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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 in producer W1 to the unit-rate production in injector W2


Expand
titleDerivation

Consider a pressure convolution equation for the BHP in producer W1 with constant flowrate production at producer W1 

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bodyq_1 = \rm const
:

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anchorCase2_PSS_p11
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p_1(t) = p_i - \int_0^t p_{u,\rm 11}(t-\tau) dq_1(\tau) - \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau) = p_i - \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau)

Consider a step-change in flowrate variation in injector W2: 

LaTeX Math Inline
bodydq_2(\tau) = \delta q_2 \cdot \delta(\tau) \, d\tau
 so that responding pressure variation 
LaTeX Math Inline
body\delta p_1
in producer W1 will be:

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anchorCase2_PSS_p11
alignmentleft
\delta p_1(t) = p_1(t)-p_i = - \int_0^t p_{u,\rm 21}(t-\tau)  \delta q_2 \cdot \delta(\tau) \,  d\tau = - p_{u,\rm 21}(t)  \delta q_2


LaTeX Math Block
anchorCase2_PSS_p11
alignmentleft
 p_{u,\rm 11}(0) \cdot q_1(t) + \int_0^t \dot p_{u,\rm 11}(t-\tau) dq_1(\tau)  = - p_{u,\rm 21}(0) \cdot q_2(t) -  \int_0^t \dot p_{u,\rm 21}(t-\tau) dq_2(\tau)



Case #2 – Constant BHP 
LaTeX Math Inline
bodyp_1 = \rm const

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