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LaTeX Math Inline
bodyt

time since the water injection rate has changed by the 

LaTeX Math Inline
body\delta q_2
value.

LaTeX Math Inline
bodyp_{u,\rm 21}(t)

cross-well pressure transient response (CTR) in producer W1 to the unit-rate production in injector W2

LaTeX Math Inline
bodyp_{u,\rm 11}(t)

drawdown pressure transient response (DTR) in producer W1 to the unit-rate production in the same well


For the finite-volume reservoir drain 

LaTeX Math Inline
body V_{\phi} < \infty
 the flowrate response factor 
LaTeX Math Inline
body\delta q_1 / \delta q_2 = f_{21} = \rm const
is constant:

LaTeX Math Block
anchorCase2_PSS
alignmentleft
\delta q_1 = -f_{21} \cdot / \delta q_2, \quad {\rm where} \ = f_{21} = \frac{c_{t,2} V_{\phi, 2}}{c_{t,1} V_{\phi, 1}} = \rm const


Expand
titleDerivation

For the finite-volume reservoir 

LaTeX Math Inline
body V_{\phi} < \infty
 the DTR and CTR are both going through the PSS flow regime:


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



LaTeX Math Block
anchorCase2_PSS_p21
alignmentleft
p_{u,\rm 21}(t) = \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
.

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