H23I-01 INVITED
Some Transient Analytical Groundwater Responses for Basis Spline Pulse Inputs
The linearised Boussinesq equation describes the transient movement of a groundwater free surface in unconfined flow. The classical solutions of the heat or diffusion equation provide many transient analytical solutions for the interaction between groundwater and surface water. The classic Glover and Balmer solution gives the depletion of a stream when groundwater is pumped from a nearby well. I will present generalisations of this for the flow of groundwater recharge into a stream, and for more complicated aquifer and stream geometries. The most commonly used transient analytical solutions are for either a delta function pulse input in the time domain, or else its integral the unit step input. Traditionally the response for an arbitrary input in the time domain is calculated as a finite difference approximation to a convolution integral, which is equivalent to finding the output corresponding to a row of delta function inputs. I will describe recent work in which the input time series is described as a sum of cubic basis spline functions, which is a smooth approximation to the original input function. In many cases of interest the output for each basis spline input pulse can be found analytically. This means that the input can be expressed as a sum of basis splines and the output can immediately be expressed as a sum of the appropriate response functions with the same coefficients. This principle is quite general, and examples will be given from various interactions between streams and groundwater.
H23I-02
Travel Time of Fluids in Porous Media: Bernoulli's Brachistochrone, Isoperimetric Estimates, and Optimal Design in Smart Hydroisolation of Horizontal Wells
Steady and transient 2,-3- D seepage in a rigid porous medium is studied by the methods of complex analysis and optimal shape design. Euler (1748) dichotomised the inverse-direct approaches. In the former, "surmised" functionals (in our case - travel time of a marked (neutral tracer) particle, seepage discharge, and acting seepage forces) are extremised under isoperimetric constraints, e.g., the "action formalis" (total head along streamlines of flow tubes) of Leibnitz (1860). Following Euler's ideas, the parameters of impervious hydraulic structures (dam, sheetpiling, well packer, seabed buried pipeline, etc.) are optimized. In the Verigin problem, the angle of inclination of a sheetpile providing minimal time along a bounding streamline is found. The maximum of the minimum of the travel time is searched between all streamlines originated in the upper pool. The calculated travel time distributions are also used in minimisation of the total volume of fluid, which arrives from the upper pool to the tailwater during a prescribed time span. In the class of arbitrary structures, time optimisation is carried out explicitly for the bounding streamline with a constraint of the wetted perimeter of a depressed structure. The found optimum (brachistochrone) is a semi-circle and the corresponding time coincides with that of a unidirectional flow between two constant head surfaces. A general theorem is proved that for an arbitrary 3-D flow tube the travel time along an arbitrary streamline is bounded from below by that in a 2-D Voshinin problem (line vortex). Generalisations to heterogeneous media (of varying hydraulic conductivity and porosity) and non-Darcian flows make possible isoperimetric estimates of breakthrough curves (i.e. a composition of marked particles as in Lagrange, 1760) in a purely advective transport of contaminants and advective dispersion with negligible transverse dispersivity. Seepage from one horizontal isobar to another with a tilted fracture intersecting the two isobars is studied for an impermeable seal of a constant length, which is placed as an expandable zonal isolation at the fracture-isobar intersection. The objective in optimization is either the total flow rate from the fracture or the travel time along the seal streamline. The optimal seal position is found for any given distance between isobaric horizons, tilt angle, pressure drop between isobars and hydraulic conductivity of the rock. For an impermeable depressed hydraulic structure three pore-pressure caused functionals (uplift and traction forces and the moment of the total force vector) are optimized in the class of semi-elliptical contours. In transient problems, a non-circular cylindrical no-flow boundary, partially buried into a porous bed and partially protruding into a sea layer where periodic surface waves cause 2-D pore pressure variations, is studied. The Polubarinova- Kochina model of transient impact on the structure delivers optimal shapes of cylinders for different frequencies of the imposed surface waves. References: Euler, L., 1748. Recherches sur les plus grands et les plus petits qui se trouvent dans les actions des forces. Mem. De l' Acad des Sciences de Berlin, t.4. Lagrange, J.L., 1760 Miscellanea Taurinensia, t.2 Leibniz G.W., 1860. Mathematische Schriften. Herasg. V. Gerchardt, t.II, III
