Consider a pressure convolution equation for the above 2-wells system with constant BHP: | LaTeX Math Block |
|---|
| anchor | Case2_PSS_p111 |
|---|
| alignment | left |
|---|
| 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) = \rm const |
The time derivative is going to be zero as the bottom-hole pressure in producer W1 stays constant at all times: | LaTeX Math Block |
|---|
| anchor | Case2_PSS_p111 |
|---|
| alignment | left |
|---|
| \dot p_1(t) = - \left( \int_0^t p_{u,\rm 11}(t-\tau) dq_1(\tau) \right)^{\cdot} - \left( \int_0^t p_{u,\rm 21}(t-\tau) dq_2(\tau) \right)^{\cdot} = 0 |
| LaTeX Math Block |
|---|
| anchor | Case2_PSS_p111 |
|---|
| alignment | left |
|---|
| 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) |
The zero-time value of DTR / CTR is zero by definition | LaTeX Math Inline |
|---|
| body | p_{u,\rm 11}(0) = 0, \, p_{u,\rm 21}(0) = 0 |
|---|
|
which leads to:| LaTeX Math Block |
|---|
| anchor | Case2_PSS_p11_temp |
|---|
| alignment | left |
|---|
| \int_0^t \dot p_{u,\rm 11}(t-\tau) dq_1(\tau) = - \int_0^t \dot p_{u,\rm 21}(t-\tau) dq_2(\tau) |
Consider a step-change in producer's W1 flowrate and injector's W2 flowrate at zero time , which can be written as | LaTeX Math Inline |
|---|
| body | dq_1(\tau) = \delta q_1 \cdot \delta(\tau) \, d\tau |
|---|
|
and | LaTeX Math Inline |
|---|
| body | dq_2(\tau) = \delta q_2 \cdot \delta(\tau) \, d\tau |
|---|
|
. Substituting this to | LaTeX Math Block Reference |
|---|
|
leads to:| LaTeX Math Block |
|---|
| anchor | Case2_PSS_p11_temp1 |
|---|
| alignment | left |
|---|
| \int_0^t \dot p_{u,\rm 11}(t-\tau) \delta q_1 \cdot \delta(\tau) \, d\tau = - \int_0^t \dot p_{u,\rm 21}(t-\tau) \delta q_2 \cdot \delta(\tau) \, d\tau |
| LaTeX Math Block |
|---|
| anchor | Case2_PSS_p11_temp |
|---|
| alignment | left |
|---|
| \dot p_{u,\rm 11}(t) \delta q_1 = - \dot p_{u,\rm 21}(t) \delta q_2 |
which leads to | LaTeX Math Block Reference |
|---|
|
. | 