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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

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.

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.

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.

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.

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.

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.

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.

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.

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.

Hydraulic Tomography (HT) has become one of the most robust methods to characterize the heterogeneity in hydraulic parameters such as hydraulic conductivity and specific storage. However, in order to obtain high resolution hydraulic parameter estimates, several pumping/injection tests with sufficient monitoring data are necessary. In highly heterogeneous media, even with large numbers of measurements, the resolution may not be enough for predicting contaminant transport behavior. In addition, during inverse modeling, the groundwater flow equation is solved numerous times, thus the computational burden could be large, especially for a large, three-dimensional, transient model.

The differential groundwater flow equation derived in Chap. 2 can be solved analytically in various geometries, provided that certain hypotheses are satisfied. In this chapter, a polar coordinate system with radial geometry, describing the radial groundwater flow towards a well, is considered. The hypotheses underlying the analytical solutions concern the aquifer’s geometry (constant thickness, homogeneity and isotropy, unlimited horizontal extension, initially horizontal potentiometric surface) and the pumping well (fully penetrating, infinitesimal radius, negligible storage, laminar flow and constant pumping rate). Steady state and transient analytical solutions, respectively describing the drawdown as a function of the distance from the well (r), or of r and time, are provided for confined, leaky and unconfined aquifers. Theis’ (and Cooper and Jacob’s approximation) and Thiem’s equations describe, respectively, the transient and steady state solutions of the groundwater flow equation for confined aquifers. Hantush and Jacob, instead, derived the transient analytical solution for leaky aquifers, while De Glee formalized the steady state solution. In the case of unconfined aquifers, the steady state solution formally coincides, except for an adjustment to the drawdown, to Thiem’s solution. The transient solution was, instead, derived by Neuman, under specific simplifying hypotheses, given that a fully rigorous description of flow in unconfined aquifers would entail the use of a nonlinear and nonhomogeneous differential equation due to the inclination of the water table with pumping and the generation of a vertical component of flow velocity.

In this chapter, the differential groundwater flow equation that governs the distribution of the flow directions and rates in an aquifer is derived. The problem is examined at a macroscopic scale, neglecting an analysis of detailed solid–liquid interface distribution, which would entail excessive analytical and computational complexity, without contributing useful information from an operational standpoint. The equation is thus determined as a combination of the equations that express the law of mass conservation (whose terms are described for a representative elementary volume), Darcy’s law, and the storage variation due to changes in hydraulic head. Unsteady state groundwater flow in each aquifer type is described by a different equation, each defining the Laplacian of the hydraulic head as a function of the aquifer’s storage and transport capacity, and of the hydraulic head’s partial derivative with respect to time. In particular, in the case of confined aquifers, flow is a function of specific yield and transmissivity, as is the case even for leaky aquifers, whose hydrodynamic behavior is, however, also affected by the leakage between aquifers (quantified by the leakage factor). A rigorous description of flow in unconfined aquifers would require a nonlinear and nonhomogeneous differential equation due to the inclination of the water table with flow; however, under simplifying conditions an approximate description, analogous to the confined aquifer equation, can be defined as a function of specific yield and transmissivity.

A mathematical optimal control method is developed to identify a hydraulic conductivity distribution in a density dependent flow field. Using a variational method, the adjoint partial differential equations are obtained for the density- dependent state equations used for the saline aquifer water flow. The adjoint equations are numerically solved in through a finite difference method. The developed method is applied to identify the hydraulic conductivity distribution through the numerical solution of an optimal control problem. To demonstrate the effectiveness of the optimal control method, three numerical experiments are conducted with artificial observation data. The results indicate that the developed method has the potential to accurately identify the hydraulic conductivity distribution in a saline water aquifer flow system.

