Unconfined Groundwater Flow Research Articles

Physics-Informed Neural Networks for solving transient unconfined groundwater flow

Neural networks excel in various machine learning applications; however, they lack the physical interpretability and constraints crucial for numerous scientific and engineering problems. This limitation hinders their ability to accurately capture and predict complex physical systems’ behavior, potentially yielding inaccurate or unreliable results. Physics-Informed Neural Networks (PINNs) are a class of machine learning models that integrate the power of neural networks with the physical laws governing natural phenomena. PINNs provide an effective tool for solving intricate physical problems, ranging from fluid dynamics to materials science, by incorporating physical constraints into the neural network architecture. PINNs can substantially enhance the accuracy and efficiency of model predictions, even in data-limited situations. This work offers insight into recent developments in the PINN field, including their mathematical formulation and training algorithms, and emphasizes their application in solving transient unconfined groundwater flow. In this context, the phreatic surface acts as a spatiotemporally varying boundary condition, and properly accounting for its position is vital for precise predictions of unconfined groundwater flow and related environmental and engineering applications. The study’s objective is to develop a reliable model for estimating the phreatic surface and the spatiotemporal distribution of piezometric heads in a vertical cross-section of an unconfined aquifer. Two cases are examined: the first involves a homogeneous and isotropic aquifer, while the second comprises a mildly heterogeneous and anisotropic one. The challenges and opportunities arising from this emerging research area are also explored, and essential directions for future research are underscored.

Natural convection effect on heat transfer in saturated soils under the influence of confined and unconfined subsurface flow

Recently, it has been proven that natural convection significantly impacts heat transfer close to geothermal systems in saturated soils even in the presence of groundwater flow. However, this phenomenon has only been investigated in the presence of confined groundwater flow. Therefore, in this study, a fully coupled hydro-thermal model was developed to quantitatively determine the natural convection influence on heat transfer in the presence of an unconfined groundwater flow and to compare the hydro-thermal response of soil under these two different scenarios. Results showed that natural convection influence depends on the coupled effects of soil permeability, groundwater velocity, and heater temperature. According to the results, in the case of a groundwater flow velocity less than 10−7 m/s, natural convection role is non-negligible when Rayleigh number is greater than 50, while for other groundwater velocities, that matters only when Buoyancy ratio is greater than 20. It was also shown that choosing a confined groundwater flow over an unconfined one led to an error of 47 % to 173 % in the explored scenarios. Furthermore, it was revealed that when the relative distance of the heater from the surface alters from 0 to 15 and 30 m, the error in the soil temperature due to neglecting natural convection can be as high as 43 %, 48 %, and 60 %, respectively.

Experimental study on non-Darcian flow through a single artificial fracture for different fracture apertures and surface roughness

Abstract This study aims to explore the influence of various geometrical and hydraulic parameters on flow behavior and hydraulic conductivity in a single artificial fracture through a series of laboratory experiments. Laboratory experiments were conducted to examine unconfined groundwater flow through an artificially constructed single fracture. The fracture model consisted of varying aperture sizes (3, 9, and 12 mm) and different surface roughness conditions (fine, medium, and coarse sand coatings). Non-Darcian turbulent flow characteristics were observed at different flow rates, and the gradient of Reynolds number versus average flow velocity increased with aperture size. Flow parameters of the Darcian, Izbash, and Forchheimer models were calculated to characterize the flow behavior. Both the Forchheimer and Izbash models were found suitable for describing the non-Darcian flow characteristics under the prevailing conditions. The study revealed that hydraulic conductivity depended on flow length for fractures with different apertures and surface roughnesses, likely due to the presence of 2-D torturous flow within the rough fracture surface. These findings contribute to a better understanding of groundwater flow in fractured rock aquifers and provide valuable insights for modeling and managing such systems.

