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This chapter covers the principles of groundwater flow. It begins with defining hydraulic conductivity and Darcy's Law for one-dimensional flow, and then generalizes to three dimensional flow. The concepts of specific discharge, average linear velocity, and intrinsic permeability are introduced. Heterogeneity and anisotropy in hydraulic conductivity fields are discussed along with methods for assigning bulk average conductivities. A range of methods for measuring or estimating hydraulic conductivity are discussed, including grain size correlations, lab tests, field tests, tracer tests, and models. Sections near the end of the chapter deal with flow in individual fractures in fractured rock, flow in the unsaturated zone, and groundwater flow with variable density water. Coastal fresh–salt interface flow is discussed and the Ghyben-Herzberg method of estimating interface elevation is covered.

Existing models are built upon to develop new ones. As a foundational model in porous media flow, the Darcy flow model has been built upon by many researchers. The groundwater flow equation evolved from the Darcy equation. Dupuit–Forchheimer built upon it to develop the simplified forms for both the confined and unconfined aquifers flow, usable for studying groundwater flow into wells. Aside from height, permeability, and availability of water in the aquifers, other factors influence groundwater flow into wells, as enshrined in the flow equation. This paper investigates the roles of storability, hydraulic conductivity, and source/sink strength in both confined and unconfined groundwater flow into wells using the Dupuit–Forchheimer assumption. In this model, the Dupuit–Forchheimer pressure assumption is substituted into the groundwater flow equations and solved using the Bessel form for separation of variable technique, and Mathematica 11.2 computational software to obtain the expressions for the pressure, which are computed and presented quantitatively. The results show that an increase in the hydraulic conductivity and storability have no effect on the flow pressures in the confined and unconfined aquifers but cause fluctuation in the pressure structure in the unconfined aquifer; the source/sink strength factor causes fluctuation in the pressure structures in both confined and unconfined aquifers flow. However, in both confined/unconfined aquifers the pressures increase as the radii of the wells increase. Importantly, the fluctuation in the pressure structures causes a loss of energy for groundwater flow into the wells.

Riverbank filtration (RBF) is a natural technology that is used for river water treatment. This research seeks to investigate the effect of pumping rate on the transport of colloids in RBF. However, this work considered Dissolved Organic Matter (DOM) as a nutrient for bacteria. The mathematical model consists of groundwater flow equation and colloids concentration equations. The equations were solved analytically using parameter expanding method and Eigen function expansion techniques. The results obtained are presented graphically and discussed. It was observed that increase in pumping rate value enhance both the hydraulic head and concentration of colloids which slightly reduces the quality of pumped water from RBF. Keywords: Riverbank filtration, analytical model, colloids, hydraulic head and pumping rat

This research concerns with optimal utilization of unconfined coastal aquifers under different constraints. This problem is a complicated, nonlinear, and constrained one. Therefore, the Genetic Algorithm nonlinear optimization method was selected to handle the present problem. Analytical solution was applied for the groundwater flow equation under assumption of steady sharp interface in homogenous aquifer. Image theory and the superposition principle were the main tools used through the analytical procedure. A Fortran program was written to apply the mathematical concepts of the problem. The method was applied to aquifer system underlain the City of Miami Beach at north of Spain.

An integrated space–time artificial neural network (ANN) model inspired by the governing groundwater flow equation was developed to test whether a single ANN is capable of modeling regional groundwater flow systems. Model-independent entropy measures and random forest (RF)-based feature selection procedures were used to identify suitable inputs for ANNs. L2 regularization, five-fold cross-validation, and an adaptive stochastic gradient descent (ADAM) algorithm led to a parsimonious ANN model for a 30 691 km2 agriculturally intensive area in the Ogallala Aquifer of Texas. The model testing at 38 independent wells during the 1956–2008 calibration period showed no overfitting issues and highlighted the model's ability to capture both the observed spatial dependence and temporal variability. The forecasting period (2009–2015) was marked by extreme climate variability in the region and served to evaluate the extrapolation capabilities of the model. While ANN models are universal interpolators, the model was able to capture the general trends and provide groundwater level estimates that were better than using historical means. Model sensitivity analysis indicated that pumping was the most sensitive process. Incorporation of spatial variability was more critical than capturing temporal persistence. The use of the standardized precipitation–evapotranspiration index (SPEI) as a surrogate for pumping was generally adequate but was unable to capture the heterogeneous groundwater extraction preferences of farmers under extreme climate conditions.

