Horizontal Aquifer Research Articles

Analytical approximation for the recession of a sloping aquifer

AbstractAn approximation is obtained for the recession of a sloping aquifer. The analytical approximation can provide a useful tool to analyze data and obtain physical properties of the aquifer. In contrast to the case of a horizontal aquifer, when plotting the time derivative of the flux versus the flux on a log scale, the result shows that the flux derivative reaches a minimum value and that the curve can have a slope of unity as often observed. Illustration of the application of the analytical results to the Mahantango Creek data is also discussed.

Reply to comment by Jeffrey Olsen et al. on “Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers”

Olsen et al. [2014] are correct in pointing out the seemingly inconsistent assumption of a constant propagation speed ρ when the derived results show that ρ is in fact time-dependent. However, using a time-dependent ρ in the perturbation expansion does not alter the major results as Olsen et al. point out in the Comment. The reason is that ρ, or its new substitute , is just an intermediary variable that relates the flux rate with the head at the boundary. Therefore, the use of the more complicated definition of ρ is not warranted since it does not result in any improvement of the zero-order solution. On the other hand, a simplified definition of the traveling wave velocity ρ is intuitively more appealing and it leads to simple and elegant results. The fact that the first-order term is identically equal to zero is a consequence of the perturbation expansion used, as stated in sec. 3.3 in Basha [2013]. In any case, the zero-order solution provides a remarkably good approximation. The choice of would have improved the prediction slightly if a first-order correction were available. However, this is a particularly difficult task because the zero-order term is itself a nonlinear implicit solution that is not easily amenable to further manipulations. Instead, it is more worthwhile to consider tackling the Boussinesq equation via a refinement of its linearized solution in a perturbation series framework.

Comment on “Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers” by H. A. Basha

Comment on “Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers” by H. A. Basha

On the periodic injection of fluid into, and its extraction from, a confined aquifer

AbstractWe consider the periodic injection and extraction of fluid from a line well in a horizontal saturated aquifer of finite thickness as part of an aquifer thermal energy storage system. We focus on the case in which the injected fluid is dense relative to the original fluid in the aquifer and we explore the competition between the driving pressure and buoyancy force in controlling the dispersal of the injected fluid through the aquifer. We show that, after each cycle, a progressively larger fraction of the injected fluid is extracted, while the remainder of the injected fluid gradually migrates away from the well such that, after time $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}t$, the position of the leading edge of the injected fluid, $x_{nose}(t)$, scales as $ x_{nose}(t)\sim x_{nose}(\tau )\sqrt{t/\tau }$, where $\tau $ is the period of injection. If the fluid is extracted from the base of the layer, then, near the well, the thickness of the injected fluid at the end of the extraction cycle tends to a constant value, which decreases with injection rate. We also show that there is a class of self-similar exchange-flow solutions that develop when a saturated porous layer of thickness $H$ is in contact with a stratified fluid reservoir, filled to thickness $F_0H<H$ with relatively dense fluid, and with original reservoir fluid above this level. We show that these solutions coincide exactly with the far-field flow produced by the injection–extraction cycles. We successfully test the models with a series of analogue experiments of both the injection–extraction flow and the exchange flow using a Hele-Shaw cell. In the case that the fluid is injected and extracted from the top of the aquifer, the value $F_0$ tends to unity in all cases, although the convergence time depends on the rate and period of injection, the buoyancy speed and the vertical extent and the porosity of the aquifer. We use the model to explore how the concentration of reservoir fluid in the produced fluid varies as the system evolves from cycle to cycle, and we also examine the time required to transport a localised but distant contaminant to the production well through the far-field exchange flow. Finally, we consider the analogous axisymmetric injection–extraction flow problem, and show, through both numerical solution of the governing equations and experiment, that, although there is no simple class of similarity solutions, the fraction of injected fluid that is extracted progressively increases in each cycle.