Increasing knowledge of the transmissivity field of a detrital aquifer by geostatistical merging of different sources of information

Transmissivity is a significant hydrogeological parameter that affects the reliability of groundwater flow and transport models. This study demonstrates the improvement in the estimated transmissivity field of an unconfined detritic aquifer that can be obtained by using geostatistical methods to combine three types of data: hard transmissivity data obtained from pumping tests, soft transmissivity data obtained from lithological information from boreholes, and water head data. The piezometric data can be related to transmissivity by solving the hydrogeology inverse problem, i.e., including the observed water head to determine the unknown model parameters (log transmissivities). The geostatistical combination of all the available information is achieved by using three different geostatistical methodologies: ordinary kriging, ordinary co-kriging and inverse problem universal co-kriging. In addition, there are eight methodological cases to be compared according to which log-transmissivity data are considered as the primary variable in co-kriging and whether two or three variables are used in inverse-problem universal co-kriging. The results are validated by using the performance statistics of the direct modelling of the unconfined groundwater flow and comparing observed water heads with the modelled ones. Although the results show that the two sets of log-transmissivity data are incompatible, the set of log-transmissivity data from the lithofacies provides a good log-transmissivity image that can be improved by inverse modelling. The map provided by inverse-problem universal co-kriging provides the best results. Using three variables, rather than two in the inverse problem, gives worse results because of the incompatibility of the log-transmissivity data sets.

Investigating steady unconfined groundwater flow using Physics Informed Neural Networks

A deep learning technique called Physics Informed Neural Networks (PINNs) is adapted to study steady groundwater flow in unconfined aquifers. This technique utilizes information from underlying physics represented in the form of partial differential equations (PDEs) alongside data obtained from physical observations. In this work, we consider the Dupuit–Boussinesq equation, which is based on the Dupuit–Forchheimer approximation, as well as a recent, more complete model derived by Di Nucci (2018) as underlying models. We then train PINNs on data obtained from steady-state analytical solutions and laboratory based experiments.Using PINNs, we predict phreatic surface profiles given different input flow conditions and recover estimates for the hydraulic conductivity from the experimental observations. We show that PINNs can eliminate the inherent inability of the Dupuit–Boussinesq equation to predict flows with seepage faces. Moreover, the inclusion of physics information from the Di Nucci and Dupuit–Boussinesq models constrains the solution space and produces better predictions than training on data alone. PINNs based predictions are robust and show a little effect from added noise in the training data. Furthermore, we compare the PINNs solutions obtained via the Di Nucci and Dupuit–Boussinesq flow models to examine the effects of higher order flow terms that are included in the Di Nucci formulation but are neglected by the Dupuit–Boussinesq approximation. Lastly, we discuss the effectiveness of using PINNs for examining groundwater flow.

Understanding the effects of subsidence on unconfined aquifer parameters by integration of Lattice Boltzmann Method (LBM) and Genetic Algorithm (GA)

Excessive exploitation of groundwater has hitherto led to a significant land subsidence in a considerable number of plains in Iran. The compaction of aquifer layers ends up with changes in aquifer parameters, including hydraulic conductivity (Kx), specific yield (Sy), and compressibility (α). Accordingly, a precise estimation of aquifer parameters, Kx, Sy, and α seems essential for future water resources planning and management. In this study, an innovative inversion solution based on the combination of lattice Boltzmann method (LBM) and genetic algorithm (GA) was developed to determine the aquifer parameters, Kx, Sy, and α in Darab plain (in Fars province, Iran), which is highly subject to land subsidence. Herein, a newly developed LBM for unconfined groundwater flow was employed by incorporating the amount of subsidence measured by synthetic aperture radar interferometry (InSAR) spanning from 2010 to 2016. In order to optimize the aquifer parameters, the whole process of inverse modeling is replicated on the annual basis from 2010 to 2016 which leads to the temporal estimation of the aquifer parameters. Due to the compaction occurring in the aquifer system, a declining temporal trend is observed in the aquifer parameters in most parts of the plain. By fitting a function to time-dependent aquifer parameters, Kx, Sy, and α, their corresponding values and consequently the amount of subsidence in the near future, i.e., 2017, are predicted. The small average relative error (~ 3.5%) between the predicted land subsidence and the InSAR measurements demonstrates the high performance of the proposed inverse modeling approach.

