Groundwater Flow Problems Research Articles

An efficient method to solve the fuzzy Crank–Nicolson scheme with application to the groundwater flow problem

Abstract In this paper, we propose a comprehensive methodological framework for solving the fuzzy groundwater flow problem in a simpler and faster way based on numerical analysis. In particular, a novel simplified matrix explicit inverse formula is proposed as an efficient method to solve the fuzzy Finite Difference numerical scheme, called Crank–Nicolson implicit scheme. The main advantage of the proposed method is that it offers an efficient and simpler solution to the algebraic tridiagonal system of equations that appeared in the fuzzy Crank–Nicolson scheme, without affecting the accuracy of the results. Without doubt, the simulation of an unconfined aquifer flow, using the Finite Difference Method, is often time-consuming due to the costly calculations required to solve the implicit Finite Difference schemes. This process becomes even more difficult when the physical problem is solved in its fuzzy form where the calculations become more complex and greatly increase. However, problem uncertainties are not considered negligible and must be included in the final calculations for a more rational management of the water body. This creates the need to find a new and faster way to solve the fuzzy implicit schemes, as proposed in this work. The simplified matrix explicit inverse formula was applied to the fuzzy Crank–Nicolson scheme to solve the fuzzy partial differential equation of Boussinesq and, hence, the simulation of the physical problem. Its results were compared with the corresponding results of the Thomas algorithm, which until today, is the most common method of solving the tridiagonal system of equations. The case under consideration refers to a sudden rise in the lake's water level, thus resulting in the aquifer recharging from the lake. The performance of the proposed method was more than satisfactory in terms of calculation accuracy and reliability where the numerical results are matched with the differences appearing from the 12 decimal point onward between the two examined methods. The proposed method tested also considered the running times, achieving much better times compared with the Thomas algorithm, where the differences ranged from a few seconds to several hours depending on the examined time and space steps.

A local–global multiscale method for highly heterogeneous stochastic groundwater flow problems

In this paper, we propose a local–global multiscale method for highly heterogeneous stochastic groundwater flow problems under the framework of reduced basis method and the generalized multiscale finite element method (GMsFEM). Due to incomplete characterization of the medium properties of the groundwater flow problems, random variables are used to parameterize the uncertainty. As a result, solving the problem repeatedly is required to obtain statistical quantities. Besides, the medium properties are usually highly heterogeneous, which will result in a large linear system that needs to be solved. Therefore, it is intrinsically inevitable to seek a computational-efficient model reduction method to overcome the difficulty. We will explore the combination of the reduced basis method and the GMsFEM. In particular, we will utilize residual-driven basis functions, which can provide highly accurate reduced order solution. This local–global multiscale method is more efficient than applying the GMsFEM or reduced basis method individually. We first construct parameter-independent multiscale basis functions that include both local and global information of the permeability fields, and then use these basis functions to construct several global snapshots and global basis functions for fast online computation with different parameter inputs. We provide rigorous analysis of the proposed method and extensive numerical examples to demonstrate the accuracy and efficiency of the local–global multiscale method.

Groundwater flow simulation in a confined aquifer using Local Radial Point Interpolation Meshless method (LRPIM)

Groundwater flow problems are generally solved using analytical or numerical methods. Though analytical solutions are exact and preferable, they are not available for complex field problems. Hence numerical methods such as Finite Element and Finite Difference methods are used to solve complex groundwater problems. These conventional mesh/ grid-based numerical methods need construction of a detailed mesh/ grid. On the other hand, the meshless approach creates a system of algebraic equations on a collection of distributed nodes in the problem area and the boundary. As a result, it is easy to incorporate any modifications to the model at a later time by simply adding nodes to the domain. In this study a weak form meshless method known as local radial point interpolation method (LRPIM) which uses radial basis functions for approximation or interpolation is developed to solve the groundwater flow problems in a confined aquifer. The results obtained from the LRPIM model has been compared with other numerical methods for benchmark and real field problems, and are found to be satisfactory. Implementation of the essential boundary conditions was relatively easier in LRPIM and gave good accuracy for the problems considered. LRPIM can potentially be used as an alternative to the other conventional methods, especially where the domain boundary is irregular or varying with time.

