Difference between revisions of "4/π stimulated well potential"

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:<math> \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}</math>
 
:<math> \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}</math>
  
−
<math>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}</math>
+
:<math>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}</math>
  
 
==See also==
 
==See also==

Revision as of 10:12, 10 September 2018

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
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