Hydraulic Conductivity Tensor Research Articles

Deep learning surrogate for predicting hydraulic conductivity tensors from stochastic discrete fracture-matrix models

Deep learning surrogate for predicting hydraulic conductivity tensors from stochastic discrete fracture-matrix models

Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model

This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions.

Hydraulic conductivity tensor of anisotropic soils: the impact on seepage flow

Natural or artificial changes in soil stratigraphic-plane (i.e., the deposition of soil and sediments into distinct layers) orientation can cause variations in hydraulic conductivity in different direction. Hydraulic conductivity must, therefore, be considered – in this case – as a nondiagonal, full tensor to appropriately represent the effect of the orientation of soil stratigraphic planes on the seepage flow pattern. This paper introduces the derivation of the formula for the three-dimensional nondiagonal hydraulic conductivity full tensor calculation in terms of azimuth and vertical angles. Furthermore, two-dimensional numerical simulations of the seepage flow beneath a concrete dam are presented to demonstrate the need to account for the nondiagonal elements of the hydraulic conductivity tensor in anisotropic soils of varying degrees of stratigraphic tilt with respect to the coordinate system.

Depth-dependent hydraulic conductivity tensor for jointed granite

Stress-dependent hydraulic conductivity tensor is formulated in a closed form on the basis of Oda's permeability tensor and Bianchi & Snow's proposal for change of the crack aperture. Using a simple cracked model, a general equation is derived to obtain the hydraulic conductivity tensor as a function of depth and the coefficient of earth pressure at rest. The general trend of the equation is consistent with the in-situ hydraulic conductivity values measured at various sites. Conductivity itself does not have any practical meaning unless the state of stress is concurrently specified. Anisotropy in the hydraulic conductivity tensor is mainly related to the state of stress given by the earth pressure coefficient rather than the crack geometry, particularly in the cases where three orthogonal sets of joints are observed. Anisotropy is magnified in two dimensions because of excessive confinement of the flow direction. Additional procedures should be performed when the results obtained from two-dimensional analyses are generalized for three-dimensional real sites. The hydraulic conductivity K0 at the earth surface and γ′H for controlling the aperture of cracks are proposed as index measures for characterizing a given site from a comprehensive point of view.

A new semi-analytical method for estimation of anisotropic hydraulic conductivity of three-dimensional fractured rock masses

• A semi-analytical method based on Snow's peameability tensor model is proposed. • The new method can predict anisotropic hydraulic conductivity of complex DFNs. • The new method can deal with engineering level problems with low computational cost. The hydraulic characteristics of rock masses usually show strong anisotropy due to the existence of oriented structure planes. A new semi-analytical method is proposed for estimation of the anisotropic hydraulic conductivity of fractured rock masses based on the discrete fracture network (DFN) description and the Snow’s model ( Snow, 1969 ). In this method, the number of intersections on a fracture is a key parameter to construct correction coefficient, which are used to adjust the contribution of each individual fracture to the macro hydraulic conductivity tensor. In this way, the initial Snow’s model is extended from infinite connectivity conditions to finite connectivity conditions. The correction coefficient is determined by using a mathematical optimization method in the condition of networks following isotropic orientation distribution and polydisperse size distribution. Then, the new method is applied to estimate the directional hydraulic conductivity of engineering rock masses with complex anisotropic oriented fracture networks. The results show that the directional hydraulic conductivities predicted by the proposed semi-analytical method are in good agreement with the referential numerical results. The proposed semi-analytical model can be used to predict the anisotropic hydraulic conductivity of complex fractured rock masses with less processing time and lower memory requirements than numerical models.

Research on Seepage Control of Jurong Pumped Storage Hydroelectric Power Station

Based on the geological and hydrogeological conditions of the Jurong Pumped Storage Hydroelectric Power Station (JPSHP), a 3D groundwater flow model was developed in the power station area, which took into account the heterogeneity and anisotropy of fractured rocks. A control inversion method for fractured rock structural planes was proposed, where larger-scale fractures were used as water-conducting media and the relatively intact rock matrix was used as water-storage media. A statistical method was used to obtain the geometric parameter values of the structural planes, so as to obtain the hydraulic conductivity tensor of the fractured rocks. Combining the impermeable drainage systems of the upper storage reservoir, underground powerhouse and lower storage reservoir, the 3D groundwater seepage field in the study area was predicted using the calibrated model. The leakage amounts of the upper storage reservoir, powerhouse and lower storage reservoir were 710.48 m3/d, 969.95 m3/d and 1657.55 m3/d, respectively. The leakage changes of the upper storage reservoir, powerhouse and lower storage reservoir were discussed under the partial and full failure of the anti-seepage system. The research results provide a scientific basis for the seepage control of the power station, and it is recommended to strengthen the seepage control of the upper and lower storage reservoirs and the underground powerhouse to avoid excessive leakage and affect the efficiency of the reservoir operation.

