New approximation for free surface flow of groundwater: capillarity correction

An existing capillarity correction for free surface groundwater flow as modelled by the Boussinesq equation is re-investigated. Existing solutions, based on the shallow flow expansion, have considered only the zeroth-order approximation. Here, a second-order capillarity correction to tide-induced watertable fluctuations in a coastal aquifer adjacent to a sloping beach is derived. A new definition of the capillarity correction is proposed for small capillary fringes, and a simplified solution is derived. Comparisons of the two models show that the simplified model can be used in most cases. The significant effects of higher-order capillarity corrections on tidal fluctuations in a sloping beach are also demonstrated.     

Analytical model for computing residence times near a pumping well

An analytical solution for calculating the residence time of fluid flowing toward a pumping well in an unconfined aquifer has been developed. The analytical solution was derived based on a radial, steady-state, Dupuit-Forchheimer flow model. The resulting integral expression involved computing the imaginary error function, for which a simple series expansion is proposed. The validity of the analytical expression is demonstrated by testing its results against numerical results for an example problem. The analytical solution compared favorably with the numerical approximation.     

A simple model of hillslope response for overland flow generation

This paper deals with the derivation of the hydrological response of a hillslope on the assumption of quick runoff by surface runoff generation. By using the simple non‐linear storage based model, first proposed by Horton, an analytical solution of the overland flow equations over a plane hillslope was derived. This solution establishes a generalization for different flow regimes of Horton's original solution, which is valid for the transitional flow regime only. The solution proposed was compared successfully with that of Horton and, for the turbulent flow regime, to the one derived from kinematic wave theory. This solution can be applied easily to both stationary and non‐stationary rainfall excess events. An analytical solution for the instantaneous response function (IRF) was also derived. Finally, simple expressions to compute peak and time to peak of IRF are proposed. Copyright © 2001 John Wiley & Sons, Ltd.     

Fluid dispersion effects on density-driven thermohaline flow and transport in porous media

This study introduces the dispersive fluid flux of total fluid mass to the density-driven flow equation to improve thermohaline modeling of salt and heat transports in porous media. The dispersive fluid flux in the flow equation is derived to account for an additional fluid flux driven by the density gradient and mechanical dispersion. The coupled flow, salt transport and heat transport governing equations are numerically solved by a fully implicit finite difference method to investigate solution changes due to the dispersive fluid flux. The numerical solutions are verified by the Henry problem and the thermal Elder problem under a moderate density effect and by the brine Elder problem under a strong density effect. It is found that increment of the maximum ratio of the dispersive fluid flux to the advective fluid flux results in increasing dispersivity for the Henry problem and the brine Elder problem. The effects of the dispersive fluid flux on salt and heat transports under high density differences and high dispersivities are more noticeable than under low density differences and low dispersivities. Values of quantitative indicators such as the Nusselt number, mass flux, salt mass stored and maximum penetration depth in the brine Elder problem show noticeable changes by the dispersive fluid flux. In the thermohaline Elder problem, the dispersive fluid flux shows a considerable effect on the shape and the number of developed fingers and makes either an upwelling or a downwelling flow in the center of the domain. In conclusion, for the general case that involves strong density-driven flow and transport modeling in porous media, the dispersive fluid flux should be considered in the flow equation.     

Data assimilation in Parker’s dynamo model

Application of the extended Kalman filter in the data assimilation problem of the one-dimensional (1D) Parker dynamo model is considered for two extreme cases: (1) when the magnetic field observations are taken uniformly across the entire surface of the liquid core of the Earth and (2) when the observations are taken at one spatial point. The algorithm allows the model solution with arbitrary initial conditions to be modified based on observations. It is shown that the redundancy of the noised data can reduce the efficiency of the recovery of the solution. Considerations on the applicability of data assimilation for the case of a higher dimension and a solution’s convergence dependent on the density of the flow of information assimilated are suggested.     

A semi‐analytical approach for predicting peak discharge of floods caused by embankment dam failures

This study presents an analytical solution of dam‐break floods in a trapezoidal channel with detailed solution procedure. An approach predicting the peak discharge of floods caused by embankment dam failures was derived from the aforementioned analytical solution with a database of 27 historical dam failures. The prediction performance of this approach has been proved by comparing with other 14 straightforward equations for estimating the peak discharge. The proposed model with a small uncertainty of predicted peak flow rates has a high coefficient of determination and a small standard error, being ranked in the top four of the 15 methods considered in this paper. The robustness and predictive capability of the proposed model are further demonstrated in two case studies, and both were considered in the previous analyses performed by other investigators. This method provides a simple and transparent tool for engineers to predict the peak discharge and is easy to implement for trial and error calculation. Copyright © 2016 John Wiley & Sons, Ltd.     

