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anchorCase2_PSS
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\delta q_1 / \delta q_2 = f_{21} =  \frac{V_{\phi, 2}}{ V_{\phi, 1}} = \rm const

while the response delay in time becomes irrelevant at long terms of the conventional production analysis.


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titleDerivation


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For the finite-volume reservoir 

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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:


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anchorCase2_PSS_p11
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p_{u,\rm 11}(t \rightarrow \infty) \rightarrow \frac{t}{c_t V_{\phi, 1}}



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


where

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bodyc_t

average drain-area  total compressibility of formation within  

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

Substituting 

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anchorCase2_PSS_p11
 and 
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anchorCase2_PSS_p21
 in 
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anchorCase2
 one arrives to
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anchorCase2_PSS
.



In case when injector W2 supports only one producer W1 then

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bodyV_{\phi, 2} = V_{\phi, 1}
 and 
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body\delta q_1 = \delta q_2
, which means that producer W1 with constant BHP and finite-reservoir volume will eventually vary its rate at the same volume as injector W2 (the response delay in time becomes irrelevant at long terms of the conventional production analysis)2.



If pressure in producer W1 is supported by several injectors 

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bodyN_{\rm inj} > 1
then over a long period of time one can assume:

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