This study develops a three-dimensional (3-D) mathematical model for describing transient hydraulic head distributions due to pumping at a radial collector well (RCW) in a rectangular confined or unconfined aquifer bounded by two parallel streams and no-flow boundaries. The streams with low-permeability streambeds fully penetrate the aquifer. The governing equation with a point-sink term is employed. A first-order free surface equation delineating the water table decline induced by the well is considered. Robin boundary conditions are adopted to describe fluxes across the streambeds. The head solution for the point sink is derived by applying the methods of finite integral transform and Laplace transform. The head solution for a RCW is obtained by integrating the point-sink solution along the laterals of the RCW and then dividing the integration result by the sum of lateral lengths. On the basis of Darcy's law and head distributions along the streams, the solution for the stream depletion rate (SDR) can also be developed. With the aid of the head and SDR solutions, the sensitivity analysis can then be performed to explore the response of the hydraulic head to the change in a specific parameter such as the horizontal and vertical hydraulic conductivities, streambed permeability, specific storage, specific yield, lateral length, and well depth. Spatial head distributions subject to the anisotropy of aquifer hydraulic conductivities are analyzed. A quantitative criterion is provided to identify whether groundwater flow at a specific region is 3-D or 2-D without the vertical component. In addition, another criterion is also given to allow for the neglect of vertical flow effect on SDR. Conventional 2-D flow models can be used to provide accurate head and SDR predictions if satisfying these two criteria.

We investigate the flow in a porous medium subjected to an oscillatory (sinusoidal) pressure gradient. Direct numerical simulation (DNS) has been performed to benchmark the analytical solutions of the unsteady Darcy equation with two different expressions of the time scale: one given by a consistent volume averaging of the Navier–Stokes equation with a steady-state closure for the flow-resistance term, another given by volume averaging of the kinetic energy equation with a closure for the dissipation rate. For small and medium frequencies, the analytical solutions with the time scale obtained by the energy approach compare well with the DNS results in terms of amplitude and phase lag. For large dimensionless frequencies ( $$\omega \tau \gtrsim 10$$ ), we observe a slightly smaller damping of the amplitude than predicted by the unsteady Darcy equation with the low-frequency time scale. This can be explained by a change in the velocity fields towards a potential flow solution. Note that at those high frequencies, the flow amplitudes remain below 1 % of those of the steady-state flows. Our DNSs, however, indicate that the time scale predicted by the steady-state closure for the flow-resistance term is too small. In general, this study supports the use of the unsteady form of Darcy’s equation with constant coefficients to solve time-periodic Darcy flow, provided the proper time scale has been found.

The effect of groundwater flow on the heat transfer performance of pile geothermal heat exchanger (PGHE) with spiral coils was simulated by a 3-D simulation model using finite element method. Different groundwater flow conditions were taken into consideration by applying different hydraulic gradients. Based on the moving ring-coils model and simulation results, an improved analytical model is developed by introducing a key parameter of effective dimensionless velocity. The calculation results shows that the improved model can better describe the heat transfer performance of PGHE with spiral coils. Both numerical and analytical results indicated that the groundwater flow has an enhancing effect on the heat transfer performance of the PGHE with spiral coils and can accelerate the heat transfer process into stability. When the groundwater flow mean velocity equal to 6.98E−06 m/s, the amount of heat exchange is higher than 26.72% than it of non-advection situation. The improved ring-coils analytical model can be used as a reliable tool for the design of pile geothermal heat exchanger with spiral coils under groundwater flow.