Efficient mass conservative numerical model for solving variably saturated groundwater flow

A multi-dimensional mass conservative numerical method, particularly suitable for limited computational resources, is developed for solving transient variably saturated groundwater flow problems. The Richards equation is discretized spatially with a finite element method and temporally with an implicit Euler scheme, in which mass-conservative and mass-lumping techniques are used to keep the numerical simulation stable. In addition, the stiffness and mass matrices involved are approximated in a way to guarantee less computational effort. To confirm the accuracy and the efficiency of this code, we verified it using benchmark tests using one, two and three-dimensional problems. The present model is also applied to a real field case problem, where its superiority is clearly demonstrated. The code achieved reliable results for each problem.

Analytical modeling of mine water rebound: Three case studies in closed hard-coal mines in Germany

Purpose.In this paper we present and validate an analytical model of water inflow and rising level in a flooded mine and examine the model robustness and sensitivity to variations of input data considering the examples of three closed hard-coal mines in Germany. Methods. We used the analytical solution to a boundary value problem of radial ground water flow to the shaft, treated as a big well, and water balance relations for the series of successive stationary positions of a depression cone to simulate a mine water rebound in the mine taking into account vertical distribution of hydraulic conductivity, residual volume of underground workings, and natural pores. Findings. The modeling demonstrated very good agreement with the measured data for all the studied mines. The maximum relative deviation for the mine water level during the measurement period did not exceed 2.1%; the deviation for the inflow rate to a mine before its flooding did not exceed 0.8%. Sensitivity analysis revealed the higher significance of the residual working volume and hydraulic conductivity for mine water rebound in the case of thick overburden and the growing significance of the infiltration rate and the flooded area size in the case of lower overburden thickness. Originality.The developed analytical model allows realistic prediction of transient mine water rebound and inflow into a mine with layered heterogeneity of rocks, irregular form of the drained area, and with the inflow/outflow to a neighboring mine and the volume of voids as a distributed parameter without gridding the flow domain performed in numerical models. Practical implications.The study demonstrated the advantages of analytical modeling as a tool for preliminary evaluation and prediction of flooding indicators and parameters of mined out disturbed rocks. In case of uncertain input data, modeling can be considered as an attractive alternative to usually applied numerical methods of modeling ground and mine water flow.

Application of physics-informed neural networks to inverse problems in unsaturated groundwater flow

ABSTRACT This paper investigates the application of Physics-Informed Neural Networks (PINNs) to inverse problems in unsaturated groundwater flow. PINNs are applied to the types of unsaturated groundwater flow problems modelled with the Richards partial differential equation and the van Genuchten constitutive model. The inverse problem is formulated here as a problem with known or measured values of the solution to the Richards equation at several spatio-temporal instances, and unknown values of solution at the rest of the problem domain and unknown parameters of the van Genuchten model. PINNs solve inverse problems by reformulating the loss function of a deep neural network such that it simultaneously aims to satisfy the measured values and the unknown values at a set of collocation points distributed across the problem domain. The novelty of the paper originates from the development of PINN formulations for the Richards equation that requires training of a single neural network. The results demonstrate that PINNs are capable of efficiently solving the inverse problem with relatively accurate approximation of the solution to the Richards equation and estimates of the van Genuchten model parameters.

Software DHI Feflow and Processing Modflow in Solving Groundwater Flow Problems for HPP Projects Design and Reconstruction in a Complex Geological and Hydrogeological Environment

Software DHI Feflow and Processing Modflow in Solving Groundwater Flow Problems for HPP Projects Design and Reconstruction in a Complex Geological and Hydrogeological Environment

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.

Bayesian inference with subset simulation in varying dimensions applied to the Karhunen–Loève expansion

AbstractUncertainties associated with spatially varying parameters are modeled through random fields discretized into a finite number of random variables. Standard discretization methods, such as the Karhunen–Loève expansion, use series representations for which the truncation order is specified a priori. However, when data is used to update random fields through Bayesian inference, a different truncation order might be necessary to adequately represent the posterior random field. This is an inference problem that not only requires the determination of the often high‐dimensional set of coefficients, but also their dimension. In this article, we develop a sequential algorithm to handle such inference settings and propose a penalizing prior distribution for the dimension parameter. The method is a variable‐dimensional extension of BUS (Bayesian Updating with Structural reliability methods), combined with subset simulation (SuS). The key idea is to replace the standard Markov Chain Monte Carlo (MCMC) algorithm within SuS by a trans‐dimensional MCMC sampler that is able to populate the discrete‐continuous parameter space. To address this task, we consider two types of MCMC algorithms that operate in a fixed‐dimensional saturated space. The performance of the proposed method with both MCMC variants is assessed numerically for two examples: a 1D cantilever beam with spatially varying flexibility and a 2D groundwater flow problem with uncertain permeability field.