Pumping tests are established for characterizing spatial average properties of aquifers. At the same time, they are promising tools to identify heterogeneity characteristics such as log-conductivity variance and correlation scales. We present the extended Generalized Radial Flow Model (eGRF) which combines the characterization of well flow in fractal geometry with an upscaled conductivity for pumping tests in heterogeneous media. We show that the eGRF is a generalization of previously solutions, such as that of Barker, Butler and Neuman. We derive effective conductivities for uniform and well flow conditions in heterogeneous log-normal media with a truncated power law correlation structure through the upscaling procedure Coarse Graining. The radial-dependent effective conductivity for well flow reflects the gradual change of heterogeneity impact on average pumping test drawdowns. We then combine upscaled conductivities with the eGRF model to determine the effective pumping test solution. We provide a proof of concept by comparing theoretical upscaling results with Monte Carlo well flow simulations in heterogeneous fractal fields. The eGRF and upscaling results are implemented and made freely available as python code for transport simulation as well as pumping test analysis.

A robust, high order modeling approach is introduced, based on the finite difference-based radial basis functions method, for solving the groundwater flow equation in the presence of an active well, in the case of a confined aquifer in a complex geological environment. The two important novelties of this work are the analytical handling of the wells’ singularities and the ability to do this accurately and efficiently in a heterogeneous medium. It is argued that the most commonly used methods for this type of problem have severe weaknesses in both the treatment of the singularities associated with the well, and in representing heterogeneities which commonly occur in geological processes. The method presented here is first applied to the groundwater flow problem in a homogeneous medium for which the analytical solution is known, to show its high order algebraic convergence. The method is then compared against the United States geological survey’s MODFLOW software on a quasi-realistic benchmark test case in a heterogeneous medium. It is shown that much fewer nodes are needed by the proposed method to yield a similar accuracy.

Governing equations of physical problems are traditionally derived from conservation laws or physical principles. However, some complex problems still exist for which these first-principle derivations cannot be implemented. As data acquisition and storage ability have increased, data-driven methods have attracted great attention. In recent years, several works have addressed how to learn dynamical systems and partial differential equations using data-driven methods. Along this line, in this work, we investigate how to discover subsurface flow equations from data via a machine learning technique, the least absolute shrinkage and selection operator (LASSO). The learning of single-phase groundwater flow equation and contaminant transport equation are demonstrated. Considering that the parameters of subsurface formation are usually heterogeneous, we propose a procedure for learning partial differential equations with heterogeneous model parameters for the first time. Derivative calculation from discrete data is required for implementing equation learning, and we discuss how to calculate derivatives from noisy data. For a series of cases, the proposed data-driven method demonstrates satisfactory results for learning subsurface flow equations.

Oscillatory pumping tests (OPTs) provide an alternative to constant-head and constant-rate pumping tests for determining aquifer hydraulic parameters when OPT data are analyzed based on an associated analytical model coupled with an optimization approach. There are a large number of analytical models presented for the analysis of the OPT. The combined effects of delayed gravity drainage (DGD) and the initial condition regarding the hydraulic head are commonly neglected in the existing models. This study aims to develop a new model for describing the hydraulic head fluctuation induced by the OPT in an unconfined aquifer. The model contains a groundwater flow equation with the initial condition of a static water table, Neumann boundary condition specified at the rim of a partially screened well, and a free surface equation describing water table motion with the DGD effect. The solution is derived using the Laplace, finite-integral, and Weber transforms. Sensitivity analysis is carried out for exploring head response to the change in each hydraulic parameter. Results suggest that the DGD reduces to instantaneous gravity drainage in predicting transient head fluctuation when the dimensionless parameter a1=ϵSyb/Kz exceeds 500 with empirical constant ϵ, specific yield Sy, aquifer thickness b, and vertical hydraulic conductivity Kz. The water table can be regarded as a no-flow boundary when a1<10-2 and P4 s, with P being the period of the oscillatory pumping rate. A pseudo-steady-state model without the initial condition causes a time-shift from the actual transient model in predicting simple harmonic motion of head fluctuation during a late pumping period. In addition, the present solution agrees well with head fluctuation data observed at the Savannah River site. Highlights. An analytical model of the hydraulic head due to oscillatory pumping in unconfined aquifers is presented. Head fluctuations affected by instantaneous and delayed gravity drainages are discussed. The effect of the initial condition on the phase of head fluctuation is analyzed. The present solution agrees well with head fluctuation data taken from field oscillatory pumping.

The Boussinesq equation is usually used to describe one-dimensional unconfined groundwater movement. Solutions of this equation are important as they provide useful insights regarding the water table response to stream level variations and allow us to quantify the exchange flow between the stream and the aquifer. Due to the nonlinearity of the Boussinesq equation, the solutions are generally obtained using numerical methods. However, for certain classes of initial and boundary conditions there are both exact and approximate analytical solution techniques. This work focuses on the latter approach. A new mathematical technique for approximate solutions of the Boussinesq equation describing flow in horizontal unconfined aquifers induced by sudden change in boundary head is presented. The method applies to the problems of recharging and dewatering of an unconfined aquifer, and approximate solutions to both problems are derived. The solutions were obtained by introducing an empirical function with four parameters which might be obtained using a numerical fitting procedure. Results based on this technique were found to be easily calculated and to be in good agreement with those obtained using numerical calculation based on Runge-Kutta approach. A benchmark between the proposed solutions and five existing approximate analytical solutions shows that the present solutions are the most accurate approximate solutions among those tested. Applications of the solutions are presented in the context of estimating aquifer hydraulic parameters.