Saltwater intrusion problems have been usually tackled through analytical models because of its simplicity, easy implementation and low computational cost. Most of these models are based on the sharp-interface approximation and the Ghyben–Herzberg relation, which neglects mixing of fresh water and seawater and implicitly assumes that salt water remains static. This paper provides insight into the validity of a sharp-interface approximation defined from a steady state solution when applied to transient seawater intrusion problems. The validation tests have been performed on a 3D unconfined synthetic aquifer, which include spatial and temporal distribution of recharge and pumping wells. Using a change of variable, the governing equation of the steady state sharp-interface problem can be written with the same structure of the steady confined groundwater flow equation as a function of a single potential variable (ϕ). We propose to approach also the transient problem solving a single potential equation (using also the ϕ variable) with the same structure of the confined groundwater flow equation. It will allow solving the problem by using the classical MODFLOW code. We have used the parameter estimation model PEST to calibrate the parameters of the transient sharp-interface equation. We show how after the calibration process, the sharp-interface approach may provide accurate enough results when applied to transient problems and improve the steady state results, thus avoiding the need of implementing a density-dependent model and reducing the computational cost. This has been proved by comparing results with those obtained using the finite difference numerical code SEAWAT for solving the coupled partial differential equations of flow and density-dependent transport. The comparison was performed in terms of piezometric heads, seawater penetration, transition zone width and critical pumping rates. Copyright © 2014 John Wiley & Sons, Ltd.

Purpose – A new coupled finite element model has been developed for the joint resolution of both the shallow water equations, that governs the free surface flow, and the groundwater flow equation that governs the motion of water through a porous media. The paper aims to discuss these issues. Design/methodology/approach – The model is based upon two different modules (surface and ground water) previously developed by the authors, that have been validated separately. Findings – The newly developed software allows for the assessment of the fluid flow in natural watersheds taking into account both the surface and the underground flow in the way it really takes place in nature. Originality/value – The main achievement of this work has dealt with the coupling of both models, allowing for a proper moving interface treatment that simulates the actual interaction that takes place between surface and groundwater in natural watersheds.

Flow nets are one of the effective tools for illustrating groundwater flow rate and direction. Flow nets of a thermal groundwater system under different reference temperatures are examined in this paper. A 3D mathematical model describing water flow and heat transport is established. The stream function and the equivalent hydraulic head are used to describe the flow field in terms of mass flux. The partial differential governing equations are given by the mass conservation equation of the fluid phase and by the advection–diffusion transport equations of heat. They are solved with the standard Galerkin finite element method. Influence of variation in temperature on thermal groundwater flow nets is illustrated with a simplified 2D vertical steady-state mathematical model. The results show that the equipotential lines are not always orthogonal to the streamlines in a geothermal system of a confined aquifer when the heat flow from below is present and when the temperature increases, the equipotential lines are clearly curved and the skew intersection angles between the streamlines and the equipotential lines decrease. For the node of coordinate of (5000, 0) in the aquifer, when the temperature difference between the top boundary and the bottom boundary is 4 and 10 °C, the skew intersection angle is 85.54° and 78.73°, respectively. When the hydraulic gradients are low, the equipotential lines are clearly curved. However, when the hydraulic gradients are high, the bending of the equipotential lines is not obvious. When the hydraulic gradient increases, the skew intersection angles change relatively little. For the same node when the hydraulic head of the left boundary is 600, 700, 800 and 1,000 m, the skew intersection angle is 78.73°, 84.3°, 86.19° and 87.71°, respectively. When the reference temperature increases, the hydraulic heads and the skew intersection angles increase.

An analytical solution for the space-time variation of contaminant concentration in one-dimensional transient groundwater flow in a homogenous semi-infinite aquifer, subjected to time-dependent source contamination, is derived. The uniform and time varying dispersion along transient groundwater flow is investigated under two conditions. First, the flow velocity distribution in the aquifer is considered as a sinusoidally varying function, and second, the flow velocity distribution is treated as an exponentially increasing function of time. It is assumed that initially the aquifer is not solute free, so the initial background concentration is considered as an exponentially decreasing function of the space variable which is tending to zero at infinity. It is assumed that dispersion is directly proportional to the square of the velocity, noting that experimental observations indicate that dispersion is directly proportional to the velocity with a power ranging from 1 to 2. The analytical solution is illustrated using an example and may help benchmark numerical codes and solutions.