An improved numerical model for groundwater flow simulation with MPFA method on arbitrary polygon grids

Groundwater models are critical for simulating subsurface hydrological processes and guiding informed policy-making for groundwater management. However, the widely applied groundwater models typically use regular-shaped grids to discretize aquifer systems and require that the directions of the grid edges are aligned with the hydraulic conductivity tensor. Such rigorous requirements for spatial discretization have constrained the models’ application in aquifer systems with anisotropic hydrogeological characteristics. To address such limitations, we develop an improved groundwater flow model based on the multipoint flux approximation (MPFA) method in this study. The new model allows us to use arbitrary-shaped polygon grids to discretize aquifer systems and relaxes the rigorous requirement of the alliance between polygon edges and hydraulic conductivity tensor. The functionality and performance of the new model are demonstrated by comparing the output between our model, MODFLOW, and analytical solution in four case studies with various hydrogeological conditions. In a real-world watershed with complex-shaped boundaries, our model outperforms the conventional groundwater model in boundaries. The modeling results show that our model can yield accurate simulation of subsurface hydrological processes in aquifer systems with complex-shaped boundaries. Furthermore, our model can provide a more flexible discretization solution to couple surface water and groundwater model in integrated hydrological model development.

Numerical analysis of a second order ensemble algorithm for numerical approximation of stochastic Stokes–Darcy equations

Numerical approximation of stochastic Stokes–Darcy equations usually requires repeated sampling of the random hydraulic conductivity tensor and then simulating flow ensembles. In this setting, we propose an efficient, second order, ensemble algorithm for fast computation of the whole set of realizations of the stochastic Stokes–Darcy model corresponding to different random hydraulic conductivity tensor samples. The ensemble algorithm only requires the solution of two linear systems that have the same constant coefficient matrices for all realizations. We give a complete long time stability and convergence analysis for the method. Numerical experiments are presented to support theoretical results and demonstrate the application of the method.

A karst networks generation model based on the anisotropic Fast Marching algorithm

Karst aquifers are difficult to model due to their complex conduit networks that are heterogeneous and anisotropic. Here, we present a heuristic algorithm for modeling karst networks by calculating minimum-effort paths based on our knowledge of karst evolution that is constrained by field observations.The methodology consists of a) using discrete fracture network models along with fracture-parameter estimates (e.g., frequency, extent, spacing, and apertures) to equivalent porous media properties (e.g., size of the representative elementary volume, hydraulic conductivity tensor), b) rendering the velocity field using the tensor field coupled with the 3D regional hydrogeology that includes likely recharge and discharge locations, and c) generating karst conduits using the Anisotropic Fast Marching Algorithm (AFMA). This approach a) avoids the need to model discrete fractures by replacing them with a tensor that incorporates small-scale, dense and interconnected fractures, b) assumes that conduits follow the minimum-effort path from sinkholes toward springs, and c) efficiently accounts for regional hydrogeology, anisotropy, and regional flow conditions within 3D conduit networks. The application of the proposed approach in Wulongdong Basin in Hubei Province, China is taken as the demonstrate example. Compared to other techniques, this methodology substantially improves computational efficiency and reduces computer memory, thus providing a useful tool for generating ensembles of possible karstic conduit networks to analyze flow and transport prediction uncertainty associated with a lack of knowledge about network geometry.

Optimization of the Groundwater Remediation Process Using a Coupled Genetic Algorithm-Finite Difference Method

In situ chemical oxidation using permanganate as an oxidant is a remediation technique often used to treat contaminated groundwater. In this paper, groundwater flow with a full hydraulic conductivity tensor and remediation process through in situ chemical oxidation are simulated. The numerical approach was verified with a physical sandbox experiment and analytical solution for 2D advection-diffusion with a first-order decay rate constant. The numerical results were in good agreement with the results of physical sandbox model and the analytical solution. The developed model was applied to two different studies, using multi-objective genetic algorithm to optimise remediation design. In order to reach the optimised design, three objectives considering three constraints were defined. The time to reach the desired concentration and remediation cost regarding the number of required oxidant sources in the optimised design was less than any arbitrary design.

Stochastic correlated hydraulic conductivity tensor calibration using gradual deformation

Quasi-point hydraulic properties (K) measured locally under laboratory or field conditions need to be upscaled to block-scale K-tensors for use in flow simulators. The upscaled model also needs to be calibrated to hydraulic head observations. The calibration must preserve spatial covariance, cross-covariance and non-linear relations between tensor components. We apply a new upscaling method that allows to compute and model the covariance between block K-tensor components. We use a gradual deformation method for calibration of simulated K-tensor fields to measured head data. Our method incorporates a new bivariate transform that preserves the non-linear relations between K-tensor components. The ensemble of calibrated realizations allows quantification of uncertainty of groundwater flow models. A comparison with PEST on a test case defining capture zones for water supply wells shows that our method calibrates better to measured heads than PEST, provides more realistic K-tensors and results in larger capture zones.