Three-dimensional mathematical representation of territorial subsurface water flow is a worthwhile contribution to the regulation and governing of underwater reserves as they provide the components of hydrological processes as well as the flow of water in an aquifer. An attempt is made in applying this type of modeling study in the Palar river basin, Thirukkazhukundram block, southern India, which is chosen as the study area. The research area is signalized by multiple aquifer system consumed for agrarian and intake purposes. This model is having two vertical layers of the zonal type which prompts subsurface water flow over an area of 229 km2 with 34 rows and 40 columns with a size of 500m2 grids for elaborate study. The model was initiated in a steady and intermittent state utilizing a finite difference approximation of the three-dimensional partial differential groundwater flow equation in the concerned aquifer from Jan 2017 to Dec 2019. The pattern is designed in calibrating constant and transitory conditions. There is always a sensible match between the worked-out and spotted heads. As per the obtained modeling output, the aquifer is noted to be stable with the acquired pumping rate. The transient model is expected to work out until the period 2022. This model is helpful in forecasting the active groundwater flow under various pumping tests also, in monitoring the release and recharge of water.

Darcy’s law is the basic law of flow, and it produces a partial differential equation is similar to the heat transfer equation when coupled with an equation of continuity that explains the conservation of fluid mass during flow through a porous media. This article, titled the groundwater flow equation, covers the derivation of the groundwater flow equations in both the steady and transient states. We look at some of the most common approaches and methods for developing analytical or numerical solutions. The flaws and limits of these solutions in reproducing the behavior of water flow on the aquifer are also discussed in the article.

In this work, the aim is to discuss a fractional model for the groundwater flow equation; as groundwater is not static, it flows into the groundwater layer and its flow can be described using the fractional differential equation with the initial and boundary conditions associated with it, and this model is derived using the law of conservation of mass and generalization of the Darcy law with taking into account the flow of water to the piezometric head as a function of the derivative of the fractional arrangement. To solve the groundwater model equation, there are many different techniques such as the Frobenius method and the finite element method. In this work, we present an approximate solution to space–time fractional derivative of the groundwater flow equation, by applying Sumudu transformation and the Adomain decomposition method (ADM). The proposed method is validated by providing two examples and simulations for these examples to complete the paper. Illustrative examples show that the proposed method can be used for similar problems and is very effective. The fractional derivative is Caputo derivative in this paper, and this problem has great importance in petroleum technology.

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.

In this paper, the groundwater flow equation within an unconfined aquifer is modified using the concept of new derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. Some properties and applications are given regarding the Caputo-Fabrizio fractional order derivative. The existence and the uniqueness of the solution of the modified groundwater flow equation within an unconfined aquifer is presented, the proof of the existence use the definition of Caputo-Fabrizio integral and the powerful fixed-point Theorem. A detailed analysis on the uniqueness is included. We perform on the numerical analysis on which the Crank-Nicolson scheme is used for discretisation. Then we present in particular the proof of the stability of the method, the proof combine the Fourier and Von Neumann stability analysis. A detailed analysis on the convergence is also achieved.

(IF 3.67; Q1); International audience; In this work, we present a novel method to interpret, at a field scale, the preferential flows generated by harmonic pumping tests, in which the pumped flowrate varies according to a sinusoidal function with a given period. The experimental protocol relies on the application of harmonic pumping tests in a karstic field located near to Montpellier (Southern France) at 4 different boreholes, each time with a shorter and a longer period, and the analysis of the hydraulic responses recorded at the 13 observation wells. A qualitative analysis of the oscillatory component in the hydraulic responses, in term of amplitude decay and phase lag, permitted to propose a preliminary model of degree of connectivity between the boreholes, through the network of conduits. Then, a quantitative interpretation of the harmonic responses was applied to image the spatial heterogeneity of the hydraulic properties (hydraulic conductivity and storage coefficient) by using a deterministic inverse algorithm called CADI. This algorithm is based on an equivalent porous medium concept and parameterized by a Cellular Automata approach in order to provide a realistic reconstruction of the karstic network. This algorithm is linked to the groundwater flow equation, reformulated in frequency domain, to simulate the amplitudes and phase shifts responses to the harmonic pumping tests. The inverse process was successfully applied on the dataset collected with both periods, in a separate and joint way. The results obtained allowed for a discussion on the efficiency of the harmonic pumping tomography for the characterization of the karstic structures.