Most groundwater equations for flow toward wells use a set of assumptions and idealizations about the aquifer–well configuration so that analytical expressions can be derived for steady-state and unsteady-state flows. In this article, the main assumption in these equations is that constant hydraulic conductivity is relaxed and instead allows radial variability. The basic question is how the hydraulic conductivity gradient affects groundwater flow. Changes in hydraulic conductivity influence groundwater flow; any local changes in the hydraulic conductivity cause local changes in hydraulic gradient and in groundwater velocity. This problem is solved using water balance equations with changes in linear radial hydraulic conductivity. Simple but more general equations for groundwater flow toward wells are derived and applied to steady-state groundwater flows in a confined aquifer. This formulation reduces to the classical Theim solution for constant hydraulic conductivity. The use of this methodology i...

In the field of computational fluid dynamics, the f inite volume method is dominant over other numerical techniques like the finite difference and finite element methods because the underlying physical quantities are conserved at the discrete level. In the presen t study, the finite volume method is used to solve an isotropic transient groundwater flow model to obtain hydrauli c heads and flow through an aquifer. The objective is to discuss the theory of finite volume method and its applications in groundwater flow modelling. To achieve this, an orthogonal grid with quadrilateral contro l volumes has been used to simulate the model using mixed boundary conditions from Bwaise III, a Kampala Surburb. Results show that flow occurs from regions of high hydraulic head to regions of low hydraulic head un til a steady head value is achieved.

A main purpose of groundwater inverse modeling lies in estimating the hydraulic conductivity field of an aquifer. Traditionally, hydraulic head measurements, possibly obtained in tomographic setups, are used as data. Because the groundwater flow equation is diffusive, many pumping and observation wells would be necessary to obtain a high resolution of hydraulic conductivity, which is typically not possible. We suggest performing heat tracer tests using the same already installed pumping wells and thermometers in observation planes to amend the hydraulic head data set by the arrival times of the heat signals. For each tomographic combinations of wells, we recommend installing an outer pair of pumping wells, generating artificial ambient flow, and an inner well pair in which the tests are performed. We jointly invert heads and thermal arrival times in 3-D by the quasi-linear geostatistical approach using an efficiently parallelized code running on a mid-range cluster. In the present study, we evaluate the value of heat tracer versus head data in a synthetic test case, where the estimated fields can be compared to the synthetic truth. Because the sensitivity patterns of the thermal arrival times differ from those of head measurements, the resolved variance in the estimated field is 6 to 10 times higher in the joint inversion in comparison to inverting head data only. Also, in contrast to head measurements, reversing the flow field and repeating the heat-tracer test improves the estimate in terms of reducing the estimation variance of the estimate. Based on the synthetic test case, we recommend performing the tests in four principal directions, requiring in total eight pumping wells and four intersecting observation planes for heads and temperature in each direction.

Water is a very important resource for any living organism and is mainly underground. The movement of groundwater is governed by the Laplace equation r 2 h = 0, where h is the hydraulic head. Much research has been on groundwater modelling and many of them are inspired by Darcy’s law. The Boundary Element Method is one of the few widely used numerical techniques for solving boundary value problems in engineering and physical sciences. In this study, a steady state groundwater ow model has been analyzed using the boundary element method. The results show that water moves from points at higher hydraulic head to points at lower hydraulic head. This result using the boundary element method is in perfect agreement with the 2014 result by the same authors using the nite volume method.

[1] An analytical treatment of aquifer flow is presented to establish a link between runoff models and aquifer properties. Recently developed one-dimensional analytical solutions for transient flows in homogeneous aquifers produce expressions for the flux between the aquifer and the surface water, and the difference between the average hydraulic head in the aquifer and the surface water. It is shown that the ratio between these large-scale variables (i.e., the aquifer-scale hydraulic conductivity) can assume one of three asymptotic values. A non-Darcian aquifer-scale flow equation is derived for the average head in the aquifer minus the surface water level. This first-order ordinary differential equation has nonconstant coefficients based in part on the aquifer-scale conductivity. The aquifer-scale equation is a generalization of linear reservoir models: when the aquifer-scale conductivity is stationary, its solution has an exponential term (like a linear reservoir) with a reservoir coefficient that depends on external factors, and a constant term. The solution applies to a wider range of problems than conventional linear reservoir models. The aquifer's characteristic time (derived from the solution) shows that dense drainage networks can make aquifers behave like linear reservoirs most of the time, while large systems never will. When the asymptotic values are used before the time period for which they become valid (i.e., shortly after a perturbation in the surface water level or recharge/extraction), the predicted fluxes can be very inaccurate, and possibly have the wrong sign. In such cases, the full analytical solutions should be used.