A semianalytical solution for the Boussinesq equation with nonhomogeneous constant boundary conditions

AbstractThe Boussinesq groundwater equation is widely used in hydrology to predict streamflow from an unconfined aquifer and derive the aquifer's saturated hydraulic conductivity and drainable porosity, and to predict water table height in drainage engineering. In this work, we solve this equation in an unconfined horizontal aquifer for nonhomogeneous boundary conditions for the water table height. The solution is found in the form of a Taylor series that has a finite radius of convergence, which is different for each initial condition. We also present an expression for the flux boundary condition at the origin as a function of the depth of the adjoining stream that automatically satisfies the boundary condition at infinity, and thus eliminates the need for a trial‐and‐error approach for the solution, which is accurate to 10−7. In order to obtain an approximation for the water table height in the region where the series solution diverges, first we computed a diagonal Padé approximation from the series coefficients, which converges in a larger interval than the series, and then we matched it with a new asymptotic approximation for large values of the independent variable. We found that the proposed matched solution is better suited to cases where the water head at the origin is close to the initial water head in the aquifer.

An extended finite element method model for carbon sequestration

SummaryA two‐phase flow eXtended Finite Element Method (XFEM) model is presented to analyse the injection and sequestration of carbon dioxide (CO2) in deep saline aquifers. XFEM is introduced to accurately approximate near‐injection well pressure behaviour with elements significantly larger than the injection well diameter. We present a vertically averaged multiphase flow model that combines XFEM to approximate the pressure field, with a Streamline Upwind/Finite Element Method/Finite Difference Method (SU‐FEM‐FDM) to approximate the distribution of CO2 in the aquifer. Near‐well enrichment functions are presented along with the solution procedure for the coupled problem. Two examples are presented: in the first, CO2 injection into a perfectly horizontal aquifer is modelled with both XFEM and FEM‐based methods. The results suggest that XFEM is able to provide low relative errors in the pressure near the well at a reduced computational cost compared with FEM. The impact and selection of the stabilization coefficient of the SU‐FEM‐FDM is also discussed. In the second example, the XFEM and SU‐FEM‐FDM model is applied to a more realistic problem of an inclined aquifer to demonstrate the ability of the model to capture the buoyancy‐driven migration of CO2 in a deep saline aquifer. Copyright © 2014 John Wiley & Sons, Ltd.

Joint inversion in coupled quasi-static poroelasticity

Geodetic surveys now provide detailed time series maps of anthropogenic land subsidence and uplift due to injection and withdrawal of pore fluids from the subsurface. A coupled poroelastic model allows the integration of geodetic and hydraulic data in a joint inversion and has therefore the potential to improve the characterization of the subsurface and our ability to monitor pore pressure evolution. We formulate a Bayesian inverse problem to infer the lateral permeability variation in an aquifer from geodetic and hydraulic data and from prior information. We compute the maximum a posteriori (MAP) estimate of the posterior permeability distribution and a Gaussian approximation of the posterior. Computing the MAP estimate requires the solution of a large-scale minimization problem subject to the poroelastic equations, for which we propose an efficient Newton-conjugate gradient optimization algorithm. The covariance matrix of the Gaussian approximation of the posterior is given by the inverse Hessian of the log posterior, which we construct by exploiting low-rank properties of the data misfit Hessian. First and second derivatives are computed using adjoints of the time-dependent poroelastic equations, allowing us to fully exploit transient data. Using three increasingly complex model problems, we find the following general properties of poroelastic inversions: Augmenting standard hydraulic well data by surface deformation data improves the aquifer characterization. Surface deformation contributes the most in shallow aquifers but provides useful information even for the characterization of aquifers down to 1 km. In general, it is more difficult to infer high-permeability regions, and their characterization requires frequent measurement to resolve the associated short-response timescales. In horizontal aquifers, the vertical component of the surface deformation provides a smoothed image of the pressure distribution in the aquifer. Provided that the mechanical properties are known, coupled poroelastic inversion is therefore a promising approach to detect flow barriers and to monitor pore pressure evolution.