H23I-03 INVITED
A Formulation to Solve Mixing-Driven Reactive Transport Problems
Several reactive transport phenomena are driven by mixing. Therefore, quantification of the mixing rate is essential for evaluating the fate of reactive solutes in rivers, lakes and aquifers. Mixing-driven chemical processes are generally highly non-linear; the resulting complexity has inhibited the development of analytical solutions for multi-component heterogeneous reactions (such as precipitation/dissolution) and led to analyze reactive transport processes mainly using advanced numerical codes. We propose a novel mixing ratios-based formulation to evaluate solute concentrations and reaction rates when equilibrium aqueous reactions and precipitation-dissolution of minerals are driven by mixing of different waters. The approach decouples solute transport and chemical speciation, so that mixing ratios are obtained solving a conservative transport problem and then used to evaluate species concentrations. One key point of the methodology is constituted by the general expression for reaction rates which clearly demonstrates that the amount of reactants evolving into products depends on the rate at which solutions mix. The methodology is a useful tool to derive analytical solutions highlighting the relative importance of involved phenomena and parameters. The derived analytical solutions may also be used as benchmark for reactive transport codes or in designing/interpreting experimental analyses devoted to study mixing-driven reactive transport processes. We apply the developed formulation to provide an analytical solution of the reactive transport process resulting from mixing different CaCO3-saturated waters in a two-dimensional set-up. The derived solution reproduces the non-linear behaviour of calcite precipitation/dissolution in the fresh-saltwater mixing zone of coastal aquifers, with modest computational effort if compared to reactive transport numerical codes. The obtained solution is thus a promising tool to investigate carbonate system evolution.
H23I-04
Analytical solutions for bacterial energy taxis (chemotaxis): traveling bacterial bands and their role in groundwater remediation
Motile bacteria may form bands that travel with a constant speed of propagation through a medium containing a dissolved substrate, to which they respond energy tactically. We generalize the analytical solution by Keller and Segel for such bands by accounting for (1) the presence of a porous medium, (2) substrate consumption described by a Monod kinetics model, and (3) an energy tactic response model derived by Rivero et al. We also comment on the potential role of traveling bacterial bands in the remediation of groundwater contamination.
H23I-05 INVITED
What use are Analytic Series Methods in Modern Groundwater Modelling?
Analytic techniques used for seepage and infiltration problems are usually series solutions or conformal transformations. In the past, these classical techniques have been restricted to regular or transformable geometries and were superseded, to some extent, by numerical schemes such as finite element, boundary integral and more recently, analytic element methods. However, in contrast to the classical series approach, I have extended these methods to cater for arbitrary boundary shapes and steady water table (i.e., free boundary) problems, steady infiltration through arbitrary vadose zones and coupled saturated-unsaturated problems. In this paper, I will summarize some of the more significant results that I have obtained using these methods and discuss the inherent and practical advantages of an analytic solution. I will also discuss the range of problems that can be solved using these methods, their limitations and then compare them with their numerical counterparts.