This paper analyzes the groundwater flow system in an unconfined downward-sloping aquifer of semi-infinite extent in response to localized transient recharge. The aquifer is in contact with a water body of constant water level at one end and receives localized transient recharge from a recharge basin of finite width. The mathematical model is based on the Boussinesq equation with Dupuit-Forchheimer assumption, in which the spatial coordinate of the recharge basin is treated as an additional parameter. Analytical solutions of the linearized Boussinesq equation are obtained using the Laplace transform technique by dividing the aquifer in a three-zone system containing both Dirichlet and Neumann boundary conditions at the hypothetical interfaces. Upward- and zero-sloping cases are deduced from the main results by appropriately adjusting the slope parameter. To assess the validity and efficiency of the linearization method, the nonlinear Boussinesq equation is also solved using a fully explicit predic...

[1] An approximate nonlinear solution of the one-dimensional Boussinesq equation is presented using the traveling wave approach. The solution pertains to a semi-infinite phreatic aquifer with a uniform water table that is subject to a time-varying water level at the stream-aquifer boundary. The advantage of the traveling wave method is in its versatility in handling transient boundary conditions while preserving the inherent nonlinearity of the physical phenomenon. The nonlinear solution is of a simple logarithmic form and describes the position of the water table as a function of time. It yields the exact solution for the special case of uniform water level rise at the boundary. Algebraic expressions that quantify the main flow processes are derived from the basic solution. These include the stream-aquifer exchange flow rates, bank storage and depletion, front position and propagation speed, and an improved working relationship for aquifer parameter estimation. A comparison with two exact solutions and numerical solutions of the Boussinesq equation validates the accuracy of the approximation and highlights the limitation of the method in specific flow conditions. The traveling wave model performs best for sharp front movements and monotonic water table profiles and provides excellent estimates of the flow rate and volume at the inlet boundary. The accuracy of the solution deteriorates for fluctuating inlet conditions and worsens in cases when there is a sharp reversal of flow conditions.

In general, groundwater flow in porous media on a microscopic level is three-dimensional. However, when considering regional flow problems i.e. groundwater flow on a macroscopic level, it is noted that because of the low ratio of aquifer thickness to horizontal length, the flow in the aquifer is practically horizontal[21]. Flow in confined aquifers is therefore described by a two-dimensional second order non-linear partial differential equation with variable coefficients on a finite domain. In this paper, we examine the case of a confined aquifer that is being pumped

The aim of this work was to convert the Thiem and the Theis groundwater flow equation to the time-fractional groundwater flow model. We first derived the analytical solution of the Theim time-fractional groundwater flow equation in terms of the generalized Wright function. We presented some properties of the Laplace-Carson transform. We derived the analytical solution of the Theis-time-fractional groundwater flow equation (TFGFE) via the Laplace-Carson transform method. We introduced the generalized exponential integral, as solution of the TFGFE. This solution is in perfect agreement with the data observed from the pumping test performed by the Institute for Groundwater Study on one of its borehole settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rateQand monitoring the piezometric head for 350 minutes.

In this study, non-Darcian flow to a partially penetrating pumping well in a leaky aquifer was investigated. The aquifer system is composed of a main aquifer with an aquitard bounded on the top of the aquifer. The storage of the aquitard was considered in this study, which is the main contribution of this study. The horizontal flow in the main aquifer is assumed to be non-Darcian, while both the vertical flows in the main aquifer and the aquitard were assumed to be Darcian due to the relative low velocities. The non-Darcian flow was described by the Izbash equation. A linearization procedure associated with Laplace transform and separate variable method were used to solve the non-Darcian flow model. Semianalytical solutions (Laplace-domain solutions) were obtained then inverted to time domain by using the Stehfest method. The results indicated that the power index n in the Izbash equation results in a smaller drawdown at late times, the flow approaches quasi steady state earlier, and the leakage h...