In this manuscript, we discuss the capabilities of a deep learning algorithm implemented with the Conventional Neural Network concept to characterize the hydraulic properties of aquifers. The algorithm called CNN-HT is designed to predict the inverse operator of hydraulic tomography using a synthetic training dataset in which the hydraulic head data associated with pumping tests are linked to hydraulic transmissivity field. This approach relies on an adaptation of the SegNet network that was initially developed to process image segmentation. The SegNet is composed of encoders and decoders networks. In the encoder, sequential operations with multiple filters, as convolution, batch normalization, max-pooling are performed to identify feature maps of the input data. In the decoder, the up-sampling, convolution, batch normalization and regression operations are used to prepare the output by recovering the loss of spatial resolution that occurred in the encoder process. In this adaptation, we used the least-square iterative formulation at the initial iteration with Jacobian matrix to resize the hydraulic head data to match the size of the output (transmissivity field). This protocol was applied to the hydraulic head data computed numerically by solving the groundwater flow equation for a given transmissivity field, generated geostatistically with Gaussian and spherical variograms. A part of this data was used for training the network and the other part to test its performance. The test step confirmed the effectiveness of this tool in reconstructing the main heterogeneities of the hydraulic properties, and its effectiveness is related to the nature and quantity of the training data. Moreover, the CNN-HT method provided inversion results of the same quality than those obtained with the Gauss-Newton algorithm using the finite difference or adjoint state method in the computation of the Jacobian matrix. However, the computational time is longer in CNN-HT but this time can be less or of the same order as that of Gauss-Newton using finite difference method.

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.

Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi-fractional confined aquifer is developed. First, a dimensionally consistent continuity equation for transient saturated groundwater flow in fractional time and in a multi-fractional, multidimensional confined aquifer is developed. For the equation of water flux within a multi-fractional multidimensional confined aquifer, a dimensionally consistent equation is also developed. The governing equation of transient saturated groundwater flow in a multi-fractional, multidimensional confined aquifer in fractional time is then obtained by combining the fractional continuity and water flux equations. To illustrate the capability of the proposed governing equation of groundwater flow in a confined aquifer, a numerical application of the fractional governing equation to a confined aquifer groundwater flow problem was also performed.

A new mathematical model is developed to describe the flow in response to a constant-head pumping (or constant-head test, CHT) in a leaky unconfined aquifer system of infinite lateral extent with considering unsaturated flow. The model consists of an unsaturated zone on the top, an unconfined aquifer in the middle, and a second aquifer (aquitard) at the bottom. The unsaturated flow is described by Richard's equation, and the flows in unconfined aquifer and second layer are governed by the groundwater flow equation. The well partially penetrates the unconfined aquifer with a constant head in the well due to CHT. The governing equations of the model are linearized by the perturbation method and Gardner's exponential model is adopted to describe the soil retention curves. The solution of the model for drawdown distribution is obtained by applying the methods of Laplace transform and Weber transform. Then the solution for the wellbore flowrate is derived from the drawdown solution with Darcy's law. The issue of the equivalence of normalized drawdown predicted by the present solution for constant-head pumping and Tartakovsky and Neuman's (2007) solution for constant-rate pumping is discussed. On the basis of the wellbore flowrate solution, the results of the sensitivity analysis indicate that the wellbore flowrate is very sensitive to the changes in the radial hydraulic conductivity and the thickness of the saturated zone. Moreover, the results predicted from the present wellbore flowrate solution indicate that this new solution can reduce to Chang’s et al. (2010a) solution for homogenous aquifers when the dimensionless unsaturated exponent approaches 100. The unsaturated zone can be considered as infinite extent in the vertical direction if the thickness ratio of the unsaturated zone to the unconfined aquifer is equal to or greater than one. As for the leakage effect, it can be ignored when the vertical hydraulic conductivity ratio (i.e., the vertical hydraulic conductivity of the lower layer over that of the unconfined aquifer) is smaller than 0.1. The present solution is compared with the numerical solution from FEMWATER for validation and the results indicate good match between these two solutions. Finally, the present solution is applied to a set of field drawdown data obtained from a CHT for the estimation of hydrogeologic parameters.

