Steady two-dimensional groundwater flow in a porous low-permeable trough is studied by the method of boundary-value problems for holomorphic functions. In the overlying highly permeable aquifer the hydraulic head varies linearly, i.e. flow is unidirectional. The exchange of groundwater between aquifer and trough does not affect the flow in the aquifer. It is assumed that through a horizontal aquifer–trough interface the head is transmitted into the trough, where the bounding effect of the bed causes circulatory seepage. Triangular troughs are studied and an isosceles form with a base angle of 38° is proved to have the highest circulation rate at a given cross-sectional area. In the class of arbitrary forms, solution to this optimal shape design problem is obtained by tackling the Schwartz and Signorini singular integrals and a unique and global maximum of the rate is found. The extreme curve coincides with an optimal soil channel of minimal seepage losses having depth to width ratio of 0.371. The global maximum differs from that one for the triangular class in 3% only that corroborates stability and robustness of the optimization criterion and makes possible isoperimetric estimates of the seepage intensity through arbitrary troughs.
Permeable reactive barriers (PRBs) are a popular technology for passive contaminant remediation in aquifers through installation of reactive materials in the pathway of a plume. Of fundamental importance are the degree of remediation inside the reactor (residence time) and the portion of groundwater intercepted by a PRB (capture width). Based on a two‐dimensional conformal mapping approach (previously used in related work), the latter is studied in the present work for drain‐and‐gate (DG) PRBs, which may possess a collector and a distributor drain (“full” configuration) or a collector drain only (“simple” configuration). Inherent assumptions are a homogeneous unbounded aquifer with a uniform far field, in which highly permeable drains establish constant head boundaries. Solutions for aquifer flow fields in terms of the complex potential are derived, illustrated, and analyzed for doubly symmetric DG configurations and arbitrary reactor hydraulic resistance as well as ambient groundwater flow direction. A series of practitioner‐friendly charts for capture width is given to assist in PRB design and optimization without requiring complex mathematics. DG PRBs are identified as more susceptible to flow divergence around the reactor than configurations using impermeable side structures (e.g., funnel‐and‐gate), and deployment of impermeable walls on drains is seen to mitigate this problem under certain circumstances.
Abstract Design of managed aquifer recharge ( MAR ) for augmentation of groundwater resources often lacks detailed data, and simple diagnostic tools for evaluation of the water table in a broad range of parameters are needed. In many large‐scale MAR projects, the effect of a regional aquifer base dip cannot be ignored due to the scale of recharge sources (e.g., wadis, streams, reservoirs). However, Hantush's (1967) solution for a horizontal aquifer base is commonly used. To address sloping aquifers, a new closed‐form analytical solution for water table mound accounts for the geometry and orientation of recharge sources at the land surface with respect to the aquifer base dip. The solution, based on the Dupiuit‐Forchheimer approximation, Green's function method, and coordinate transformations is convenient for computing. This solution reveals important MAR traits in variance with Hantush's solution: mounding is limited in time and space; elevation of the mound is strongly affected by the dip angle; and the peak of the mound moves over time. These findings have important practical implications for assessment of various MAR scenarios, including waterlogging potential and determining proper rates of recharge. Computations are illustrated for several characteristic MAR settings.
Steady, 2D Darcian seepage in a homogeneous isotropic porous medium under an impervious structure is studied by the methods of complex analysis. The geometry of the structure is studied focusing on the travel time of a marked (neutral tracer) particle from the upper pool to the tailwater. In the Verigin problem, the angle of inclination of a sheetpile resulting in minimal time along the bounding streamline is π∕2 . If the maximum of the minimum of the travel time is searched between all streamlines originated in the upper pool, then the optimal angles are found to be 0.404π and 0.596π . The minimization of the total volume of fluid that arrives from the upper pool to the tailwater during a prescribed time span is also considered. For arbitrary geometry, structure optimization with respect to travel time is carried out explicitly for the bounding streamline with a constraint on the wetted perimeter of a depressed structure. The minimal-time shape is found to be the Voshinin semicircular structure, which is mathematically generated by a line vortex.
The known formulae of Freeze and Cherry, Polubarinova–Kochina, Vedernikov for flow rate during 2-D seepage into horizontal drains and axisymmetric flow into cavities are examined and generalized. The case of an empty drain under ponded soil surface is studied and existence of drain depth providing minimal seepage rate is presented. The depth is found exhibiting maximal difference in rate between a filled and an empty drain. 3-D flow to an empty semi-spherical cavity on an impervious bottom is analysed and the difference in rate as compared with a completely filled cavity is established. Rate values for slot drains in a two-layer aquifer are 'inverted' using the Schulgasser theorem from the Polubarinova–Kochina expressions for corresponding flow rates under a dam. Flow to a point sink modelling a semi-circular drain in a layered aquifer is treated by the Fourier transform method. For unsaturated flow the catchment area of a single drain is established in terms of the quasi-linear model assuming the isobaric boundary condition along the drain contour. Optimal shape design problems for irrigation cavities are addressed in the class of arbitrary contours with seepage rate as a criterion and cavity cross-sectional area as an isoperimetric restriction. © 1998 John Wiley & Sons, Ltd.
We consider solutions to two boundary values problems for the Poisson equation on plane domains. We prove several estimates for integrals of solutions using geometric characteristics of domains.
Summary Transport phenomena in porous media depend strongly on three‐dimensional pore structures. Macropore networks enable water and solute to move preferentially through the vadose zone. A complete representation of their geometry is important for understanding soil behaviour such as preferential flow. Once we know the geometrical, topological and scaling attributes of preferential flow paths, we can begin computer simulations of water movement in the soil. The box‐counting method is used in three dimensions (i.e. cube‐counting algorithm) to characterize the mass fractal dimension of macropore networks using X‐ray computed tomography (CT) matrices. We developed an algorithm to investigate the mass fractal dimension in three dimensions and to see how it compares with the co‐dimensions obtained using the box‐counting technique in two dimensions. For that purpose, macropore networks in four large undisturbed soil columns (850 mm × 77 mm diameter) were quantified and visualized, in both two and three dimensions, using X‐ray CT. We observed an increasing trend between the fractal dimension and macroporosity for the four columns. Moreover, similar natural logarithm functions were obtained for the four cores by a least squares fit through plots of mass fractal dimension against macroporosity.