Is the forward problem of ground water hydrology always well posed?

Complex aquifer systems are often modeled with quasi-three-dimensional models, which consider two-dimensional horizontal flow in the aquifers and one-dimensional vertical flow through aquitards. When the aquifer system consists of a phreatic aquifer and one or more semiconfined aquifers connected by aquitards, the discrete model consists of a nonlinear system of algebraic equations, because the transmissivity of the phreatic aquifer depends on the phreatic head. If the water extraction is very high, the phreatic aquifer can be depleted and the equations of the model must be modified accordingly. There are not simple and general criteria to state if the phreatic aquifer is depleted before solving the system of equations. Therefore, the iterative procedures (e.g., relaxation methods), used to find the solution to the forward problem, must handle these particular conditions and can suffer several problems of convergence. These problems can be caused by the choice of the initial head values or of the relaxation coefficient of the iterative algorithms; however, they can also be caused by the nonexistence or nonuniqueness of the solution to the system of nonlinear equations. The study of existence and uniqueness of the general problem is very difficult and, therefore, we consider a simplified problem, for which the discrete model can be handled analytically. The results of the numerical experiments show that the solution to the forward problem can be nonunique. Only for some cases it is possible to invoke physical arguments to eliminate tentative solutions.     

Steady Couette flow of a simple fluid between porous plates

Summary Steady Couette flow ofNoll's simple fluid between two porous plates has been considered. It is seen that such flow, though not strictly lineal, is viscometric and unique solution for the main flow exists, in general, under Lipschitz's condition on the shearing function. Special cases have been discussed.     

Self-consistent problem of induced polarization of electrokinetic origin

In the first part of the paper, with some constraints, we find the analytical solution of the self-consistent problem of induced polarization (IP) for an electrokinetically polarized sphere. The stationary (on long time intervals) solution of the self-consistent problem is a set of the potential fields that are interconnected with each other: the exciting electric field, the extraneous hydrodynamical field (electroosmotic flow of a viscous incompressible fluid), and the resulting electromagnetic IP field. The extraneous field is the field of the osmotic flow of a charged liquid and the field of the charges that emerge due to the membrane effect in the narrowed segments of the pore channels. The calculations show that the IP fields derived by solving the self-consistent problem and by the Seigel-Komarov phenomenological approach are different. In the second part of the paper, by generalization of the obtained analytical solution, we formulate the self-consistent IP problem for isotropic σ-η media of arbitrary shape, which are bounded by a smooth surface. The problem can be solved by the numerical methods.     

A numerical method for two-phase flow in fractured porous media with non-matching grids

We propose a novel computational method for the efficient simulation of two-phase flow in fractured porous media. Instead of refining the grid to capture the flow along the faults or fractures, we represent the latter as immersed interfaces, using a reduced model for the flow and suitable coupling conditions. We allow for non matching grids between the porous matrix and the fractures to increase the flexibility of the method in realistic cases. We employ the extended finite element method for the Darcy problem and a finite volume method that is able to handle cut cells and matrix-fracture interactions for the saturation equation. Moreover, we address through numerical experiments the problem of the choice of a suitable numerical flux in the case of a discontinuous flux function at the interface between the fracture and the porous matrix. A wrong approximate solution of the Riemann problem can yield unphysical solutions even in simple cases.     

Subharmonic Dynamo Action in the Roberts Flow

The paper deals with the dynamo action of the Roberts flow, that is, a flow depending periodically on two cartesian coordinates, X and Y , but being independent of the third one, Z . In particular the case is considered in which the magnetic fields, which are periodic in X, Y and Z , have period lengths in the XY -plane being integer multiples of that of the flow. Two approaches are used. Firstly, the equations governing the magnetic field are reduced to a matrix eigenvalue problem, which is solved numerically. Secondly, a mean magnetic field is defined by averaging over proper areas in the XY -plane, corresponding equations are derived, in which the induction effect of the flow occurs as an anisotropic f -effect, and analytic solutions are given. The results are of particular interest for the Karlsruhe dynamo experiment, which works with a Roberts type flow consisting of 52 cells inside a cylindrical volume. In order to check the reliability of predictions concerning self-excitation based on the mean-field approach, analogous predictions are derived for a rectangular box containing 50 cells, and are compared with results obtained with the help of direct solutions of the eigenvalue problem mentioned. It turns out that the simple mean-field approach in general underestimates the requirements for self-excitation. The corresponding results agree with those obtained in the subharmonic approach only if the side length L of the box, its height H and the edge length l of a spin generator satisfy $ L \gg H \gg l $ . In Appendix B, some comments on previous results concerning $\cal {ABC}$ dynamos are made in the light of the subharmonic formalism used in the paper.     