Joint Optimization of Measurement and Modeling Strategies With Application to Radial Flow in Stratified Aquifers

AbstractWhen applying environmental models, the choice of model complexity and the design of field campaigns depend on each other and on the modeling/prediction goal. We propose jointly optimizing model complexity and data collection (design) by maximizing the expected performance for the modeling goal. We use ensembles of highly resolved virtual realities and of less complex modeling variants that differ in their degrees of upscaling and simplified parameterization. For each design under consideration, we simulate hypothetical measurement data (subject to noise) with all realizations of all models. To mimic model calibration with hypothetical data, we identify pairs of best fitting realizations between virtual reality and each model variant for each design. Then, we emulate model choice by selecting (across the model variants, for each design and for each virtual reality) the pair that shows the best predictive match in the modeling goal. Finally, we identify the model/design combination that offers, on average over all virtual realities, the best predictive match. As a test application, we consider a heterogeneous, stratified aquifer, in which the stratification enhances hydraulic anisotropy on the macroscale. We define two different modeling goals: (a) estimating the hydraulic conductivity tensor upscaled to the full aquifer thickness and (b) predicting the pumping rate needed to dewater a construction pit. Our results indicate that jointly optimizing observation networks and model selection can reduce the prediction uncertainty of parameters at lower experimental costs. We also show that the involved trade‐offs between model complexity and required design depend on the target quantity.

Estimation of the representative elementary volume of a fractured till: a field and groundwater modeling approach

Fractured till is often represented as an equivalent porous medium (EPM) in groundwater models. Knowledge of the representative elementary volume (REV) is necessary for proper application of an EPM model. While REV estimation and hydraulic conductivity tensor determinations are common in fractured rock studies, they are rarely applied to materials with a permeable matrix such as fractured till. This study uses field fracture measurements, model simulations, and the FracKFinder toolbox to estimate the REV and determine hydraulic conductivity tensors for the fractured, late Wisconsinan till of the Dows Formation in central Iowa (USA), at depths of 1.0–1.5, 2.0–2.5, and 3.3–3.7 m. Results indicate that the REV of the till is directly related to both fracture density and transmissivity. REV size ranges from 4 to 5 m3 at depths of 1–2 m, where fractures are densest (P32 27.5 m2/m3) and most transmissive, to 2–3 m3 at depths of 3.3–3.7 m where fractures are least dense (P32 24.4 m2/m3) and least transmissive. Fracture hydraulic conductivity is five, four, and three orders of magnitude greater than matrix hydraulic conductivity at the shallowest to deepest depths of investigation, respectively. The results also suggest that hydraulic conductivity values estimated from permeameters and large-core laboratory experiments may not be of sufficient volume to represent the REV of fractured till.

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.

An efficient ensemble algorithm for numerical approximation of stochastic Stokes–Darcy equations

We propose and analyze an efficient ensemble algorithm for fast computation of multiple realizations of the stochastic Stokes–Darcy model with a random hydraulic conductivity tensor. The algorithm results in a common coefficient matrix for all realizations at each time step making solving the linear systems much less expensive while maintaining comparable accuracy to traditional methods that compute each realization separately. Moreover, it decouples the Stokes–Darcy system into two smaller sub-physics problems, which reduces the size of the linear systems and allows parallel computation of the two sub-physics problems. We prove the ensemble method is long time stable and first-order in time convergent under a time-step condition and two parameter conditions. Numerical examples are presented to support the theoretical results and illustrate the application of the algorithm.

A Double Scale Methodology to Investigate Flow in Karst Fractured Media via Numerical Analysis: The Cassino Plain Case Study (Central Apennine, Italy)

A methodology to evaluate the hydraulic conductivity of the karst media at a regional scale has been proposed, combining pumping tests and the hydrostructural approach, evaluating the hydraulic conductivity of fractured rocks at the block scale. Obtaining hydraulic conductivity values, calibrated at a regional scale, a numerical flow model of the Cassino area has been developed, to validate the methodology and investigate the ambiguity, related to a nonunique hydrogeological conceptual model. The Cassino plain is an intermontane basin with outstanding groundwater resources. The plain is surrounded by karst hydrostructures that feed the Gari Springs and Peccia Springs. Since the 1970s, the study area was the object of detailed investigations with an exceptional density of water-wells and piezometers, representing one of the most important karst study-sites in central-southern Italy. Application of the proposed methodology investigates the hydraulic conductivity tensor at local and regional scales, reawakening geological and hydrogeological issues of a crucial area and tackling the limits of the continuum modelling in karst media.

