Groundwater flow in a medium with periodic inclusions

A study is made of the problem of two-dimensional confined steady flow through a porous reservoir whose percolation coefficient is a step function, while the homogeneity zones are rectangular inclusions (blocks) and a matrix (fractures). The velocity distributions, streamlines, isochrones of the motion of marked particles, and their time of passage through the elementary cell are found for this doubly periodic structure under conditions of continuity of the fluid head and the normal component of the flow on the phase contact boundary. Conclusions concerning the convective component of the longitudinal and transverse dispersion are drawn on the basis of these essentially two-dimensional hydrodynamic characteristics. The exact solutions obtained are compared with the results of numerical calculations carried out by the finite-difference method.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 139–148, September–October, 1995.     

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.     

Estimates for Integral Means of Hyperbolically Convex Functions

We prove the Mejia-Pommerenke conjecture that the Taylor coefficients of hyperbolically convex functions in the disk behave like O(log^?2(n)/n) as n → ∞ assuming that the image of the unit disk under such functions is a domain of bounded boundary rotation. Moreover, we obtain some asymptotically sharp estimates for the integral means of the derivatives of such functions and consider an example of a hyperbolically convex function that maps the unit disk onto a domain of infinite boundary rotation.     

Analytic solution of an ℝ-linear conjugation problem in the case of hyperbolic interface

We derive an explicit analytical solution for the conjugation problem in a heterogeneous infinite planar structure consisting of two dissimilar homogeneous media with two branches of a hyperbola as an interface. We investigate both cases involving real and complex coefficients of the corresponding boundary condition.     

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.     

An accurately solvable problem of the mutual effect of inclusions in the theory of heterogeneous media

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 21–29, January–February, 1990.     

A generalized Milne–Thomson theorem for the case of parabolic inclusion

Complex analysis methods are applied to determine a velocity field of seepage in a heterogeneous infinite planar medium consisting of two dissimilar homogeneous components with a parabolic interface. New cases with arbitrary singularities of the principal part of a required complex potential are considered.