Boundary conditions effect on linearized mud-flow shallow model

The occurrence of roll-waves in mud-flows is investigated based on the formulation of the marginal stability threshold of a linearized onedimensional viscoplastic (shear-thinning) flow model. Since for this kind of non-Newtonian rheological models this threshold may occur in a hypocritical flow, the downstream boundary condition may have a nonnegligible effect on the spatial growth/decay of the perturbation. The paper presents the solution of the 1D linearized flow of a Herschel and Bulkley fluid in a channel of finite length, in the neighbourhood of a hypocritical base uniform flow. Both linearly stable and unstable conditions are considered. The analytical solution is found applying the Laplace transform method and obtaining the first-order analytical expressions of the upstream and downstream channel response functions in the time domain. The effects of both the yield stress and the rheological law exponent are discussed, recovering as particular cases both power-law and Bingham fluids. The theoretical achievements may be used to extend semi-empirical criteria commonly employed for predicting roll waves occurrence in clear water even to mud-flows.     

Modeling groundwater–surface water interactions using the Dupuit approximation

Global errors in head and/or discharge may be introduced when groundwater flow to a stream is modeled using the Dupuit approximation. We consider a simple case of steady groundwater flow in the vertical plane to a horizontal stream bed in direct connection with the aquifer, and compare solutions to the exact problem with Dupuit solutions where common representations of the stream are chosen. In all cases considered, adopting the Dupuit approximation introduces global errors into the mathematical model, and the magnitude of the errors depends on the regional flow conditions. This behavior makes calibration of a model difficult and limits the predictive abilities of the model under conditions of changed regional flow. The global errors and their dependence on flow conditions can be minimized, but not eliminated by treating the resistance of a fictitious leaky stream bed as an effective parameter.We propose an alternate Dupuit model of groundwater–surface water interaction and demonstrate, for the case considered, that adding a second effective parameter allows us to eliminate global errors in head and discharge, and eliminate the dependence of the effective values on the flow field. Explicit expressions are provided to evaluate the two effective properties. We propose that the results be used as a general guideline for modeling groundwater–surface water interaction at streams.     

Stochastic space transforms in subsurface hydrology — Part 2: Generalized spectral decompositions and plancherel representations

In earlier publications, certain applications of space transformation operators in subsurface hydrology were considered. These operators reduce the original multi-dimensional problem to the one-dimensional space, and can be used to study stochastic partial differential equations governing groundwater flow and solute transport processes. In the present work we discuss developments in the theoretical formulation of flow models with space-dependent coefficients in terms of space transformations. The formulation is based on stochastic Radon operator representations of generalized functions. A generalized spectral decomposition of the flow parameters is introduced, which leads to analytically tractable expressions of the space transformed flow equation. A Plancherel representation of the space transformation product of the head potential and the log-conductivity is also obtained. A test problem is first considered in detail and the solutions obtained by means of the proposed approach are compared with the exact solutions obtained by standard partial differential equation methods. Then, solutions of three-dimensional groundwater flow are derived starting from solutions of a one-dimensional model along various directions in space. A step-by-step numerical formulation of the approach to the flow problem is also discussed, which is useful for practical applications. Finally, the space transformation solutions are compared with local solutions obtained by means of series expansions of the log-conductivity gradient.     

Effect of natural advection on stabilization of contaminant plume in natural traps at underground disposal of liquid wastes

The migration of a contaminant from a zone of injection disposal of hazardous liquid waste in a deep-seated aquifer is considered. Because of its higher density, the polluted groundwater will accumulate under the effect of gravity in aquifer dips (depressions). A 2D-model of variable-density groundwater flow is used to determine the conditions under which the gravity force will prevent polluted groundwater from leaving depressions driven by regional current. As the result, such depressions can serve as natural traps for polluted waters. The required conditions are based on simple analytical relationships, derived from the analysis of a theoretical model of variable-density groundwater flow in an inclined confined aquifer. The obtained technique is used to estimate the efficiency of such a trap at the site of injection disposal of liquid radioactive waste from Mining and Chemical Combine in Krasnoyarsk region. The analytical estimates of the trap with the use of the proposed technique are shown to be in good agreement with the results of numerical simulation of contaminant migration.     