Objectives: Since 1856, Darcy's equation is used to analyse the ground water flow. This equation can be applied only in laminar regime of flow. No suitable equation is available to analyse other regimes of flow. During the pumping the water from well, the water around well flows in converging pattern towards well. The analysis of flow of water around wells is a problem in converging boundary through porous soils or sands. Generally, Dupuit's equation is used to analyse the flow around the well. One of the drawback in Dupuit's equation is it can be used only in laminar flow, since, this equation used Darcy's concept. No equation is available to analyse other than laminar flow in converging pattern towards well. Therefore, this situation motivated to think to design a sector of well model to conduct experiments on porous medium flow in converging boundary for all regimes of flow to estimate the discharge of a well. Methodology: In order to achieve the objective, a converging permeameter (a sector of a well model) is planned, designed and experimented. In this study, experiments have been conducted on porous medium in converging flow permeameter for all regimes of flow. From the experimentation of converging boundary, data like discharge, hydraulic gradient and velocity are calculated. Further, a new non-dimensional form of equation is also derived relating hydraulic gradient with Reynolds number. Findings: A set of graphs are drawn between hydraulic gradient and velocity, using Darcy's range of velocities and corresponding hydraulic gradient. These graphs are compared with Darcy's graph and the validity of Darcy's equation is verified. The trends of the present graph is coinciding with that of Darcy's graph such that the validation of this experimentation is checked. Another set of graphs are also drawn with hydraulic gradient versus velocity for all regimes of flow. A polynomial second degree equation (i=a1 Vb 2 +k'Vb) has been proposed relating hydraulic gradient with bulk velocity in laminar and turbulent regimes. Hence this project result proves that, using this equation and corresponding constants discharge of a well can be estimated even if the flow is laminar and turbulent. Applications: The relation between hydraulic gradient and Reynolds number is obtained as i=0.0004 Re0.5. Substituting hydraulic gradient in this equation from field observations, Reynolds number can be find out. From the Reynolds number seepage velocity can be find out and corresponding discharge may be calculated in a well for any regime of flow which superfluous Dupuit's equation.

In order to study the water flow in the drainage layer of highway under steady-state condition, one-dimensional (1D) Boussinesq equation-based model with Dupuit-Forchheimer assumption was established and the semi-analytical solutions to predict the water-table height were presented. In order to validate the model, a two-dimensional (2D) saturated flow model based on the Laplace equation was applied for the purpose of the model comparison. The water-table elevations predicted by 1D Boussinesq equation-based model and 2D Laplace equation-based model match each other well, which indicates that the horizontal flow in drainage layer is dominated. Also, it validates the 1D Boussinesq equation-based model which can be applied to predict the water-table elevation in drainage layer. Further, the analysis was conducted to examine the effect of infiltration rate, hydraulic conductivity and slope of drainage layer on the water-table elevation. The results show that water-table falls down as the ratio of Is to K decreases and the slope increases. If the aquifer becomes confined by the top of drainage layer due to the larger ratio of Is to K or smaller slope, the solution presented in this work can also be applied to approximate the water-table elevation in unconfined sub-section as well as hydraulic head in the confined sub-section.

An analytical solution for the space-time variation of contaminant concentration in one-dimensional uniform groundwater flow in a homogenous semi-infinite porous formation (e.g., aquifer) subjected to time-dependent source contamination is derived. The temporally dependent dispersion in the aquifer is investigated under two conditions. First, the temporally dependent dispersion distribution in the aquifer is considered as a sinusoidally varying function and, second, the temporally dependent dispersion distribution is treated as an exponentially increasing function of time. It is assumed that initially the aquifer is not solute free; i.e., the aquifer is not clean and the initial concentration is an exponentially decreasing function of the space variable and is tending to zero toward infinity. The concept that dispersion is directly proportional to the seepage velocity is employed. The analytical solution is illustrated using an example and may help benchmark a numerical code and solution.

