A solution of the differential equation of longitudinal dispersion in porous media

TLDR: In this paper, a more direct method is presented for solving the differential equation governing the process of dispersion in a semi-infinite medium having a plane source at z = 0.

Abstract: Published papers indicate that most investigators use the coordinate transformation (x ut) in order to solve the equation tor dispersion of a moving fluid in porous media. Further, the boundary conditions O=0 at x=«> and 0=00 at x= «> for Z>0 are used, which results in a symmetrical concentration distribution. This paper presents a solution of the differential equation that avoids this transformation, thus giving rise to an asymmetrical concentration distribution. It is then shown that this solution approaches that given by symmetrical boundary conditions, provided the dispersion coefficient D is small and the region near the source is not considered. INTRODUCTION In recent years considerable interest and attention have been directed to dispersion phenomena in flow through porous media. Scheidegger (1954), deJong (1958), and Day (1956) have presented statistical means to establish the concentration distribution and the dispersion coefficient. A more direct method is presented here for solving the differential equation governing the process of dispersion. It is assumed that the porous medium is homogeneous and isotropic and that no mass transfer occurs between the solid and liquid phases. It is assumed also that the solute transport, across any fixed plane, due to microscopic velocity variations in the flow tubes, may be quantitatively expressed as the product of a dispersion coefficient and the concentration gradient. The flow in the medium is assumed to be unidirectional and the average velocity is taken to be constant throughout the length of the flow field. BASIC EQUATION AND SOLUTION Because mass is conserved, tl^e governing differential equation is determined to be d<7 (1) v where D=dispersion coefficient C= concentration of solute in the fluid u= average velocity of fluid or superficial velocity/ porosity of medium x= coordinate parallel to flow y,z coordinates normal to flow 2=time. In the event that mass transfer takes place between the liquid and solid phases, the differential equation becomes _ 5(7 d(7 &F where F is the concentration of the solute in the solid phase. The specific problem considered is that of a semiinfinite medium having a plane source at z=0. Hence equation 1 becomes Initially, saturated flow of fluid of concentration, (7=0, takes place in the medium. At t Q, the concentration of the plane source is instantaneously changed to (7=(70 . Thus, the appropriate boundary conditions are <7(co,£)=0; *> The problem then is to characterize the concentration as a function of x and t. To reduce equation 1 to a more familiar form, let (4) A-l 586211 61 2 A-2 FLUID MOVEMENT IN EARTH MATERIALS Substituting equation 4 into equation 1 gives The boundary conditions transform to = Co exp ( It is thus required that equation 5 be solved for a timedependent influx of fluid at 2=0. The solution of equation 5 may be obtained readily by use of Duhamel's theorem (Carslaw and Jaeger, 1947, p. 19): If C=F(x,y,z,t) is the solution of the diffusion equation for semi-infinite media in which the initial concentration is zero and its surface is maintained at concentration unity, then the solution of the problem in which the surface is maintained at temperature <j>(t) is