Performance Analysis of the χMD Matrix Solver Package for MODFLOW-USG.

The χMD matrix solver package is incorporated into USGS groundwater modeling software, such as MODFLOW-NWT, MODFLOW-USG, and MT3D. The solver is used to solve matrices assembled through numerical discretization of the groundwater flow equation, and solute transport equations. χMD has demonstrated its higher robustness, faster execution speed, and more efficient memory usage compared to the existing solvers for many types of groundwater flow problems. χMD uses preconditioned iterative Krylov-subspace methods and consists of preconditioning and acceleration modules. Because the solver package uses a variety of preconditioning features including level-based incomplete lower-upper (ILU) factorization method with a drop tolerance scheme, users must choose optimal preconditioning parameters to improve execution speed and robustness. In order to examine how the preconditioning parameters, ILU factorization level, and drop tolerance values affect the overall performance of the matrix solver, we evaluated five different groundwater model applications using MODFLOW-USG that include different numerical complexities. For those five cases, the number of discretization nodes varied from 10,000 cells to 730,300 cells. From the analysis, we found that the preconditioning parameters greatly affect execution times and memory usage of the preconditioning and acceleration procedures. In addition, a combination of the ILU level between five to seven and the drop tolerance value between 10-2 and 10-3 usually resulted in shorter overall execution time. Our study suggests that the users can elicit higher performance and robustness of the χMD matrix solver using this combination of the parameters and enhance computational efficiency of solving groundwater and solute transport problems.

Educational interactive LabVIEW simulations of field hydraulic conductivity tests below water table

AbstractVisual artifacts used in graphical designs of the mechanisms of groundwater flow problems are very influential in learning and teaching. The merits of the classical methods of problem‐solving using only mathematics become much less superior to those linked with interactive graphics. In this study, an interactive educational LabVIEW program is designed to graphically simulate the constructional and mathematical procedures of the two widely used field methods in groundwater engineering, the Ernst auger‐hole method, and the Bouwer–Rice slug test method, for field measurements of the hydraulic conductivity of soils below the water table. The simulations included the basic constructional steps in carrying out the auger hole and the slug test field methods. Virtual geometric wells, boreholes, and pipes are graphically and interactively designed to conduct the required field tests. The simulations of the program enabled the students to get acquainted with what should be done in the field to measure soil hydraulic conductivity and enhanced understanding of how to perform a field test. Students were able to try different field hydrological parameters, and by successively changing any hydrological field parameter, they were able to visualize graphically and interactively its influence inside the classroom. This particular step improved the student's practical standards. Conditions of validity of each field test are completely fulfilled by the program. Results of the interactive program are found to be visually interesting and extremely accurate, which made this educational program quite interesting in teaching and learning.

An Improved numerical manifold method model for groundwater flow problems

This paper presents an improved numerical manifold method (NMM) model for groundwater flow problems. Hermite interpolation is adopted to construct NMM approximation in order to enhance the accuracy and convergence. Therefore, the new model can achieve accurate solution with C1 continuity in coarse mesh. Compared with conventional high-order NMM, the proposed method is free from linear dependance problem with even better accuracy and convergence. In addition, a local-refined mesh is used to interpolate detail information of the solution without significantly increasing computation time. The accuracy and convergence of the improved NMM model for groundwater flow problems are demonstrated by several examples.

A fully interior penalty discontinuous Galerkin method for variable density groundwater flow problems

Discontinuous Galerkin (DG) methods due to their robustness properties, e.g. local conservation, low numerical dispersion, and well-capturing strong shocks and physical discontinuities, are well-suited for the simulation of Variable Density Flow (VDF) in porous media. This paper aims at introducing, in a unified format, the general class of Interior Penalty DG (IPDG) methods to solve the VDF equations. A combination of symmetric, non-symmetric and incomplete IPDG methods is used to discretize both head and concentration variables. Compatibility analysis is performed to prevent the loss of accuracy of the IPDG methods in simulations of coupled flow and transport equations. An accurate technique is used for time integration, based on a non-iterative procedure and adaptive time stepping with embedded error control. Several benchmarks are investigated to validate the proposed DG scheme and to examine its performance in simulating VDF problems. The new DG scheme reproduces better the experimental data than the conventional SEAWAT model. Its results are in excellent agreement with a recent semi-analytical solution of the Henry problem, dealing with seawater intrusion under convection-dominating conditions. The performance of the DG scheme is examined by simulating the challenging problem of natural convection in porous enclosure. The method is compared against a finite element solution obtained with COMSOL multi-physics. The numerical experiments indicate clearly that high-order DG method is much more appropriate than standard conforming Galerkin method in simulating VDF problems while at the same time, guaranteeing a better precision and high-fidelity solutions. The proposed numerical method can be extended to 3D problems.

