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The flowrate response 

LaTeX Math Inline
body\delta q_1
in producer W1 to the flowrate variation 
LaTeX Math Inline
body\delta q_2
 in injector W2:

LaTeX Math Block
anchor1Case2
alignmentleft
\delta q_1 = - \frac{p_{u,\rm 21}(t)}{p_{u,\rm 11}(t)} \cdot \delta q_2

...

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



Pseudo-steady state flow

LaTeX Math Block
anchorQ7LHOCase2_PSS
alignmentleft
\delta q_1 = -\frac{c_{t,2} V_{\phi, 2}}{c_{t,1} V_{\phi, 1}} \cdot \delta q_2


Expand
titleDerivation


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}}

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
.



Steady state flow

LaTeX Math Block
anchorDWLRD
alignmentleft
\delta q_1 = -\frac{c_{t,2} V_{\phi, 2}}{c_{t,1} V_{\phi, 1}} \cdot \delta q_2

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