Protecting groundwater resources plays an important role in watershed management. For this purpose, studies on groundwater flow dynamics incorporating surface water–groundwater interactions have been conducted including analytical, numerical, and experimental models. In this research, a stream–aquifer system was considered to understand the physical behavior of surface water–groundwater interactions. Interactions in a stream–aquifer system were incorporated into the mathematical modeling by defining the stream head as a boundary condition for the groundwater flow equation. This boundary was chosen as a sloping stream boundary, which is an approach in representing the natural conditions of the stream and may be used to define continuous interactions between stream and aquifer. A semi-analytical solution for transient 2D groundwater flow was developed for the considered problem. Isotropic, homogeneous, and finite aquifer assumptions were made in order to define the aquifer characteristics. Then, a series of laboratory experiments was conducted to simulate this stream–aquifer system. Finally, a numerical model was developed by using Visual MODFLOW to verify analytical and experimental results. Numerical results matched with both analytical solutions and the experimental observations.

The hydraulic diffusivity gives a measure of diffusion speed of pressure disturbances in groundwater system; large values of hydraulic diffusivity lead to fast propagation of signals in aquifer. This research provides a novel design and derives spectral representation to determine hydraulic diffusivity using spectral analysis of groundwater levels coupled with time-dependent boundary adjacent to marine system and no flow boundary in aquifer system. To validate the proposed method, water levels of fluctuated boundary and groundwater well in a sandy confined aquifer were collected. The hydraulic diffusivity is then obtained by an inverse process in the non-linear complex form of spectral relationship. The method essentially is constructed on the conceptual design of natural forcing transmitted in large aquifer. It is unlike the conventional field pumping test which is only used to determine hydraulic properties of groundwater in small range around the well. Hydraulic diffusivity of the confined aquifer is determined using real observation and then checked by comparing to the published range. It suggests that without local aquifer test to estimate hydraulic diffusivity in a coastal aquifer using spectral representation with its relevant flow system and boundary has become feasible.

Analytical simulation of groundwater flow is a necessary and useful technique to predict the various behavior patterns of a groundwater system. The main aim of the present study is to derive new analytical solutions to compute the unsteady flows inside an aquifer between two parallel streams of constant and varying heads. The problems are solved by means of Laplace transform method and the solution results are verified with the results of MODFLOW. It is observed that the obtained results agreed very well with the results of MODFLOW. The solutions are carried out for two cases of ascending and descending water levels and the obtained results are compared with each other. In addition, the sensitivity of hydraulic heads to aquifer parameters and how locations of water divide change by change in aquifer parameters are investigated. In sensitivity analysis of hydraulic heads to changes in recharge rate with different values of hydraulic conductivity, thickness, and length of the aquifer, it is shown that among these parameters the length of the aquifer is the most important parameter affecting the hydraulic heads. Furthermore, the sensitivity of flow rates to recharge rates and water level change rates are analyzed.

The prediction of long-term dynamics of transitional environments, e.g., lagoon evolution, salt-marsh growth or river delta progradation, is an important issue to estimate the potential impacts of different scenarios on such vulnerable intertidal morphologies. The numerical simulation of the combined accretion and consolidation, i.e., the two main processes driving the dynamics of these environments, however, suffers from a significant geometric non-linearity, which may result in a pronounced grid distortion using standard grid-based discretization methods. The present work describes a novel numerical approach, based on the Virtual Element Method (VEM), for the long-term simulation of the vertical dynamics of transitional landforms. The VEM is a grid-based variational technique for the numerical discretization of Partial Differential Equations (PDEs) allowing for the use of very irregular meshes consisting of a free combination of different polyhedral elements. The model solves the groundwater flow equation, coupled to a geomechanical module based on Terzaghi's principle, in a large-deformation setting, taking into account both the geometric and the material non-linearity. The use of the VEM allows for a great flexibility in the element generation and management, avoiding the numerical issues connected with the adoption of strongly distorted meshes. The numerical model is developed, implemented and tested in real-world examples, showing an interesting potential for addressing complex environmental situations.

[1] Heterogeneity is the single most salient feature of hydrogeology. An enormous amount of work has been devoted during the last 30 years to addressing this issue. Our objective is to synthesize and to offer a critical appraisal of results related to the problem of finding representative hydraulic conductivities. By representative hydraulic conductivity we mean a parameter controlling the average behavior of groundwater flow within an aquifer at a given scale. Three related concepts are defined: effective hydraulic conductivity, which relates the ensemble averages of flux and head gradient; equivalent conductivity, which relates the spatial averages of flux and head gradient within a given volume of an aquifer; and interpreted conductivity, which is the one derived from interpretation of field data. Most theoretical results are related to effective conductivity, and their application to real world scenarios relies on ergodic assumptions. Fortunately, a number of results are available suggesting that conventional hydraulic test interpretations yield (interpreted) hydraulic conductivity values that can be closely linked to equivalent and/or effective hydraulic conductivities. Complex spatial distributions of geologic hydrofacies and flow conditions have a strong impact upon the existence and the actual values of representative parameters. Therefore it is not surprising that a large body of literature provides particular solutions for simplified boundary conditions and geological settings, which are, nevertheless, useful for many practical applications. Still, frequent observations of scale effects imply that efforts should be directed at characterizing well-connected stochastic random fields and at evaluating the corresponding representative hydraulic conductivities.