Global existence of strong solutions to a groundwater flow problem

In this paper we study the initial boundary value problem for the system $\Delta v= u_{x_1},\ u_t-\mbox{div}\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\right)=-\nabla u\cdot\mathbf{q}$, where $\mathbf{q}=(-v_{x_2}, v_{x_1})^T$, $\mathbf{q}\otimes\mathbf{q}=\mathbf{q}\mathbf{q}^T$. This problem has been proposed as a model for a fluid flowing through a porous medium under the influence of gravity and hydrodynamic dispersion. For each $T>0$ we obtain a so-called strong solution $(v, u)$ in the function space $L^\infty(0,T; \left(W^{1,\infty}(\Omega)\right)^2)$, where $\Omega$ is a bounded domain in $\mathbb{R}^2$. The key ingredient in our approach is the decomposition $A^2=\mbox{tr} (A)A-\mbox{det}(A) I$ for any $2\times 2$ symmetric matrix $A$. By exploring this decomposition, we are able to derive an equation of parabolic type for the function $\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\cdot\nabla u\right)^j, j\geq 1$. With the aid of this equation we obtain a uniform bound for $\nabla u$.

On Solving Groundwater Flow and Transport Models with Algebraic Multigrid Preconditioning.

Iterative solvers preconditioned with algebraic multigrid have been devised as an optimal technology to speed up the response of large sparse linear systems. In this work, this technique was implemented in the framework of the dual delineation approach. This involves a single groundwater flow linear solution and a pure advective transport solution with different right-hand sides. The new solver was compared with other preconditioned iterative methods, the MODFLOW's GMG solver, and direct sparse solvers. Test problems include two- and three-dimensional benchmarks spanning homogeneous and highly heterogeneous and anisotropic formations. For the groundwater flow problems, using the algebraic multigrid preconditioning speeds up the numerical solution by one to two orders of magnitude. The algebraic multigrid preconditioner efficiency was preserved for the three dimensional heterogeneous and anisotropic problem unlike for the MODFLOW's GMG solver. Contrarily, a sparse direct solver was the most efficient for the pure advective transport processes such as the forward travel time simulations. Hence, the best sparse solver for the more general advection-dispersion transport equation is likely to be Péclet number dependent. When equipped with the best solvers, processing multimillion grid blocks by the dual delineation approach is a matter of seconds. This paves the way for its routine application to large geological models. The paper gives practical hints on the strategies and conditions under which algebraic multigrid preconditioning would remain competitive for the class of nonlinear and/or transient problems.

Applying deep learning algorithms to enhance simulations of large-scale groundwater flow in IoTs

Deep learning for enhancing simulation IoTs groundwater flow is a good solution for gaining insights into the behavior of aquifer systems. In previous studies, corresponding results give a basis for the rational management of groundwater resources. The users generally require special skills or knowledge and massive observations in representing the field reality to perform the deep learning algorithms and simulations. To simplify the procedures for performing the numerical and large-scale groundwater flow simulations, we apply the deep learning algorithms which combine both the numerical groundwater model and large-scale IoTs, groundwater flow measuring equipment and various complex groundwater numerical models. The mechanism has the capability to show spatial distributions of in-situ data, analyze the spatial relationships of observed data, generate meshes, update users’ databases with in-situ observed data, and create professional reports. According to the numerical simulation results, we revealed that the deep learning algorithms are high computational efficiency, and we can enhance precise variance estimations for large-scale groundwater flow problems. The findings help users to best apply the deep learning algorithms in an easier way, get more accurate simulation results, and manage the groundwater resources rationally.

