Analytic Solution to a Problem of Seepage in a Chequer-Board Porous Massif

A study is made of steady two-dimensional seepage in a porous massif composed by a double-periodic system of white and black chequers of arbitrary conductivity. Rigorous matching of Darcy's flows in zones of different conductivity is accomplished. Using the methods of complex analysis, explicit formulae for specific discharge are derived. Stream lines, travel times, and effective conductivity are evaluated. Deflection of marked particles from the natural direction of imposed gradient and stretching of prescribed composition of these particles enables the elucidation of the phenomena of transversal and longitudinal dispersion. A model of pure advection is related with the classical one-dimensional vective dispersion equation by selection of dispersivity which minimizes the difference between the breakthrough curves calculated from the two models.     

Dipolic Flows Relevant to Aquifer Storage and Recovery: Strack’s Sink Solution Revisited

Steady, 2-,3-D Darcian flows generated by a dipole (a pair of horizontal or vertical injection–abstraction wells closely placed one above another), with circulation of fresh water inside an interface confined lens or “bubble” underneath an impermeable caprock, surrounded by a static saline groundwater, are analytically studied. For 2-D dipole, the complex potential domain is a plane with a horizontal cut. This domain is conformally mapped onto a reference half-plane where the Keldysh–Sedov formula is used to obtain the complex physical coordinates. Explicit closed-form expressions for the vase-shaped interface, flow net, isohypses, magnitudes of the Darcian velocity and Riesenkampf’s resultant force are obtained, depending on the dipole moment, its position with respect to the caprock, and the ratio of densities of the two fluids. It is shown that for sufficiently small injection-pumping rates the fresh water “vase” separates from the caprock and becomes a circle, inside which streamlines are Newtons’ loops of monodiametral degenerate hyperbolae (cubics). Two numerical codes, MT3DMS and SEAWAT, are also used for delineation of isoconcentric lines, which qualitatively corroborate the analytical solutions in delineation of the “bubble” in the part where the sharp interface model predicts stable free boundaries and evidencing “dimples” on the boundary of the “bubble” where the saline water overlies the fresh one. For 3-D dipole not bounded by the caprock, the analytical fresh water “bubble” is a sphere and solution follows, mutatis mutandis, from the textbook formulae for flow of an ideal fluid past an impermeable sphere. The Stokes streamlines inside the sphere are sixtics; isotachs are plotted in an axial section. Stability of the soil matrix near the wells is also discussed.     

Steady Flow from an Array of Subsurface Emitters: Kornev’s Irrigation Technology and Kidder’s Free Boundary Problems Revisited

Kornev’s (Subsurface irrigation, Selhozgiz, Moscow-Leningrad, 1935) subsurface irrigation with a periodic array of emitting porous pipes is analytically modeled as a steady potential Darcian flow from a line source generating a phreatic surface. The hodograph method is used. The complex potential strip is mapped onto the triangle of the inverted hodograph. An analogy with the Deemter (Theoretische en numerieke behandeling van ontwaterings-en infiltratie stromings problemen (in Dutch). Theoretical and numerical treatment of flow problems connected to drainage and irrigation. Ph.D. dissertation, Delft University of Technology, 1950) drainage problem and Kidder (J Appl Phys 27(8):867–869, 1956) free-surface flow toward an array of oil wells underlain by a “wavy” oil–water interface is drawn. For a half-period of Kornev’s flow, the “wavy” phreatic surface has an inflection point. The “waviness” of the phreatic surface is controlled by the spacing between emitters, the strength of line sources, and the pipe pressure and radius. Numerical modeling with HYDRUS involved two factors which constrained the saturated–unsaturated flow: the positive pressure head at the outlet of the modeled domain and lateral no-flow boundaries, with a qualitative corroboration of analytical solutions for potential (fully saturated) and purely unsaturated flows. HYDRUS is also applied to a generalized Philip’s regime of an unsaturated flow past a subterranean hole, which is impermeable at its top and leaks at the bottom.     

