Exactly averaged stochastic equations for flow and transport in random media Page: 4 of 22
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distributions for the random field 6 (x +h) doesn't depend on the arbitrary vector h. Let
f (x) be a non-random source density function. We introduce the random Green's
function g(x,y) for the problem described (1), (2) and (4), so that for almost all
realizations of field a (x) the function g (x, y) satisfies the following equations:
ax a, (x)agxy = -8(x -y) (5)
g (x, y)= 0 for jx-oo (6)
In the general case we can now write the solution for the problem (1), (2) and (4):
u (x)= g (x,y)f (y)dy3 (7)
where d y3 = d y, d y2 d y3 and the integration is over the entire unbounded 3-D space.
We introduce the averaged fields over the ensemble of realizations of the random
function 6 (x):
U (x)= (u (x)), V (x)= (v (x)), G (x, y)= (g (x, y)) (8)
As long as 6 (x) is a stochastically homogeneous field, the mean Green's function
G (x, y) is invariant over translation in space, and therefore, depends only on the
difference x - y . Hence, after averaging the equation (7) over the ensemble, we have:
U (x)= fG(x-y)f (y)dy3 (9)
Then we can write the averaged equation over the ensembles of equation (1):
BV (x )
= f (x) (10)
After averaging the equation (2), we have:
V (x)= fi (x-y)f (y)dy3 (11)
where
F(x-y)=- (a,(x) ag a( )(12)
We shall call the vector F(x -y) the mean Green's velocity. By substituting (11) into
equation (10) we find the relation of compatibility for the components F, (x - y):
Dr'1(x-y) (3
-S(x-y) (13)4
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Shvidler, Mark & Karasaki, Kenzi. Exactly averaged stochastic equations for flow and transport in random media, article, November 30, 2001; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc741365/m1/4/: accessed April 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.