Geology Reference
In-Depth Information
9.1.1 Example boundary value problems.
If the filtration rate q(t) is a constant, say q o , Equation 9.21 then takes the
form
dx/dt = {q o / }df w (S w , w / nw )/dS w (9.22)
Since the derivative df w (S w , w / nw )/dS w is also constant along trajectories (as
a result of Equation 9.20), depending only on the arguments S w and
w /
nw , it
follows that Equation 9.22 can be integrated in the form
x - {(q o / )df w (S w ,
w /
nw )/dS w } t = constant
(9.23)
That S w is constant when x - { ...} t is constant can be expressed as
S w (x,t) = G(x - {(q o / )df w (S w , w / nw )/dS w }t) (9.24)
where G is a general function. Note that the method by which we arrived at
Equation 9.24 is known as the method of characteristics (Hildebrand, 1948).
9.1.2 General initial value problem.
We now explore the meaning of Equation 9.24. Let us set t = 0 in
Equation 9.24. Then, we obtain
S w (x,0) = G(x) (9.25)
In other words, the general saturation solution to Equation 9.17 for constant q(t)
= q o satisfying the initial condition S w (x,0) = G(x), where G is a prescribed
initial function, is exactly given by Equation 9.24 !
Thus, it is clear that the finite difference numerical solutions offered by
some authors are not really necessary because problems without capillary
pressure can be solved analytically. Actually, such computational solutions are
more damaging than useful because the artificial viscosity and numerical
diffusion introduced by truncation and round-off error smear certain
singularities (or, infinities) that appear as exact consequences of Equation 9.17.
Such numerical diffusion, we emphasize, appears as a result of finite difference
and finite element schemes only, and can be completely avoided using the more
labor-intensive method of characteristics. For a review of these ideas, refer to
Chapter 13 of Chin (2002). It is shown how capillary pressure effects become
important when singularities appear and how their modeling is crucial to correct
strength and shock position prediction.
To examine how these singularities arise in the solution of Equation 9.17,
take partial derivatives of Equation 9.24 with respect to x, so that
S w (x,t)/ x = {G'}{1 - t (q o / )d 2 f w /dS w 2
S w (x,t)/ x}
(9.26)
Solving for S w (x,t)/ x, we obtain
S w (x,t)/ x = G'/{1 + t (q o / ) (G') d 2 f w /dS w 2 }
(9.27)
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