Traveling wave solution of the Boussinesq equation for groundwater flow in horizontal aquifers

Key Points Simplified model of the water table response to transient stream level changes Algebraic equations for the stream‐aquifer exchange flow rates and volumes An improved working relationship for aquifer parameter estimation

Analysis of groundwater-surface water interaction in horizontal unconfined aquifer bounded by a stream and groundwater divide using a non-linear numerical model

In this study, a non-linear numerical model was developed to analyse the interaction between the drainage water moving from an unconfined aquifer and a surface water body. This interaction can occur during or after a rainfall event, which can cause the groundwater table to change. At first, a laboratory-scale aquifer model with stream and groundwater divide boundaries was established to examine horizontal unconfined aquifer condition. Both steady-state and transient experiments were performed by setting the water table at a steady-state using a constant recharge rate, and then allowing the aquifer to drain. The transient water table data showed highly non-linear variations. Then the experimental data were recreated using a non-linear numerical model, which used non-linear one dimensional Boussinesq equation and Picard iteration for solving non-linear term of the Boussinesq equation. The numerical model was able to predict the observed groundwater table data. The proposed numerical model is a useful and cost effective tool for managing groundwater-surface water management problems.

Study on Geomechanical Stability of the Aquifer -caprock System During CO2 Sequestration by Coupled Hydromechanical Modelling

Carbon dioxide (CO2) sequestration in deep aquifers is one of the options for reducing the emission of greenhouse gases to the atmosphere. It is efficient to inject super critical CO2 into an aquifer if the geomechanical stability of the aquifer -caprock system can be maintained during the injection. An axisymmetric horizontal aquifer -caprock system is modeled including hydromechanical coupling in order to study influence of various factors - such as injection rate, permeability and stiffness of the caprock and aquifer - on the stability and especially the integrity of the caprock. Here the local safety factor is considered as an index of caprock integrity. It is shown that the effective stress state of the caprock close to the injection well could become critical for shear failure quite early in the injection. A buildup of gas pressure in the aquifer causes caprock deformation, inducing an increase in deviatoric stress within the caprock. Larger injection rates and lower aquifer permeability amplify such phenomena, increasing the risk of losing caprock integrity. Increased aquifer stiffness, in contrast, restricts the caprock deformation, and lowers the risk.

Spreading and convective dissolution of carbon dioxide in vertically confined, horizontal aquifers

Injection of carbon dioxide (CO2) into saline aquifers is a promising tool for reducing anthropogenic CO2 emissions. At reservoir conditions, the injected CO2 is buoyant relative to the ambient groundwater. The buoyant plume of CO2 rises toward the top of the aquifer and spreads laterally as a gravity current, presenting the risk of leakage into shallower formations via a fracture or fault. In contrast, the mixture that forms as the CO2 dissolves into the ambient water is denser than the water and sinks, driving a convective process that enhances CO2dissolution and promotes stable long‐term storage. Motivated by this problem, we study convective dissolution from a buoyant gravity current as it spreads along the top of a vertically confined, horizontal aquifer. We conduct laboratory experiments with analog fluids (water and a mixture of methanol and ethylene glycol) and compare the experimental results with simple theoretical models. Since the aquifer has a finite thickness, dissolved buoyant fluid accumulates along the bottom of the aquifer, and this mixture spreads laterally as a dense gravity current. When dissolved buoyant fluid accumulates slowly, our experiments show that the spreading of the buoyant current is characterized by a parabola‐like advance and retreat of its leading edge. When dissolved buoyant fluid accumulates quickly, the retreat of the leading edge slows as further dissolution is controlled by the slumping of the dense gravity current. We show that simple theoretical models predict this behavior in both limits, where the accumulation of dissolved buoyant fluid is either negligible or dominant. Finally, we apply one of these models to a plume of CO2 in a saline aquifer. We show that the accumulation of dissolved CO2 in the water can increase the maximum extent of the CO2 plume by several fold and the lifetime of the CO2 plume by several orders of magnitude.

