Stochastic theories of subsurface flow and transport have changed the way we think about heteroge... more Stochastic theories of subsurface flow and transport have changed the way we think about heterogeneity but have not had much impact on practical groundwater modeling. Most numerical models still provide no information on prediction uncertainty. This gap between theory and practice is due largely to the excessive computational demands of available numerical methods for solving stochastic problems. The two primary alternatives, Monte Carlo and classical perturbation methods, both require solving large numbers of equations on fine grid spacings that are smaller than the log conductivity correlation length, no matter how smooth the mean flow and variance dynamics of interest are. This results in computational times that are many orders-of-magnitude greater than those required for conventional deterministic simulations. In addition, classical perturbation methods need much more real memory than comparable deterministic simulations. In this paper, we present an innovative approach for stochastic groundwater modeling-one that finally allows taking advantage of the scale disparity between the mean distributions and small-scale process. The new approach, which is called the nonstationary spectral method, implicitly separates the problem into rapidly varying random (stationary) part and a slowly varying deterministic (nonstationary) part and only the latter needs to be solved numerically. A general analysis and a specific numerical example both demonstrate that the nonstationary spectral method is much more efficient than the available alternatives. The methodological innovation dramatically increases the size and range of the problems that can be solved using stochastic methods and represents a major step forward in our effort to make stochastic uncertainty analysis a routine part of groundwater modeling.
Advances in Water Resources, 2004
Perturbation methods are of interest to hydrologists because they provide a way to incorporate up... more Perturbation methods are of interest to hydrologists because they provide a way to incorporate upscaling and accuracy assessment capabilities into practical groundwater models. In particular, these methods may be used to obtain approximate solutions for the ensemble moments of solute concentration and for related effective properties in randomly heterogeneous porous media. Although the many perturbation methods proposed in the groundwater literature seem to be rather different they are actually closely related. For example, we show here through a systematic analysis that the second-order asymptotic expansion approach [Flow and Transport in Porous Formations, Springer-Verlag, Berlin, 1989], random Green's function methods [Dispersion in Heterogeneous Geological Formations, Kluwer Academic Publishers, the Netherlands, 2001], and Taylor series methods [Water Resour. Res. 32(1) (1996) 85; Water Resour. Res. 28(12) (1992) 3269; Water Resour. Res. 21(3) (1985) 359] all give identical expressions for the ensemble mean, for random fluctuations about this mean, and for related closure covariances. The Eulerian truncation approach [Water Resour. Res. 14(2) (1978) 263; Water Resour. Res. 19(1) (1983) 161; Phys. Fluids 31 (1988) 965; Water Resour. Res. 27(7) (1991) 1598] gives expressions similar to the other methods except that its closure covariances depend on the unknown ensemble mean rather than the known deterministic solution (the solution obtained when random velocities are replaced by their means). This reflects the fact that Eulerian truncation works with perturbations taken about the mean while the other methods work with perturbations taken about the deterministic solution. The ensemble mean accounts for the large-scale effects of small-scale variability while the deterministic solution does not. Eulerian truncation solutions can be more difficult to compute than other alternatives but they can also be more accurate. This is demonstrated for the case of Gaussian velocities with a general truncation analysis and with a specific numerical example. The results provided here suggest that Eulerian truncation is superior to alternative perturbation methods of comparable computational complexity.
Journal of Hydrology, 2015
In this paper, we present a real-world demonstration of a generalized hierarchical approach for m... more In this paper, we present a real-world demonstration of a generalized hierarchical approach for modeling complex groundwater systems, the hierarchical patch dynamics paradigm (HPDP). In particular, we illustrate how the HPDP approach enables flexible and efficient simulation of a complex contaminant capture system at one of the largest groundwater pump-and-treat remediation operations in Michigan. The groundwater flow system at the site exhibits a multi-scale pattern that is difficult to simulate using standard modeling tools because of the complex interaction between ambient hydrologic stresses and on-site remediation operations. The hierarchical modeling system was calibrated to water level measurements collected from 208 monitoring wells located both on-site and in its immediate proximity and flux measurements from 6 trenches on-site. Systematic hierarchical simulations, including forward and reverse particle tracking as well as integrated water budget analyses, were performed to study the ongoing remediation. The hierarchical modeling results show that some contamination leaked off-site because of small-scale inefficiencies in the design of the remediation system. Thus, the HPDP approach provides an opportunity to analyze complex hydrological field environments in a pragmatic, time-efficient manner. Published by Elsevier B.V. 1.1. Local grid refinement Local grid refinement is the process of subdividing relatively large-sized cells in a numerical model into cells of smaller spatial