This paper analyses the classical problem of transient surface–groundwater interaction in a stream–aquifer system under rather realistic conditions. The downward sloping unconfined aquifer is in contact with a constant water level at one end, and a fully penetrating stream at the other end whose water level is rising at a uniform speed. Furthermore, the aquifer is replenished by a vertical time-varying recharge. Closed form analytical expressions for hydraulic head and flow rate in the aquifer are obtained by solving the linearized Boussinesq equation using Laplace transform method. Effects of aquifer parameters on transient water table and flow rate are illustrated with a numerical example. To assess the efficiency of the linearization method, analytical solutions are compared with numerical solutions of the corresponding non-linear equation.

With reservoir drawdown, the groundwater table in the adjacent aquifer falls down correspondingly. It is useful to calculate the groundwater table variation as a function of time during reservoir drawdown for hydraulic and hydrological purposes. The Boussinesq equation with a moving boundary is applied to analyze the groundwater table variation in the unconfined aquifer during reservoir drawdown.This approach assumes a negligible seepage face. Because the moving boundary condition in the mathematical formulation precludes analytical solutions even for the linearized Boussinesq equation, we have transformed the Boussinesq equation into an advection–diffusion equation to address the negligible seepage face and the moving boundary condition. On the basis of the Laplace transformation, we yield an analytical solution of a fixed boundary problem, which is further simplified to upper and lower polynomial solutions for convenient practical use. The polynomial approximate solutions are satisfactorily compared with a number of numerical simulations of the nonlinear Boussinesq equation. The results indicate that the polynomial solutions match well with the numerical solution, but demonstrate that the replacement of the sloped reservoir–aquifer interface by a vertical interface may cause errors of up to 10% of the height of the reservoir drawdown in the prediction of the groundwater table location. On the basis of the polynomial solutions, a methodology is provided to determine the ratio of hydraulic conductivity to specific yield along with a chart for convenient practical use. The limitation of the present study is that the presented solution tends to underestimate the groundwater table with seepage face neglected for rapid drawdown, high specific yield, low hydraulic conductivity, or mildly sloped interface cases.

Analytical solution is obtained to predict the contaminant concentration with presence and absence of pollution source in finite aquifer subject to constant point source concentration. A longitudinal dispersion along unsteady groundwater flow in homogeneous and finite aquifer is considered which is initially solute free that is, aquifer is supposed to be clean. The constant source concentration in intermediate portion of the aquifer system is considered with pulse type boundary condition and at the other end of the aquifer, concentration gradient is supposed to be zero. The Laplace Transformation Technique (LTT) is used to obtain the analytical solution of the formulated solute transport model with suitable initial and boundary conditions. The time varying velocities are considered. Analytical solutions are perhaps most useful for benchmarking the numerical codes and models. It may be used as the preliminary predictive tools for groundwater management.

A comparative study is made among Laplace Transform Technique (LTT) and Fourier Transform Technique (FTT) to obtain one-dimensional analytical solution for conservative solute transport along unsteady groundwater flow in semi-infinite aquifer. The time-dependent source of contaminant concentration is considered at the origin and at the other end of the aquifer is supposed to be zero. Initially, aquifer is not solute free which means that the solute concentration exits in groundwater system and it is assumed as a uniform concentration. The aquifer is considered homogeneous and semi-infinite. The time-dependent velocity expressions are considered. The result may be used as preliminary predictive tools in groundwater management and benchmark the numerical code and solutions.