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.

(IF 3.91 [2018]; Q1); International audience; The self-potential method can be used to detect and monitor anomalous seepages in dams and embankments. In such a case, an electrical field of electrokinetic nature (i.e., associated with pore water flow) can be measured using a set of non-polarizable electrodes typically located at the ground surface or in some wells. This field can be in turn related to the pattern of groundwater flow. We built an experimental dam to investigate to which extent the self-potential method can help characterizing seepages in dams. We first use the finite element method to simulate the ground water flow in a heterogeneous porous and permeable material by solving the groundwater flow equation. The resulting groundwater flow solution is then used to compute the electrical potential distribution by solving the corresponding elliptic partial differential equation. In a preliminary experiment, we could not measure any self-potential anomaly associated with the infiltration of water in the dam. Our numerical simulations showed that the magnitudes of the self-potential anomalies were controlled by (1) the nature of the flow regime (viscous laminar versus inertial laminar flow regimes) and (2) the presence of insulating Polyvinyl Chloride (PVC) tubes located at the end of the preferential flow channels in the structure of the dam. Thanks to these numerical simulations, we added sand at the entrance of the infiltration area in order to reduce the effects of the PVC tubes and to restrain the flow regime to the viscous laminar flow regime. New experiments allowed for detecting a self-potential anomaly with an amplitude of around −9 mV consistent with that obtained through numerical modelling with a finite element simulator. This comparison was used to test the accuracy of the modelling approach and define the strengths and weaknesses of the self-potential method to determine preferential seepages in earth dam structures.

Accurate modeling of hydraulic properties such as transmissivity and interbed specific storages is significant for reliable predictions of land subsidence modeling. Calibration of land subsidence model is a challenge because of the strong non-linearity of groundwater flow equation especially when it accounting for the interbed drainage process. Pumping well drawdown and land subsidence data are very important signals for identification of aquifer hydraulic properties. In this work, it is proposed that the ensemble Kalman filter is used to calibrate the transmissivity and interbed elastic and inelastic specific storages using both drawdown and subsidence data for the first time. A synthetic example demonstrated that the characterization of transmissivity and specific storages is improved, and the uncertainties of predictions of both drawdown and subsidence are reduced, when additional dynamic observation data are used for inverse modeling. Issues such as how to account for interferometric synthetic aperture radar data, which may be encountered using the EnKF for real case studies, are discussed.

A low-dimensional model that describes both saturated and unsaturated flow processes in a single equation is presented. Subsurface flow processes in the groundwater, the vadose zone, and the capillary fringe are accounted for through the computation of aggregated hydrodynamic parameters that result from the integration of the governing flow equations from the bedrock to the land surface. The three-dimensional subsurface flow dynamics are thus described by a two-dimensional equation, allowing for a drastic reduction of model unknowns and simplification of the model parameterizations. This approach is compared with a full resolution of the Richards equation in different synthetic test cases. Because the model reduction stems from the vertical integration of the flow equations, the test cases all use different configurations of heterogeneity for vertical cross-sections of a soil-aquifer system. The low-dimensional flow model shows strong consistency with results from a complete resolution of the Richards equation for both the water table and fluxes. The proposed approach is therefore well suited to the accurate reproduction of complex subsurface flow processes.

This work is an initial study of a numerical method for identifying multiple leak zones in saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modelled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes, since leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we therefore modify and employ an iterative regularizing method proposed in [1] and [2]. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint, to get a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a Finite element method (FEM), and the numerical results indicate that the leak zones can be identified with the proposed method.

A hybrid discrete-continuum numerical model, Variable-Density Flow and Solute Transport—Conduit Flow Process (VDFST-CFP), is developed to simulate seawater intrusion to a coastal karst aquifer with a conduit network. The Darcy-Weisbach equation is applied to simulate the nonlaminar groundwater flow in the conduit system that is conceptualized as pipes, while the Darcy equation is used for laminar groundwater flow in the continuum porous medium. Density-dependent groundwater flow with appropriate additional density terms in the conduit is analytically derived. The flow and transport equations are coupled, and numerically solved by the finite difference method with an implicit iteration procedure. Two synthetic benchmarks are developed to compare the VDFST-CFP model results with other numerical models, such as the variable-density SEAWAT, constant-density continuum MODFLOW/MT3DMS, and constant-density discrete-continuum CFPv2/UMT3D models. The VDFST-CFP model compares reasonably well with the other model results in both conduit and porous medium domains, and well describes water and salt exchange between the two systems. Under turbulent flow conditions within the conduit, the Darcy-Weisbach equation calculates the flow rate more accurately without overestimation by the Darcy equation. Sensitivity analysis indicates that conduit diameter, friction factor, matrix hydraulic conductivity, and effective medium porosity are important parameters in the VDFST-CFP model. The pros and cons of the VDFST-CFP model are discussed, including the model assumptions and simplifications, limitations of the discrete-continuum modeling method, and the convergence criteria. In general, the newly developed VDFST-CFP model provides a new numerical modeling method for simulating seawater intrusion in a coastal karst aquifer with conduits.

