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: $\bar P = \frac{\int P dx}{\int dx}$

$\bar P = \frac{ \int \limits_{0}^{x_e/2} \left ( \frac{q B \mu}{2ky_eh} x + P_{wf} \right ) dx}{\int dx} = \left. \frac{q B \mu}{2ky_eh} \frac{x}{2} \right|_{x=0}^{x=x_e/2} + P_{wf} = \frac{q B \mu x_e}{8ky_eh} + P_{wf}$

$J_D=\frac{q B \mu}{2 \pi k h} \frac{1}{( \bar P - P_{wf})} =\frac{q B \mu}{2 \pi k h} \frac{8ky_eh}{q B \mu x_e} = \frac{4y_e}{\pi x_e}=\frac{4}{\pi}$

See also

optiFrac
fracDesign
Production Potential

Nomenclature

$A$ = cross-sectional area

$A$ = oil formation volume factor

$h$ = thickness

$k$ = permeability

$x$ = length

$x_e$ = drinage area length

$y_e$ = drinage area width

$P$ = pressure

$P_i$ = initial pressure

$\bar P$ = average pressure

$q$ = flow rate

Greek symbols

$\mu$ = oil viscosity