Theoretical solution of the unsteady unsaturated flow problems in soils

Abstract

He mathematical description of the state of a moving fluid is effected by means of functions which give the distribution of the fluid velocity v = v(x, y, z, t) and of any two thermodynamic quantities pertaining of the fluid. These two quantities can be taken to be the pressure p(x, y, z, t) and the density q(x, y, z, t). The above quantities v, p and q are functions of the coordinates x, y, z and of the time t.Dealing with flow of water through soils the following three general laws of mechanics but in hydrodynamic terms are applied to all kinds of water movement: 1) The dynamical equation of motion which specifies the dynamical reactions of the fluid to pressure gradients or external forces (Darcy's equation replacing the Navier-Stokes relations which are the hydrodynamical restatement of Newton's first law of motion), 2) The equation of continuity which is the analytical statement of the law of the conservation of matter taken from mechanics; and 3) The equation of state which defines thermody-namically the nature and the character of the fluid (relation between the pressure, density and temperature of the fluid)The flowing water has to satisfy the above equations at any point of the considered medium and at any time. This leads to a non-linear partial differential equation in which the independent variables are the time t and the position in space, and the dependent variable is the pressure of the water or the volumetric water content of the soil. The general equation for the three-dimensional flow through soils is derived, it is of such nature that a solution exists for t 0 and is uniquely determined if two relationships are defined, together with the specified state of the system, at the initial time t = 0, and at the boundaries. The two required relations are that of pressure versus permeability and pressure versus volumetric water content of the soil.Because of the strong non-linearity the governing partial differential equation is approximated by finite-differences at discrete mesh points in the solution domain and integrated for initial and boundary conditions corresponding to the gra The stability and convergence of the solution of the approximate difference equation to that of the differential equation gives justification to the use of an implicit difference scheme which generates a system of simultaneous non-linear equations that has to be solved for each time increment. For n mesh points the two boundary conditions provide two equations and the repetition of the recurrence formula provides n—2 equations, the total being n equations for each time increment. The solution of the system is obtained by matrix inversion and particularly with the back substitution techniqueThe procedures for the required numerous calculations for the numerical integration are programmed in Fortran Coding Language and fast answers (consisting of transient values for the pressure distribution, water content and flow rate) are obtained with the aid of an electronic digital computer (IBM).The inclusion of inertia terms which will alter the governing equation from parabolic (diffusion type) to hyperbolic (wave type) is suggested, in order to obtain more accurate results during the first period of absorption for the infiltration and capillary rise cases. © 1965 Taylor & Francis Group, LLC.