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titleDerivation


dq_0 (\tau) = \delta q_0 \cdot writen as 

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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
 and varying injection rate at injector W2 
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bodyq_2(t)
:

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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 01}(t-\tau) dq_0(\tau) = p_i - \int_0^t p_{u,\rm 01}(t-\tau) dq_0(\tau)

Consider a step-change in injector's W0 flowrate 

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body \delta q_0
 at zero time 
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body\tau = 0
, which can be written as: 
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body
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q_0(\tau) = \delta q_0 \cdot H(\tau)

where 

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bodyH(\tau)
is Heaviside step function:

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H(\tau) = \begin{cases} 0, &  \tau <0 \\  1, &\tau \geq 0\end{cases}

The differential

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bodydq_0
 then can be written as:

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d q_0(\tau) = q_0'(\tau) d\tau = \delta q_0 \cdot H'(\tau) \,  d\tau = \delta q_0 \cdot \delta(\tau) \,  d\tau
.
The responding pressure variation 
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body\delta p_1
 in producer W1 will be:
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\delta p_1(t) = p_1(t)-p_i = - \int_0^t p_{u,\rm 21}(t-\tau)  \delta q_0 \cdot \delta(\tau) \,  d\tau = - p_{u,\rm 01}(t) \cdot  \delta q_0
which leads to 
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anchorCase1
.

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