Analytical solutions for seepage near material boundaries in dam cores: The Davison-Kalinin problems revisited

Steady Darcian seepage through a dam core and adjacent shells is analytically studied. By conformal mappings of the pentagon in the hodograph plane and triangle in the physical plane flow through a low-permeable dam core is analyzed. Mass-balance conjugation of flow in the core and downstream highly-permeable shell of the embankment is carried out by matching the seepage flow rates in the two zones assuming that all water is intercepted by a toe-drain. Seepage refraction is studied for a wedge-shaped domain where pressure and normal components of the Darcian velocities coincide on the interface between the core and shell. Mathematically, the problem of R-linear conjugation (the Riemann-Hilbert problem) is solved in an explicit form. As an illustration, flow to a semi-circular drain (filter) centered at the triple point (contact between the core, shell and impermeable base) is studied. A piece-wise constant hydraulic gradient in two adjacent angles making a two-layered wedge (the dam base at infinity) is examined. Essentially 2-D seepage in a domain bounded by an inlet constant head segment, an outlet seepage-face curve, a horizontal base and with a straight tilted interface between two zones (core and shell) is investigated. The flow net, isobars, and isotachs in the core and shell are reconstructed by computer algebra routines as functions of hydraulic conductivities of two media, the angle of tilt and the hydraulic head value at a specified point.     

Comments on On an Exact Analytical Solution of the Boussinesq Equation,Transport in Porous Media 39, 339–345, 2000

Transport in Porous Media -     

Analytical solution to 2D problem for an anticline-diverted brine flow with a floating hydrocarbon trap

Explicit rigorous solution to a steady-state J. Bear (Dynamics of Fluids in Porous Media. Elsevier, New York, 1972) problem of brine flow in an anticline homogeneous isotropic rock with a gas cap under the anticline crest is obtained by the methods of complex analysis. The stagnant hydrocarbon volume is separated from the subjacent moving brine by a sharp interface, which is a free boundary “hanging” in the formation with the loci of the anticline roof-attachment and roof-detachment points, as well as of an inflexion point, a priori unknown. Mathematically, a conformal mapping of the complex potential strip and integral representation of the Hilbert problem for the inversed complex Darcian velocity are used to obtain the physical coordinate, complex velocity and complex potential as functions of an auxiliary variable. The interfaces are plotted for various incident brine flow rates, angles of dipping anticline flanks and gas pressure. For a gas trap comparisons with the interface calculated by the Dupuit–Forchheimer approximation are carried out.     

Optimal design of fibers subject to steady heat conduction

Steady, 2D heat conduction is studied for a double-periodic lattice of homogeneous fibers. Sink-source approximation models the thermal contact zones. The rest of the fiber surface is adiabatic. Optimal shape is found by the method of boundary-value problems of holomorphic functions. The cross-sectional area is a criterion of optimization, microscale heat flow through the fiber and temperature at a fiducial point within the cell is constraints.     

Analytical solutions of seepage theory problems. Inverse method, variational theorems, optimization and estimates (a review)

The results of analytical studies of the problems arising in connection with the prediction of ground water flow in civil engineering, hydrogeology and irrigation engineering are reviewed. Numerical techniques have become of ever greater significance in solving practical problems of seepage theory since the introduction of powerful computers in the sixties. However, even so analytical methods have proved to be necessary not only to develop and test the numerical algorithms but also to gain a deeper understanding of the underlying physics, as well as for the parametric analysis of complex flow patterns and the optimization and estimation of the properties of seepage fields, including in situations characterized by a high degree of uncertainty with respect to the porous medium parameters, the mechanisms of interaction between the fluid and the matrix, the boundary conditions and even the flow domain boundary itself. The review covers studies of ground water dynamics directly related to the problems of flow in domains with incompletely specified boundaries related to the authors interests. Mathematically, these problems reduce to boundary value problems for partial differential equations of elliptic type in domains with unknown boundaries found using specified boundary conditions. These are either deduced from the physical model of the process (the depression surface being an example) or determined by structural considerations (such as the underground shape of a dam or embankment). Kazan’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–19, March–April, 1998.