The Kelvin line-source equation, used to analyze thermal response tests, describes conductive heat transfer in a homogeneous medium with a constant temperature at infinite boundaries. The equation is based on assumptions that are valid for most ground-coupled heat pump environments with the exception of geological settings where there is significant groundwater flow, heterogeneous distribution of subsurface properties, a high geothermal gradient or significant atmospheric temperature variations. To address these specific cases, an alternative method to analyze thermal response tests was developed. The method consists in estimating parameters by reproducing the output temperature signal recorded during a test with a numerical groundwater flow and heat transfer model. The input temperature signal is specified at the entrance of the ground heat exchanger, where flow and heat transfer are computed in 2D planes representing piping and whose contributions are added to the 3D porous medium. Results obtained with this method are compared to those of the line-source model for a test performed under standard conditions. A second test conducted in waste rock at the South Dump of the Doyon Mine, where conditions deviate from the line-source assumptions, is analyzed with the numerical model. The numerical model improves the representation of the physical processes involved during a thermal response test compared to the line-source equation, without a significant increase in computational time.

In this study, finite difference method is used to solve the equations that govern groundwater flow to obtain flow rates, flow direction and hydraulic heads through an aquifer. The aim therefore is to discuss the principles of Finite Difference Method and its applications in groundwater modelling. To achieve this, a rectangular grid is overlain an aquifer in order to obtain an exact solution. Initial and boundary conditions are then determined. By discretizing the system into grids and cells that are small compared to the entire aquifer, exact solutions are obtained. A flow chart of the computational algorithm for particle tracking is also developed. Results show that under a steady-state flow with no recharge, pathlines coincide with streamlines. It is also found that the accuracy of the numerical solution by Finite Difference Method is largely dependent on initial particle distribution and number of particles assigned to a cell. It is therefore concluded that Finite Difference Method can be used to predict the future direction of flow and particle location within a simulation domain.

The main point of this paper is to propose a non-local three-dimensional hydraulic conductivity full tensor upscaling algorithm and code. The algorithm is capable of transforming very refined cell conductivity models into coarse block conductivity models for quick and accurate solution of the groundwater flow equation. Flow rate and hydraulic head gradient are the variables used to relate the outputs from the fine scale model to the outputs from the coarse scale model. The flows and gradients computed at the coarse scale blocks should match the average values of the corresponding quantities observed at the fine scale. The algorithm is geared towards its use in conjunction with finite-difference codes for the solution of the groundwater flow equation capable of handling full tensor hydraulic conductivities. The finite-difference formulation of the groundwater flow equation requires specifying the hydraulic conductivity at the block interface in order to compute the discharge crossing the interface. Most finite-difference codes take as input conductivities at the blocks and then perform some type of averaging to come up with the interblock conductivity. Typically the harmonic mean of the conductivities of neighboring blocks is used. We propose computing directly the interblock conductivity during the upscaling process thus avoiding the need to average already averaged (upscaled) values of adjacent blocks. This approach also circumvents the problem associated with averaging conductivity tensors when their principal directions are not aligned with the Cartesian axes. The proposed algorithm is successfully demonstrated in four synthetic examples with spatially isotropic, anisotropic and bimodal conductivities fields at the fine scale. The computer code is provided with explanation of the input parameters and output results.

Lattice Boltzmann (LB) models can be used to simulate flow in porous media at scales much larger than pore size. LB-based models for such macroscopic scale porous media flow simulations are an extension of standard LB models. There are at least two alternative approaches for implementing such models. In the first approach the local velocity is altered during the collision step by incorporating an external force, F, equivalent to the damping effect of solid particles in porous media. The porous media can be permeable or impermeable depending upon the external forcing term. A sink term is introduced in the LB model to simulate a pumping well and this model is further applied to solve transient ground water well problems for confined aquifers.Directly solving the ground water flow equation with an LB model by exploiting its ability to solve the diffusion equation is another strategy. The second order transient ground water flow equation is analogous to the diffusion equation and mass diffusivity is analogous to hydraulic diffusivity. This diffusion model is used to solve transient ground water problems. Simulated results accurately match analytical solutions of the transient ground water flow equation.