Analytical single-potential, sharp-interface solutions for regional seawater intrusion in sloping unconfined coastal aquifers, with pumping and recharge

Summary Contamination of groundwater by intruding seawater is a major problem in many parts of the world. This work derives novel analytical solutions for addressing this problem by extending the Girinskii–Strack concept of discharge potential to represent regional, steady-state, sharp-interface seawater intrusion in a sloping unconfined coastal aquifer. The aquifer has hydraulic conductivity K, is recharged uniformly at the rate r, receives an inflow at its land boundary and has a collector trough (or well gallery idealised as line sink) located at the distance lw from the coastline that penetrates the aquifer and draws groundwater at the rate qw. The theory rests on the approximation of a linearised gravity-part of the hydraulic potential. Analytical solutions for the discharge potential are derived, through which the hydraulic head, the flow depth and the sharp interface, and particularly the location of the interface toe, lT, are also determined. These solutions simplify to known results for the horizontal aquifer case. The utility (and robustness to uncertainty regarding the base slope) of the results in applications is demonstrated in the regional example of the Akrotiri Coastal Aquifer, Cyprus. Recasting the problem in non-dimensional form provides a relationship for the dependence of the relative interface toe location lT/lw on the appropriately normalised difference between the groundwater flow just up-gradient of the collector trough, qo, and the pumping rate, qw, given values of the base slope, r/K and Hsea/lw, where Hsea is the sea-surface elevation above the aquifer base at the coastline. That relationship frames a common groundwater management problem in coastal aquifers subject to a certain exploitation scheme, with lT as decision variable. In an exploratory demonstration study, non-dimensional sets of performance curves are calculated for the regional Akrotiri aquifer. In general, the derived analytical solutions can be used for first-order assessments of seawater intrusion vulnerability and management possibilities across a wide range of current regional coastal aquifer conditions and/or projected water demand, groundwater management and climatic change scenarios.

Errata for “Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains” by Sushil K. Singh

Errata for “Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains” by Sushil K. Singh

Discussion of “Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains” by S. K. Singh

Discussion of “Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains” by S. K. Singh

Upscaling of CO 2 vertical migration through a periodic layered porous medium: The capillary-free and capillary-dominant cases

We present an upscaled model for the vertical migration of a CO 2 plume through a vertical column filled with a periodic layered porous medium. This model may describe the vertical migration of a CO 2 plume in a perfectly layered horizontal aquifer. Capillarity and buoyancy are taken into account and semi-explicit upscaled flux functions are proposed in the two following cases: (i) capillarity is the main driving force and (ii) buoyancy is the only driving force. In both cases, we show that the upscaled buoyant flux is a bell-shaped function of the saturation, as in the case of a homogeneous porous medium. In the capillary-dominant case, we show that the upscaled buoyant flux is the harmonic mean of the buoyant fluxes in each layer. The upscaled saturation is governed by the continuity of the capillary pressure at the interface between layers. In the capillary-free case, the upscaled buoyant flux and upscaled saturation are determined by the flux continuity condition at the interface. As the flux is not continuous over the entire range of saturation, the upscaled saturation is only defined where continuity is verified, i.e. in two saturation domains. As a consequence, the upscaled buoyant flux is described by a piecewise continuous function. Two analytical approximations of this flux are proposed and this capillary-free upscaled model is validated for two cases of heterogeneity. Upscaled and cell averaged saturations are in good agreement. Furthermore, the proposed analytical upscaled fluxes provide satisfactory approximations as long as the saturation set at the inlet of the column is in a range where analytical and numerical upscaled fluxes are close.

