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titleDerivation

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

LaTeX Math Inline
bodyq_1 = \rm const
 and varying injection rate at injector W2:

LaTeX Math Block
anchorCase2_PSS_p11
alignmentleft
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:

LaTeX Math Block
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


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