The Multi‐Scale Dynamics of Groundwater Depletion

AbstractUnsustainable depletion of aquifer storage and diversion of groundwater from downstream users continue to be serious problems, even though their adverse effects are widely recognized. Groundwater depletion involves interactions between economically motivated pumping decisions and physical constraints. Here, we investigate these interactions by using optimal control techniques to describe the pumping decisions of economic agents who share an aquifer. Our approach relies on a multi‐scale description of unconfined groundwater flow, applied to a computational experiment that illustrates some important impacts of aquifer development. We show that cooperative groundwater management can provide higher economic benefits, with less storage depletion, than an uncooperative alternative. However, demand for water still drives pumping decisions, even when users cooperate. In many aquifers the benefits of pumping are necessarily accompanied by a reduction in aquifer outflow and by depletion of the groundwater reserve, for both cooperative and uncooperative management. These benefits decrease significantly when pumping is limited by well yield constraints or when the diversion of aquifer outflow is restricted. Our overall analysis emphasizes the importance of viewing aquifer management as part of a larger resource allocation problem that considers the conflicting needs of well operators, downstream users, and future consumers of the resource.

The solution of unconfined groundwater flow with an innovative lattice Boltzmann method

Lattice Boltzmann method (LBM) has emerged as a fast, precise, and efficient numerical solution to solve differential equations. There seems to be a dearth of research regarding the solution for groundwater flow in unconfined aquifer using LBM. Accordingly, in this study, an innovative numerical solution based on LBM was introduced to solve groundwater flow in unconfined aquifers, taking into account D2Q9 scheme. The solutions obtained from the proposed LBM were compared to results stemmed from three different unconfined groundwater problems with known solutions. Both steady and transient conditions for groundwater flow were considered in simulations. It was deduced that the proposed LBM could simulate the unconfined groundwater flow satisfactorily.

Fractional governing equations of transient groundwater flow in unconfined aquifers with multi-fractional dimensions in fractional time

Abstract. In this study, a dimensionally consistent governing equation of transient unconfined groundwater flow in fractional time and multi-fractional space is developed. First, a fractional continuity equation for transient unconfined groundwater flow is developed in fractional time and space. For the equation of groundwater motion within a multi-fractional multidimensional unconfined aquifer, a previously developed dimensionally consistent equation for water flux in unsaturated/saturated porous media is combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. Combining the fractional continuity and groundwater motion equations, the fractional governing equation of transient unconfined aquifer flow is then obtained. Finally, two numerical applications to unconfined aquifer groundwater flow are presented to show the skills of the proposed fractional governing equation. As shown in one of the numerical applications, the newly developed governing equation can produce heavy-tailed recession behavior in unconfined aquifer discharges.

Hydraulic-Head Formulation for Density-Dependent Flow and Transport.

Density-dependent flow and transport solutions for coastal saltwater intrusion investigations, analyses of fluid injection into deep brines, and studies of convective fingering and instabilities of denser fluids moving through less dense fluids typically formulate the groundwater flow equation in terms of pressure or equivalent freshwater head. A formulation of the flow equation in terms of hydraulic head is presented here as an alternative. The hydraulic-head formulation can facilitate adaptation of existing constant-density groundwater flow codes to include density-driven flow by avoiding the need to convert between freshwater head and hydraulic head within the code and by incorporating density-dependent terms as a compartmentalized "correction" to constant-density calculations already performed by the code. The hydraulic-head formulation also accommodates complexities such as unconfined groundwater flow and Newton-Raphson solution schemes more readily than the freshwater-head formulation. Simulation results are presented for four example problems solved using an implementation of the hydraulic-head formulation in MODFLOW.

Polynomial approximate solutions of an unconfined Forchheimer groundwater flow equation

We consider a one-dimensional, unconfined groundwater flow equation for the horizontal propagation of water. This equation was derived by using a particular form of the Forchheimer equation in place of Darcy’s Law. Such equations can model turbulent flows in coarse and fractured porous media. For power-law head, exponential head, power-law flux and exponential flux boundary conditions at the inlet, the problems can be reduced, using similarity transformations, to boundary-value problems for a nonlinear ordinary differential equation. We construct quadratic and cubic approximate solutions of these problems. We also numerically compute solutions using a new modification of a method of Shampine, which exploits scaling properties of the governing equation. The polynomial approximate solutions replicate well the numerical solutions and they are easy to use. Last, we compare the predicted wetting front positions from our quadratic and cubic polynomials to predictions based on Adomian polynomials of the same degrees. The work demonstrates the value of polynomial approximate solutions for validating numerical solutions and for obtaining good approximations for water profiles and the extent of water propagation. This work also presents a new application of Shampine’s method for this type of groundwater flow equation. We note that this paper introduces additional classes of approximate solutions for the Forchheimer equation. Up to this date, not many solutions are known, especially for the transient cases considered here.