Groundwater as an important, life-sustaining resource for humankind is being threatened by massive over-extraction and wide-spread contamination. Wise development and protection of this crucial resource requires a thorough understanding of groundwater flow in the subsurface. This paper presents a novel direct method to estimate important hydraulic parameters characterizing steady state groundwater flows for confined and unconfined isotropic aquifers. The method is appropriate for application in aquifers where horizontal flow dominates. The governing equations for the direct estimation method are superposed extensions of the well-known Thiem equation governing steady-state radial flow toward a pumping well under confined and unconfined conditions. This new approach has the following advantages over conventional methods: (1) simultaneously provides estimates of both hydraulic conductivity and hydraulic gradient, (2) can be applied using historical data without the need to conduct a pumping test, and (3) is a simple analytical method that can be applied easily. Verification of the direct estimation method is achieved by applying it to hypothetical homogeneous and heterogeneous aquifers simulated by three-dimensional finite element models. The usefulness of the method is also demonstrated with data from an actual field site. Two dimensional steady state flow in confined and unconfined aquifers.Assessment of aquifer parameters using long term pumping and piezometric head data.Concurrent estimation of aquifer permeability and groundwater discharge rate.

We obtain a solution to a boundary-value problem of a flow of spherical form particle for stationary system of equations of viscous non-isothermal gaseous medium including the Stokes equation, heat conductivity equation, and state equation with account taken of dependence of viscosity, heat conductivity, and density of gaseous medium on temperature.

A new model of groundwater flowing within a confined aquifer was proposed using the concept of local derivative with fractional order. The derivative used in this model obeys all the properties of a local derivative and has a fractional order. The new groundwater flow equation was solved analytically via three different analytical methods. The first method is the well-known method of separation of variable. The problem with this method is the introduction of the Eigen-value that does not have physical meaning. The second method was achieved using novel integral equation called Atangana-transform and this method yields exact solution. An alternative method based on the modified Boltzmann transformation also yields to exact solution. Some numerical simulations were done to express the efficiency of the model

Use of heat-as-a-tracer is a common method to quantify surface water-groundwater interactions (SW-GW). However, the method relies on assumptions likely violated in natural systems. Numerical studies have explored violation of fundamental assumptions such as heterogeneous streambed properties, two-dimensional groundwater flow fields and uncertainty in thermal parameters for the 1D heat-as-a-tracer method. Few studies to date have modeled complex, fully three-dimensional groundwater flows to address the impacts of non-uniform, 3D flow vectors on use of heat-as-a-tracer to quantify SW-GW interactions. COMSOL Multiphysics was used to model scenarios in a fully three-dimensional flow field in homogeneous, isotropic sand with a sinusoidal temperature upper boundary where vertical flows are deliberately disrupted by large and varied horizontal flows from two directions. Resulting temperature time series from multiple depths were used to estimate vertical Darcy flux and compared with modeled fluxes to assess the performance of the 1D thermal methods to quantify multi-dimensional groundwater flows. In addition, apparent effective thermal diffusivity was calculated from synthetic temperature time series, and compared to model input diffusivity. Both increasingly non-uniform and non-vertical groundwater flow fields resulted in increasing errors for both the temperature-derived flux and temperature-derived effective thermal diffusivity. For losing (downward) flow geometries, errors in temperature-derived effective thermal diffusivity were highly correlated with errors in temperature-derived flux and were used to identify how and when underlying assumptions necessary for heat-as-a-tracer for quantifying groundwater flows were violated. Specifically, non-uniform flow fields (with flow lines that converge or diverge) produced the largest errors in simulated fluxes. This article is protected by copyright. All rights reserved.

Summary A semi-analytical solution was presented for groundwater flow due to pumping in a leaky sloping fault-zone aquifer surrounded by permeable matrices. The flow in the aquifer was descried by a three-dimensional flow equation, and the flow in the upper and lower matrix blocks are described by a one-dimensional flow equation. A first-order free-water surface equation at the outcrop of the fault-zone aquifer was used to describe the water table condition. The Laplace domain solution was derived using Laplace transform and finite Fourier transform techniques and the semi-analytical solutions in the real time domain were evaluated using the numerical inverse Laplace transform method. The solution was in excellent agreement with Theis solution combined with superposition principle as well as the solution of Huang et al. (2014). It was found that the drawdown increases as the sloping angle of the aquifer increases in early time and the impact of the angle is insignificant after pumping for a long time. The free-water surface boundary as additional source recharges the fault aquifer and significantly affect the drawdown at later time. The surrounding permeable matrices have a strong influence on drawdown but this influence can be neglected when the ratio of the specific storage and the ratio of the hydraulic conductivity of the matrices to those of the fault aquifer are less than 0.001.