H23I-06
Unbounded, Exact Solution for 3-D Topography Driven Groundwater Flow
An exact analytical solution is presented for saturated groundwater flow to provide improved understanding of the renewal rate of deep and shallow groundwater and the long-term management of groundwater resources. The solution is derived under the assumptions that the hydraulic potential of the groundwater surface follows the topography and imposes a steady boundary condition for driving the groundwater flow. This assumption is justified in most areas of humid climate. The solution is applicable on a wide range of spatial scales and accounts for decaying permeability with depth, stratified aquifers as well as anisotropy. The flow problem is solved by representing the topography with a three-dimensional spectral scaling solution based on harmonic functions that are independent in x- and y-directions. In most areas the Fourier-series, representing the topography, give a nearly perfect image of the ground surface elevation. The topography is found to be fractal and this imposes a fractal nature of the groundwater flow that is altered by the additional geometrical scales. The groundwater flow solution, based on the Fourier-spectrum, depends on the decay with depth and anisotropy in hydraulic conductivity and stratifications due to quaternary deposits, layered sediments etc. Prior analytical solutions are limited to either two-dimensional flows or harmonic functions uniform in the x- and y- directions, hence making them unable to predict three-dimensional subsurface flows beneath a realistic landscape. However, the most important advantage of this new method is the ability to analyse the impact of different geometrical scales on the groundwater flow. Analyses indicate that in a homogeneous subsurface, shallow groundwater flows would be approximately equally controlled by all scales of topography. Although shorter topographical wavelengths control the surface water flux, their impact decreases faster with depth in relation to longer wavelengths. This induces an increasing importance of large-scale topography with depth. However, the hydraulic conductivity tends to decay with depth and this counteracts the effect of the large-scale topography on the groundwater flow more effectively than the smaller landscape scales. For the depth-dependent hydraulic conductivity applicable to the Fennoscandian bedrock, we find a depth-limitation of the flow cells that tends to reduce the importance of the larger wavelengths on the fluxes at all depths.
H23I-07 INVITED
A new method for constructing analytic elements for groundwater flow.
The analytic element method is based upon the superposition of analytic functions that are defined throughout the infinite domain, and can be used to meet a variety of boundary conditions. Analytic elements have been use successfully for a number of problems, mainly dealing with the Poisson equation (see, e.g., Theory and Applications of the Analytic Element Method, Reviews of Geophysics, 41,2/1005 2003 by O.D.L. Strack). The majority of these analytic elements consists of functions that exhibit jumps along lines or curves. Such linear analytic elements have been developed also for other partial differential equations, e.g., the modified Helmholz equation and the heat equation, and were constructed by integrating elementary solutions, the point sink and the point doublet, along a line. This approach is limiting for two reasons. First, the existence is required of the elementary solutions, and, second, the integration tends to limit the range of solutions that can be obtained. We present a procedure for generating analytic elements that requires merely the existence of a harmonic function with the desired properties; such functions exist in abundance. The procedure to be presented is used to generalize this harmonic function in such a way that the resulting expression satisfies the applicable differential equation. The approach will be applied, along with numerical examples, for the modified Helmholz equation and for the heat equation, while it is noted that the method is in no way restricted to these equations. The procedure is carried out entirely in terms of complex variables, using Wirtinger calculus.
H23I-08
Transient Analytic Element Solutions for Flexible Aquifer Test Analyses
We present three extensions to the 2D Laplace transform analytic element method (LT-AEM), introduced by Furman and Neuman (2003), which exemplify the types of problems that are easily solved using the LT-AEM, and are useful for performing flexible aquifer test analyses. First, we give the equation for a simplified leaky aquifer- aquitard LT-AEM system, similar to that used by Hantush (1960); in this example the source term is proportional to the drawdown in the aquifer (dual-domain flow is another example). Secondly, we present an approximate unconfined integrodifferential equation, as initially proposed by Boulton (1954) and generalized by Herrera, et al (1978). This solution illustrates how problems defined by convolution integrals are easily handled using LT-AEM (leaky systems can also be represented using convolution integrals). Finally, we present a damped-wave generalization of the diffusion equation that arises from considering a more general form of Darcy's law. The effects of inertia in the aquifer can be considered and may be important near sources in very course materials (e.g., gravel packed envelopes surrounding pumping wells). This final example shows how higher-order time derivatives may be handled in a simple and elegant fashion using LT-AEM techniques; solving the wave equation is as straightforward as solving the diffusion equation in Laplace space. Each of the LT-AEM problems presented here can be solved using any developed LT-AEM element (e.g., point, line, or area sources) or any combination of them, with little modification to the method used to solve the standard diffusion equation.