Fluid flow and effective conductivity calculations on numerical images of bentonite microstructure

Hydraulic conductivities of compacted water-saturated bentonite were computed based on the real microstructure. The Homogenization of Periodic Media approach employed fully acknowledges the heterogeneous and multiscale microstructure of clay, as well as locally varying physical flow properties. Consequently, three levels of description were considered : the microscopic level of clay particles, the mesoscopic level of clay aggregates, mineral grains and inter-aggregate porosity, and the macroscopic level of the sample subjected to fluid pressure gradients in the laboratory. Starting from the local description of fluid flow, the expression of the effective hydraulic conductivity tensor was derived. The soft and dense gels and the open voids may form a connected flow path or remain occluded. The local problems were solved on the microstructure obtained from a digitalized micrograph by image analysis. The contribution to macroscopic flow by the soft and dense gels was investigated in various configurations, and comparisons were made with hydraulic conductivity data for MX-80 bentonite.

Numerical homogenization of the Richards equation for unsaturated water flow through heterogeneous soils

AbstractHomogenized equations and the corresponding effective constitutive relations are generally necessary for numerically modeling large‐scale unsaturated flow processes in soils. Recently, based on the Kirchhoff transformation and the two‐scale convergence theory, a homogenization method for the Richards equation with the Mualem‐van Genuchten model has been proposed, with a constant model parameter α relating to the inverse of the air‐entry pressure and the soil pore size distribution. The homogenized model is computationally efficient and convenient to use because of its explicit expression. In this study, we generalize this method, allowing α to be a spatially distributed random field and proposing a homogenized Richards equation in the mixed form (θ/h) under the condition that the effective hydraulic conductivity tensor is diagonal. This generalization eliminates the limitation of a constant α in practical applications; the proposed homogenized model is meaningful in most situations because the flow problems are influenced mainly by the diagonal terms of conductivity and the off‐diagonal terms are often neglected. Two‐dimensional numerical tests are conducted in soil profiles with different degrees of spatial heterogeneity structure to illustrate that the homogenized model can capture the fine‐scale flow behaviors on coarse grids effectively. Homogenization for the Richards equation with other two commonly used constitutive relations—the Brooks‐Corey model and the Gardner‐Russo model—is also illustrated in this study.

Helical Flow and Transient Solute Dilution in Porous Media

Helical flow can occur in porous media if the hydraulic conductivity tensor is anisotropic. We study the structure of steady-state flow fields in three-dimensional anisotropic porous media formed by two homogeneous layers, one of which is anisotropic. We simulate transient transport of a conservative scalar in such flow fields by a hybrid streamline/smoothed particle hydrodynamics method and analyze dilution. We use stretching and folding metrics to characterize the flow field and the dilution index of a conservative scalar divided by the volume of the domain to quantify plume dilution. Based on the results of detailed numerical simulations, we conclude that nonlinear deformation triggers dilution and that plume dilution is controlled by two parameters: the contrast between the principal directions of the anisotropic layer, and the orientation of the hydraulic conductivity tensor with respect to the main flow direction. Furthermore, we show that in this kind of flow fields transverse dispersion is responsible for an increase in plume dilution, while the effect of longitudinal dispersion is negligible.

Integrating aerial geophysical data in multiple-point statistics simulations to assist groundwater flow models

The process of accounting for heterogeneity has made significant advances in statistical research, primarily in the framework of stochastic analysis and the development of multiple-point statistics (MPS). Among MPS techniques, the direct sampling (DS) method is tested to determine its ability to delineate heterogeneity from aerial magnetics data in a regional sandstone aquifer intruded by low-permeability volcanic dykes in Northern Ireland, UK. The use of two two-dimensional bivariate training images aids in creating spatial probability distributions of heterogeneities of hydrogeological interest, despite relatively ‘noisy’ magnetics data (i.e. including hydrogeologically irrelevant urban noise and regional geologic effects). These distributions are incorporated into a hierarchy system where previously published density function and upscaling methods are applied to derive regional distributions of equivalent hydraulic conductivity tensor K. Several K models, as determined by several stochastic realisations of MPS dyke locations, are computed within groundwater flow models and evaluated by comparing modelled heads with field observations. Results show a significant improvement in model calibration when compared to a simplistic homogeneous and isotropic aquifer model that does not account for the dyke occurrence evidenced by airborne magnetic data. The best model is obtained when normal and reverse polarity dykes are computed separately within MPS simulations and when a probability threshold of 0.7 is applied. The presented stochastic approach also provides improvement when compared to a previously published deterministic anisotropic model based on the unprocessed (i.e. noisy) airborne magnetics. This demonstrates the potential of coupling MPS to airborne geophysical data for regional groundwater modelling.