4/π stimulated well potential

Brief

Stimulated well drainage

4/π is the maximum possible stimulation potential for steady state linear flow in a square well spacing.

Math & Physics

Steady state flow boundary conditions:

$P |_{x=x_e/2} = P |_{x=-x_e/2} = P_i$

$\frac{dp}{dt} =0\ for \ \forall x$

From Darcy's law:

$\frac{q}{2}=\frac{kA}{B \mu}\ \frac{dP}{dx}$

$A =y_e*h$

$dP=\frac{q B \mu}{2ky_eh} dx$

Integration gives: $P-P_{wf}=\frac{q B \mu}{2ky_eh} x$

Since average pressure is: $\bar P = \frac{\int P dx}{\int dx}$

$\bar P = \frac{ \int \limits_{0}^{x_e/2} \left ( \frac{q B \mu}{2ky_eh} x + P_{wf} \right ) dx}{\int dx} = \frac{q B \mu}{2ky_eh} \frac{x}{2} | + P_{wf}$

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