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titleDerivation

For the finite-volume reservoir 

LaTeX Math Inline
body V_{\phi,1} \leq V_{\phi,2} < \infty
 the DTR and CTR are both going through the PSS flow regime at late transient times:


LaTeX Math Block
anchorCase2_PSS_p11
alignmentleft
p_{u,\rm 11}(t \rightarrow \infty) \rightarrow \frac{t}{c_{t,1} V_{\phi, 1}}



LaTeX Math Block
anchorCase2_PSS_p21
alignmentleft
p_{u,\rm 21}(t \rigtharrow \infty) \rightarrow \frac{t}{c_{t,2} V_{\phi,2}}


where

LaTeX Math Inline
bodyc_{t,1}

average drain-area  total compressibility of formation around producer W1

LaTeX Math Inline
bodyc_{t,2}

average drain-area  total compressibility of formation around injector W2



Substituting 

LaTeX Math Block Reference
displaytextCase2_PSS_p11
 and 
LaTeX Math Block Reference
displaytextCase2_PSS_p21
 in 
LaTeX Math Block Reference
displaytextCase2
 one arrives to
LaTeX Math Block Reference
displaytextCase2_PSS
.


If pressure in producer W1 is supported by several injectors then:

LaTeX Math Block
anchor1
alignmentleft
\delta q_1 =\sum_k f_{k1} \delta q_k

which makes one of the key assumptions in Capacitance Resistance Model (CRM).


See also

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Capacitance Resistance Model (CRM)