Flood routing model incorporating intensive streambed infiltration

Flood routing models are critical to flood forecasting and confluence calculations. In the streams that dry up and disconnect from groundwater, the streambed infiltration is intensive and has a significant effect on flood wave movement. Streambed infiltration should be considered in flood routing. A flood routing model incorporating intensive streambed infiltration is proposed. In the model a streambed infiltration simulation method based on soil infiltration theory is developed. In this method the Horton equation is used to calculate infiltration capacity. A trial-and-error method is developed to calculate infiltration rate and determine whether the flood wave can travel downstream. A formula is derived to calculate infiltration flow per unit length. The Muskingum-Cunge method with streambed infiltration flow as lateral outflow is used for flood routing. The proposed model is applied to the stream from the downstream of the Yuecheng Reservoir to the Caixiaozhuang Hydrometric Station in the Zhangwei River of the Haihe River Basin. Simulation results show that the accuracy of the model is high, and the infiltration simulation method can represent infiltration processes well. The proposed model is simple and practical for flood simulation and forecasting, and can be used in river confluence calculations in a rainfall-runoff model for arid and semiarid regions.     

Two types of longitudinal dune fields and possible mechanisms for their development

Longitudinal dune fields characterized by nearly uniform interdune spacing are distinguished from longitudinal dune fields characterized by fairly variable interdune spacing and high frequencies of dune coalescence. The empirical and theoretical evidence indicating that the former may be due to helical air currents aligned with the dunes is reviewed. Hypotheses arguing that the latter may arise indirectly from horizontal pressure gradients or bidirectional wind regimes are discussed. Evenly spaced linear sand banks aligned with tidal currents may be shown mathematically to result from energy optimalization within two-dimensional, sand-transporting flow regimes, and a similar simple or non-rotational flow model is considered for the problem of desert longitudinal dunes. An initial complex or rotational flow analysis is undertaken to discern the likely significance of roll vortices in desert sediment transport. An ‘evolutionary timescale’ is estimated for the formation of desert longitudinal dune fields. A simple analysis is performed for the effect of regional sand mass change on longitudinal dune field ordering. Recommendations are made for future empirical and theoretical research.     

A practical method for analysis of river waves and for kinematic wave routing in natural channel networks

The behaviour of river waves is described using a simplified dimensionless form of the momentum equation in conjunction with the continuity equation. Three dimensionless parameters were derived based on a quantitative linear analysis. These parameters, which depend on the Froude number of the steady uniform flow and the geometric characteristics of the river, permit quantification of the influence of inertia and pressure in the momentum equation. It was found that dynamic and diffusion waves occur mainly on gentle channel slopes and the transition between them is characterized by the Froude number. On the other hand, the kinematic wave has a wide range of applications. If the channel slope is greater than 1%, the kinematic wave is particularly suitable for describing the hydraulics of flow. Since slopes in natural channel networks are often greater than 1%, an analytical solution of the linearized kinematic wave equation with lateral inflow uniformly distributed along the channel is desirable and was therefore derived. The analytical solution was then implemented in a channel routing module of an existing simple rainfall–runoff model. The results obtained using the analytical solution compared well with those obtained from a non‐linear kinematic wave model. Copyright © 2000 John Wiley & Sons, Ltd.     

Flowing partially penetrating well: solution to a mixed-type boundary value problem

A new semi-analytic solution to the mixed-type boundary value problem for a flowing partially penetrating well with infinitesimal skin situated in an anisotropic aquifer is developed. The solution is suited to aquifers having a semi-infinite vertical extent or to packer tests with aquifer horizontal boundaries far enough from the tested area. The problem reduces to a system of dual integral equations (DE) and further to a deconvolution problem. Unlike the analogous Dagan's steady-state solution [Water Resour. Res. 1978; 14:929–34], our DE solution does not suffer from numerical oscillations. The new solution is validated by matching the corresponding finite-difference solution and is computationally much more efficient. An automated (Newton–Raphson) parameter identification algorithm is proposed for field test inversion, utilizing the DE solution for the forward model. The procedure is computationally efficient and converges to correct parameter values. A solution for the partially penetrating flowing well with no skin and a drawdown–drawdown discontinuous boundary condition, analogous to that by Novakowski [Can. Geotech. J. 1993; 30:600–6], is compared to the DE solution. The D–D solution leads to physically inconsistent infinite total flow rate to the well, when no skin effect is considered. The DE solution, on the other hand, produces accurate results.     

Fundamental Control Functions and Error Analysis in Variational Data Assimilation

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The equation for the error of the optimal solution (analysis) is derived through the errors of the input data (background and observation errors). The numerical algorithm is developed to compute the sensitivity coefficients for the analysis error using the fundamental control functions. Application to the variational data assimilation problem for a model of ocean thermodynamics is considered.