Steady-state response of annular wedge-shaped aquifers to arbitrarily-located multiwells with regional flow

Wedge-shaped aquifers are sometimes cut by crossing streams so that the areal geometry can be delineated by an annular wedge of infinite radial extent. Despite their worldwide spread, the problem of groundwater flow in such geometrical entities has not received much attention in analytical studies. This paper presents a closed-form analytical solution for the aquifer head distribution in response to a system of multiwells superimposed on a background regional flow. The aquifer is isotropic, homogeneous and in prefect hydraulic connection with the surrounding streams. A steady-state Green’s function is constructed for this laterally bounded aquifer system. The formulation is general in the sense that the aquifer vertex angle as well as number of wells and their positions can be arbitrary chosen. Closed-form expression for the stream function is obtained by integration of the Cauchy-Riemann equations. Three types of hydrogeological boundary conditions are considered, combining streams of constant and linearly varying head and no-flow barrier. The results are applicable to both confined and unconfined flows. Extremely simple expressions are derived to quantify the amount of water exchange across stream-aquifer interfaces. Flow nets are given for a number of hypothetical test cases, demonstrating different aspects of the generated flow field. Sensitivity maps are prepared for the stream depletion rates due to a single pumping well and the role of aquifer vertex angle on the model response is investigated. Multiple stagnation points in the flow field are identified on a semi-analytical basis. This helps to assess how the position of a single stagnation point is affected by changes in the gradient of regional flow. For relatively high gradients, it is observed that the radial position of the stagnation point coincides with that of the well itself. Finally, the formulation is extended to account for rainfall recharge over an unconfined aquifer having the same areal extent. The aquifer response is found to agree well with other analytical and numerical results.

Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability

The pollution of groundwater, essential for supporting populations and agriculture, can have catastrophic consequences. Thus, accurate modeling of water pollution at the surface and in groundwater aquifers is vital. Here, we consider a density-driven groundwater flow problem with uncertain porosity and permeability. Addressing this problem is relevant for geothermal reservoir simulations, natural saline-disposal basins, modeling of contaminant plumes and subsurface flow predictions. This strongly nonlinear time-dependent problem describes the convection of a two-phase flow, whereby a liquid flows and propagates into groundwater reservoirs under the force of gravity to form so-called “fingers”. To achieve an accurate numerical solution, fine spatial resolution with an unstructured mesh and, therefore, high computational resources are required. Here we run a parallelized simulation toolbox ug4 with a geometric multigrid solver on a parallel cluster, and the parallelization is carried out in physical and stochastic spaces. Additionally, we demonstrate how the ug4 toolbox can be run in a black-box fashion for testing different scenarios in the density-driven flow. As a benchmark, we solve the Elder-like problem in a 3D domain. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute the mean, variance, and exceedance probabilities for the mass fraction. We use the solution obtained from the quasi-Monte Carlo method as a reference solution.

Multiresolution Approach to Condition Categorical Multiple‐Point Realizations to Dynamic Data With Iterative Ensemble Smoothing

AbstractA new methodology is presented for the conditioning of categorical multiple‐point statistics (MPS) simulations to dynamic data with an iterative ensemble smoother (ES‐MDA). The methodology relies on a novel multiresolution parameterization of the categorical MPS simulation. The ensemble of latent parameters is initially defined on the basis of the coarsest‐resolution simulations of an ensemble of multiresolution MPS simulations. Because this ensemble is non‐multi‐Gaussian, additional steps prior to the computation of the first update are proposed. In particular, the parameters are updated at predefined locations at the coarsest scale and integrated as hard data to generate a new multiresolution MPS simulation. The performance of the methodology was assessed on a synthetic groundwater flow problem inspired from a real situation. The results illustrate that the method converges towards a set of final categorical realizations that are consistent with the initial categorical ensemble. The convergence is reliable in the sense that it is fully controlled by the integration of the ES‐MDA update into the new conditional multiresolution MPS simulations. Thanks to a massively reduced number of parameters compared to the size of the categorical simulation, the identification of the geological structures during the data assimilation is particularly efficient for this example. The comparison between the estimated uncertainty and a reference estimate obtained with a Monte Carlo method shows that the uncertainty is not severely reduced during the assimilation as is often the case. The connectivity is successfully reproduced during the iterative procedure despite the rather large distance between the observation points.

Gaussian Radial Basis Function for Unsteady Groundwater Flow

The radial basis function (RBF) collocation methods are meshless with spectrally accurate; its approximation is an extremely powerful tool for solving various types of partial differential equations (PDEs). In this paper, the Gaussian RBF (G-RBF) collocation method is applied to solving unsteady groundwater flow governed by nonlinear elliptic PDEs. During the whole solution process, no iterations are required for a kind of nonlinear PDEs. Numerical results are compared with the well-known Kansa’s method, also namely as Multiquadrics radial basis function (MQ-RBF). The results obtained show good performance of the G-RBF as solution to unsteady groundwater flow problems.