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The response delay in time still exists but in usual time-scales of production analysis it becomes negligible and one can consider

LaTeX Math Block Reference
anchorCase2_PSS
 consant as constant in time.


Expand
titleDerivation


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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 V_{\phi, 1}}



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


where

LaTeX Math Inline
bodyc_t

average drain-area  total compressibility of formation within  

LaTeX Math Inline
bodyV_{\phi,1}
 which is jointly drained by  producer W1 and injector W2 

Substituting 

LaTeX Math Block Reference
anchorCase2_PSS_p11
 and 
LaTeX Math Block Reference
anchorCase2_PSS_p21
 in 
LaTeX Math Block Reference
anchorCase2
 one arrives to
LaTeX Math Block Reference
anchorCase2_PSS
.


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