In this paper our primary aim is to define the changes of air and internal wall temperature in underground spaces in time domain. As an additional aim the change of heat flux through the wall in time domain has been calculated. Based on the heat balance, the dynamic basic equation of the space has been defined. The basic equation is a differential equation which contains the internal heat sources and the heat capacity of the space. For solving the basic equation, the initial condition, the time-varying boundary condition of the third kind and the Fourier's conductivity differential equation are necessary. The convolution integral of the solution function has been obtained by the use of the integral-differential equation acquired by substituting the temperatures and heat fluxes into the basic equation. The solution of the acquired equation can be obtained in a numerical way. Our new mathematical approach to the solution of the physical model makes it possible to investigate the air and wall temperatures, as well as the heat flow through the wall in underground spaces.

The flow of a thin plastic layer between two rigid plates approaching each other in the normal direction is considered. The kinematics of plastic layer flow is studied. An evolution equation describing the free boundary of the flow region is derived. The similarity solutions to this equation are analyzed. It is shown that the evolution equation can be reduced to a particular case of the nonlinear heat conduction equation. New exact particular solutions to the evolution equation are obtained using the variable separation method and the method of self-similar transformations.

Most previous solutions for groundwater flow induced by localized recharge assumed either aquifer incompressibility or two-dimensional flow in the absence of the vertical flow. This paper develops a new three-dimensional flow model for hydraulic head variation due to localized recharge in a rectangular unconfined aquifer with four boundaries under the Robin condition. A governing equation describing spatiotemporal head distributions is employed. The first-order free-surface equation with a source term defining a constant recharge rate over a rectangular area is used to depict water table movement. The solution to the model for the head is developed with the methods of Laplace transform and double-integral transform. Based on Duhamel's theorem, the present solution is applicable to flow problems accounting for arbitrary time-dependent recharge rates. The solution to depth-average head can then be obtained by integrating the head solution to elevation and dividing the result by the aquifer thickness. The use of a rectangular aquifer domain has two merits. One is that the integration for estimating the depth-average head can be analytically achieved. The other is that existing solutions based on aquifers of infinite extent can be considered as special cases of the present solution before the time when the aquifer boundary had an effect on head predictions. With the help of the present solution, the assumption of neglecting the vertical flow effect on the temporal head distribution at an observation point outside a recharge region can be assessed by a dimensionless parameter related to the aquifer horizontal and vertical hydraulic conductivities, initial aquifer thickness, and the shortest distance between the observation point and the edge of the recharge region. The validity of assuming aquifer incompressibility is dominated by the ratio of the aquifer specific yield to its storage coefficient. In addition, a sensitivity analysis is performed to investigate the head response to the change in each of the aquifer parameters.

Groundwater, being a vital component of the natural water resource system, needs continuous monitoring and dynamic management strategies. That said, we require computationally inexpensive groundwater flow models for repetitive solutions with desirable accuracy under budgetary limitation(s). Natural aquifer systems inherit strong heterogeneity at local scales. In this work, we have proposed ordinary kriging-based sequential algorithm for generating replicates of randomly distributed heterogeneous hydraulic conductivity field (Monte Carlo method-based algorithm) conditioned by field values from sampled locations in an irregular-unstructured grid system. Finite Volume method-based groundwater models often encounter difficulties with the representation of point source/sink terms operating within the domain. In this paper, we have proposed an irregular-unstructured grid Finite Volume discretization technique for overcoming the singularity of point source/sink term to yield a consistent output with different grid dimensions. Furthermore, full-system groundwater models often come with a substantial computational burden. Hence, reduction in model order cuts down the computational expenses (in terms of CPU time and usage) to a significant level. We have also put forth a model order reduction methodology for three different illustrative pumping tests. The proposed framework for the model order reduction projects the governing groundwater flow equation onto a set of identified patterns or orthonormal basis functions, applying the Galerkin Projection method to compute a vector of time-dependent coefficients. We have performed pattern identification by Singular Value Decomposition (SVD) of snapshots of full-system model simulation data at selected time instants within the pumping test time domain. The numerical results of the proposed reduced-order models show a good approximation of the full-system models at a comparatively lesser computational time. The accuracy and efficiency of the models attempt to ensure their potential applicability for identifying groundwater dynamics.