Mathematical modeling of two-dimensional unconfined flow in aquifers

Derivation of general equation for two-dimensional aquifer flow is given. In this derivation we perform a volume balance instead of a mass balance and obtained analytical solutions of two-dimensional saturated flow under various condition. We also constructed transient unconfined groundwater flow equation by combining continuity equation with the Darcy law and provide an analytical solution.

Efficient Uncertainty Quantification for Unconfined Flow in Heterogeneous Media with the Sparse Polynomial Chaos Expansion

In this study, we explore an efficient stochastic approach for uncertainty quantification of unconfined groundwater flow in heterogeneous media, where a sparse polynomial chaos expansion (PCE) surrogate model is constructed with the aid of the feature selection method. The feature selection method is introduced to construct a sparse PCE surrogate model with a reduced number of basis functions, which is accomplished by the least absolute shrinkage and selection operator-modified least angle regression and cross-validation. The training samples are enriched sequentially with the quasi-optimal samples until the results are satisfactory. In this study, we test the performance of the sparse PCE method for unconfined flow with the presence of random hydraulic conductivity and recharge, as well as pumping well. Numerical experiments reveal that, even with large spatial variability and high random dimensionality, the sparse PCE approach is able to accurately estimate the flow statistics with greatly reduced computational efforts compared to Monte Carlo simulations.

Nonlinear model reduction of unconfined groundwater flow using POD and DEIM

Nonlinear groundwater flow models have the propensity to be overly complex leading to burdensome computational demands. Reduced modeling techniques are used to develop an approximation of the original model that has smaller dimensionality and faster run times. The reduced model proposed is a combination of proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). Solutions of the full model (snapshots) are collected to represent the physical dynamics of the system and Galerkin projection allows the formulation of a reduced model that lies in a subspace of the full model. Interpolation points are added through DEIM to eliminate the reduced model's dependence on the dimension of the full model. POD is shown to effectively reduce the dimension of the full model and DEIM is shown to speed up the solution by further reducing the dimension of the nonlinear calculations. To show the concept can work for unconfined groundwater flow model, with added nonlinear forcings, one-dimensional and two-dimensional test cases are constructed in MODFLOW-OWHM. POD and DEIM are added to MODFLOW as a modular package. Comparing the POD and the POD-DEIM reduced models, the experimental results indicate similar reduction in dimension size with additional computation speed up for the added interpolation. The hyper-reduction method presented is effective for models that have fine discretization in space and/or time as well as nonlinearities with respect to the state variable. The dual reduction approach ensures that, once constructed, the reduced model can be solved in an equation system that depends only on reduced dimensions.

Quantitative Analysis of Groundwater Recharge in an Arid Area, Northwest China

The Mixing Cell Model (MCM) is a useful tool that can be applied to areas with limited hydrogeological data, such as arid areas in northwest China, to transform available groundwater hydrochemical data into quantitative information about an aquifer. In this study, we used the MCM to quantify water circulation in the study area and to analyze information such as the supply source composition and proportion of the confined aquifer, the main supply aquifer for local drinking water. The MCM simulation results showed that the confined aquifer in the study area is mainly recharged by leakage of water from the upper unconfined aquifer and lateral flow from the eastern and southern tablelands. Unconfined groundwater and lateral flow contributed to 67.69% and 32.31% of the recharge, respectively. The groundwater circulation model of the study area provided quantitative information about water circulation in different parts of the study area, represented by different cells known as A–F. The information from this model provides a scientific basis for the sustainable use and development of water resources in different parts of the study area.