Modeling of Hydrological Processes Using Unstructured and Irregular Grids: 2D Groundwater Application

To better handle landscape heterogeneities in distributed hydrological modeling, an earlier work proposed a discretization based on nested levels, which leads to fully unstructured modeling meshes. Upon such a discretization, traditional numerical solutions must be adapted, especially to describe lateral flow between the unstructured mesh elements. In this paper, we illustrated the feasibility of the numeric solution of the diffusion equation, representing groundwater flow, using unstructured meshes. Thus, a two-dimensional (2D) groundwater model (BOUSS2D), adapted to convex unstructured and irregular meshes was developed. It is based on the approximation of the 2D Boussinesq equation using numeric techniques suitable for nonorthogonal grids. The handling of vertical and horizontal aquifer heterogeneities is also addressed. The fluxes through the interfaces among joined mesh elements are estimated by the finite volume method and the gradient approximation method. Comparisons between the BOUSS2D predictions and analytical solutions or predictions from existing codes suggest the acceptable performance of the BOUSS2D model. These results therefore encourage the further development of hydrological models using unstructured meshes that are capable of better representing the landscape heterogeneities.

Closure to “Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains” by S. K. Singh

Closure to “Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains” by S. K. Singh

Coupled hydromechanical modeling of CO 2 sequestration in deep saline aquifers

Sequestration of carbon dioxide (CO 2) in deep saline aquifers has emerged as an option for reducing greenhouse gas emissions to the atmosphere. The large amounts of supercritical CO 2 that need to be injected into deep saline aquifers may cause large fluid pressure increases. The resulting overpressure may promote reactivation of sealed fractures or the creation of new ones in the caprock seal. This could lead to escape routes for CO 2. In order to assess the probability of such an event, we model an axisymmetric horizontal aquifer–caprock system, including hydromechanical coupling. We study the failure mechanisms, using a viscoplastic approach. Simulations illustrate that, depending on boundary conditions, the least favorable moment takes place at the beginning of injection. Initially, fluid pressure rises sharply because of a reduction in permeability due to desaturation. Once CO 2 fills the pores in the vicinity of the injection well and a capillary fringe is fully developed, the less viscous CO 2 displaces the brine and the capillary fringe laterally. The overpressure caused by the permeability reduction within the capillary fringe due to desaturation decreases with distance from the injection well. This results in a drop in fluid pressure buildup with time, which leads to a safer situation. Nevertheless, in the presence of low-permeability boundaries, fluid pressure continues to rise in the whole aquifer. This occurs when the radius of influence of the injection reaches the outer boundary. Thus, caprock integrity might be compromised in the long term.

WATER TABLE FLUCTUATIONS IN A SLOPING AQUIFER: ANALYTICAL EXPRESSIONS FOR WATER EXCHANGE BETWEEN STREAM AND GROUND-WATER WITH SURFACE INFILTRATION

This paper presents an analytical solution of a linearized Boussinesq’s equation to obtain exact expressions for hydraulic head and flow rate in an unconfined downward sloping aquifer. The aquifer is in contact with a constant piezometric head at the left boundary and a stream at the right boundary whose water level is rising at a constant rate. The aquifer also receives a constant vertical recharge from the land surface due to rain infiltration. The governing equation is solved analytically using the Laplace transform to obtain the expressions for hydraulic head and flow rate. The proposed analytical solutions are verified with an earlier known solution for a horizontal aquifer with zero surface infiltration and found to be in complete agreement. The combined effect of slope and recharge on water table fluctuation and flow rate is studied by considering a numerical example. Model results establish the dependence of the water table fluctuations and flow rate on bed slope and surface recharge rate. The analytical expressions derived here can also represent an asymptotic scenario of either sudden rise or very slow rise in the stream water.

Generalized Analytical Solutions for Groundwater Head in a Horizontal Aquifer in the Presence of Subsurface Drains

Generalized analytical solutions for groundwater head in horizontal aquifers in the presence of parallel subsurface drains are obtained considering a transient rate of recharge as a power series (polynomial) function and depth-dependent rate of evapotranspiration. A function, new to analytical drainage studies, is proposed for correctly representing the depth-dependent rate of evapotranspiration. The solutions are obtained considering the practical situation of drains placed at a shallow depth in a considerable depth of aquifer. Two conditions of large and small saturated thicknesses in comparison to the changes in groundwater head are considered. A mathematical criterion is proposed to distinguish between large and small saturated thicknesses.