Abstract
In the analytical solution of the Buckley-Leverett problem, Welge1 recommends that to locate a front, one should draw a tangent to the fractional saturation curve from the origin. In this paper I establish that this procedure will be correct only for the case of a zero initial condition. For a nonzero initial condition, a mass-conserving front will be located farther down the flow direction. The implications of this finding for error analysis in comparing numerical solutions to the analytical one are discussed.
Introduction
To establish the accuracy of a numerical solution to the Buckley-Leverett equation, one normally seeks a comparison with the analytical solution. Difficulties arise, however, when a zero initial saturation over the space domain, normally imposed on the analytical solution, is to be expressed numerically while incorporating a nonzero boundary condition. For example, the finite-element method using a "Chapeau" basis function by necessity generates a ramp initial condition. The objective of this paper is to provide a modification to Welge's1 method for an appropriate location of the front on the basis of mass conservation for a condition where some water greater than the residual water saturation is initially present.
The analytical solution to the Buckley-Leverett equation is known to yield a multiple-value saturation profile that is resolved by locating a front on the basis of mass conservation. This was suggested by Buckley and Leverett.2 A quick way of locating the front was provided by Welge,1 and is also discussed in Ref. 3. Welge's method locates the front accurately for the particular case when the initial saturation is zero (or a constant residual water saturation) over the entire space domain. In the more general case of a nonzero initial condition (i.e., initial saturation greater than residual saturation), his method needs modification. One such method is presented in this paper.
Development of the Modified Technique
The Buckley-Leverett equation is given byEquation 1
whereqt=total volumetric flow rate (L3/T),fw=fractional flow of wetting phase,Sw=saturation of wetting phase,t=time (T),x=space coordinate (L),A=cross-sectional area normal to flow (L2), andf=porosity.
Introducing ut=qt/Af, the total interstitial flow velocity, Eq. 1 may be written asEquation 2
It was shown by Buckley and Leverett2 that the solution to Eq. 2 may be generated by computing the displacement, ?x, experienced by any saturation, Sw.Equation 3
Owing to the bell-shaped property of dfw/dSw as a function of saturation Sw, the solution of Eq. 3 generates a triple-value function, ?x(Sw,t). The physical incompatibility of the multiplicity of Sw at a given x on the advanced saturation profile of the wetting phase was resolved by Buckley and Leverett2 by locating a front while maintaining conservation of mass. Welge1 rightfully pointed out the computational effort in computing the area every time a solution is required. He established that the mass-conserving front location may be arrived at by drawing (in the fw vs. Sw plane) a tangent from the origin (Sw=Swr, fw=0) to the fw(Sw) curve. The saturation at the point of tangency is the saturation at which the front is to be located.
I now show that Welge's method will yield the correct front location only in the special case of zero initial condition - i.e., when Sw(x,0)=Sw for all x. For the more general case of a nonzero (over and above Sw) initial condition, Welge's method will be modified.
A nonzero initial condition affects the solution in two respects.
On Dankwerts' transformation for two variable coupled systems
