Page 176 - Percolation Models for Transport in Porous Media With

10.2 OPTIMAL TREATMENT REGIME 173

Introduce the following notation
X= { 1n(Raq/r)/1n(Raq/rw), i = 1
{10.26)
(r- 1 - R;;q )/(r;;I - R;;q 1 ), i = 2
1
Using {10.16) and {10.23), we transform {10.22) to get

{10.27)

Further, based on {10.24), {10.25), and (10.27), we obtain a universal equation for
both problems 1

Qo K j dx {10.28)
Q = ° K[we(CJPJQ)- 1 x + T(Raq)]
0
If we assume that at a distance Raq, the temperature of the medium is close to its
initial value T(Raq) :::::l To, K(T(Raq)) = K(To) :::::l Ko, for a specific dependence
{10.24), the relationship (10.28) is an equation for Q with a fixed We.
The relationship {10.24) was obtained experimentally for sandy-argillaceous
rocks described in chapter 9 and has the following form

Ko+B"(T-To), T<Tc
{ (10.29)
K(T) = Ko + B"(T- To)+ D"(T- Tc) 2 , T > Tc

Here Ko is the permeability at the initial temperature T0 ; B", D" are the param-
eters to be determined from experiment; Tc is the critical temperature, starting
from which the dependence K(T) becomes nonlinear. After substituting (10.29)
in (10.28) and integrating, we obtain a transcendental equation that relates the
production rate Q to the power of energy release We

ln[1 + We/(E"Q)], We< We
Qowe {
Q2 E" = 2 ( w - L" E" Q)F" (10.30)
ln(1 + L") + F" arctan [ ( 2 ~ L")E" Q +We] ,

where

2
L" = (T.e - ,.,o)B"K 0 - 1 ·, F" = 4E" D" (1 + L") - 1 ;
2 2K
.~,
CJPJ o
We= QoE"L" ln- {1 + L"), E" = KoCJPJ/B"
2
1
The experimental dependence (10.29) is a good approximation for describing
the dependence K(T) in a large temperature range (from several degrees Celsius
to the boiling point of the fluid). After determining the parameters Te, Ko, B",