Annual Report 2000 - WIT

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138
In the previous section we derived an equation for the triggering time š{õ Ÿ of
a harmonic pressure perturbation. Using this equation we shall derive another one,
which will describe the triggering time ĝšŠõ°Ÿ of a step-function pressure perturbation.
From our earlier discussion we know, that the triggering time ˆ roughly corresponds to
the frequency
éä ‘“¤£è0ˆ?È (20)
Thus,
ó íNMÄíPOA‘ˆ?È (21)
From the other hand, we know š é Ÿ ö ̦¨é that generally . Now we can also use this
relationship to compute at the frequency , if is known at any arbitrary frequency
é_ä
é :
È (22)
ˆt‘
šéä«ŸK‘
š é Ÿ
Using this equation and equation (20) we obtain:
È (23)
Substituting this equation into equations for of the previous section we obtain the following
results. In the general case of an anisotropic heterogeneous poroelastic medium
š é Ÿ ‘
È (24)
ˆ ‘Q£!»P×]Ú
In the case of an isotropic poroelastic medium this equation is reduced to the following
one:
(25)
È
Ê1×
ÊÛÚ
» ‘
Inversion for the permeability of heterogeneous media
In the case of an isotropic poroelastic medium equation (25) can be directly used to
reconstruct the 3-D heterogeneous field of the hydraulic diffusivity. In turn, equation
(24) shows, that in the case of an anisotropic medium it is impossible to reconstruct a
3-D distribution of the diffusivity tensor. The only possibility is the following. Let us
assume that the orientation and the principal components proportion is constant in the
medium. Then, the tensor of hydraulic diffusivity can be expressed as
»P×]Ú(šŠõƒŸ ‘SR š{õ Ÿ5Tg×]Ú¨• (26)
where T×…Ú is a nondimensional constant tensor of the same orientation and principalcomponent
proportion as the diffusivity tensor, and R is the heterogeneously distributed
magnitude of this tensor. This tensor can be found using the global SBRC estimate of