longitudinal dispersion in nonuniform isotropic porous media
longitudinal dispersion in nonuniform isotropic porous media
longitudinal dispersion in nonuniform isotropic porous media
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Q =<br />
52<br />
(3.1)<br />
where P'cos8 is the projection of the mean pressure gradient along the<br />
pore aX1S. The frequency that the a cross sectional plane<br />
(perpendicular to the x-axis) will <strong>in</strong>tersect a capillary tube of length<br />
£ and orientation S is proportional to £cosS. The probability for<br />
<strong>in</strong>tersect<strong>in</strong>g a given pore with<strong>in</strong> the differential range £ to (£+d£),<br />
S to (S+dS) , a to (a+da) 1S<br />
dS 2£cosSs<strong>in</strong>Sg(a)f(£)d£dad8<br />
where g(a) is the probability density for pore radius and f(£) is the<br />
probability density for pore length. The 2 normalizes the probability<br />
density function. The variables £ and a range from 0 to 00 and S ranges<br />
from 0 to n/2. Assum<strong>in</strong>g a given cross section is a representative<br />
sample of pores, the total flux through the cross section is equal to<br />
the average flux through a pore multiplied by the total number of pores<br />
<strong>in</strong> the cross section, N. The total flux, Q, pass<strong>in</strong>g through the cross<br />
section is N times the probability-weighted average of equation (3.1)<br />
where the triple <strong>in</strong>tegral represents the arithmetic average flux<br />
through a pore. The seepage velocity through the <strong>porous</strong> medium is<br />
given by Q/A , where A 1S the total area of pores <strong>in</strong> the cross<br />
p p<br />
(3.2)
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