A comparison of the modern Lie scaling method to classical scaling techniques

Abstract. In the past 2 decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics, including but not limited to overland flow, groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non-dimensionalization, and the Vaschy–Buckingham-Π theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is made with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables, which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

Reduced order modeling of the Newton formulation of MODFLOW to solve unconfined groundwater flow

Projection-based model reduction techniques have been shown to be very effective for reducing the computational burden of high-dimensional groundwater simulations, but only applied to confined groundwater flow. A new methodology is proposed that reduces the dimension of a discretized, transient, unconfined groundwater-flow model. This unconfined model reduction technique is based on Galerkin projection and the Newton formulation of MODFLOW. The method is implemented following the standard package design and code structure that MODFLOW employs for all its features. When the package is invoked within MODFLOW it can collect snapshots, produce a basis, construct the reduced model and propagate the reduced model forward in time. The new formulation accurately represents the water-table surface under a variety of nonlinear settings, such as intraborehole flow from a Multi-Node Well. The unconfined model reduction is applied to four test cases to illustrate its flexibility in handling nonlinear features. Several test cases are discussed to demonstrate the unconfined model reduction applicability. The final test case applies the new model reduction methodology to a scoping MODFLOW model of Santa Barbara, CA composed of 113,578 cells, which requires solving 113,578 equations per time step, and reduces it to 127 equations.

Groundwater response to tidal fluctuation and rainfall in a coastal aquifer

Summary This paper presents an analytical solution generated by linearizing the one-dimensional non-linear Boussinesq equation to characterize the variation of groundwater level induced by tidal waves and rainfall in a coastal unconfined sloping aquifer. The area of the coastal aquifer is divided into 2 zones – the coastal zone of rainfall and the inland zone of non-rainfall. The derivation of the solution is based on the Dupuit–Forchheimer assumptions of a wide range of problems in unconfined groundwater flow and the analytical solutions of coastal and inland water tables of both zones are obtained. The results indicate that the effects of the bottom slope of the aquifer and the beach slope on the groundwater fluctuation are substantial in the unconfined aquifer. Large rainfall intensity with tidal waves will enhance the fluctuation significantly, but less effect on the fluctuation is found for the cases of steeper sloping bottom under the same rainfall condition.

A New Package in MODFLOW to Simulate Unconfined Groundwater Flow in Sloping Aquifers

The nonhorizontal-model-layer (NHML) grid system is more accurate than the horizontal-model-layer grid system to describe groundwater flow in an unconfined sloping aquifer on the basis of MODFLOW-2000. However, the finite-difference scheme of NHML was based on the Dupuit-Forchheimer assumption that the streamlines were horizontal, which was acceptable for slope less than 0.10. In this study, we presented a new finite-difference scheme of NHML based on the Boussinesq assumption and developed a new package SLOPE which was incorporated into MODFLOW-2000 to become the MODFLOW-SP model. The accuracy of MODFLOW-SP was tested against solution of Mac Cormack (1969). The differences between the solutions of MODFLOW-2000 and MODFLOW-SP were nearly negligible when the slope was less than 0.27, and they were noticeable during the transient flow stage and vanished in steady state when the slope increased above 0.27. We established a model considering the vertical flow using COMSOL Multiphysics to test the robustness of constrains used in MODFLOW-SP. The results showed that streamlines quickly became parallel with the aquifer base except in the narrow regions near the boundaries when the initial flow was not parallel to the aquifer base. MODFLOW-SP can be used to predict the hydraulic head of an unconfined aquifer along the profile perpendicular to the aquifer base when the slope was smaller than 0.50. The errors associated with constrains used in MODFLOW-SP were small but noticeable when the slope increased to 0.75, and became significant for the slope of 1.0.

Second-order shallow flow equation for anisotropic aquifers

Summary Transient unconfined ground-water flow is described using the well-known Boussinesq equation, in which the Dupuit assumptions are implicit. When these assumptions fail, one must recur to the next level of approximation, which is the second-order theory for shallow flow in porous media, developed by Dagan (1967) for isotropic aquifers. When the soil is highly anisotropic Dagan’s second-order theory can become invalid. Here we present the generalized second order theory that account for anisotropy. An analytical solution for the second-order theory with anisotropy is presented for the linearized equation that is used to illustrate this effect on the bank storage problem.