Difference between revisions of "4/π stimulated well potential"
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From [[Darcy's law]]: | From [[Darcy's law]]: | ||
| − | :<math>\frac{q}{2}=\frac{kA}{ | + | :<math>\frac{q}{2}=\frac{kA}{\mu}\ \frac{dP}{dx}</math> |
:<math> A =y_e*h</math> | :<math> A =y_e*h</math> | ||
| − | :<math>dP=\frac{q | + | :<math>dP=\frac{q \mu}{2ky_eh} dx</math> |
| − | Integration gives: <math>P-P_{wf}=\frac{q | + | Integration gives: <math>P-P_{wf}=\frac{q \mu}{2ky_eh} x</math> |
Since average pressure is: <math>\bar P = \frac{\int P dx}{\int dx}</math> | Since average pressure is: <math>\bar P = \frac{\int P dx}{\int dx}</math> | ||
| − | :<math> \bar P = \frac{ \int \limits_{0}^{x_e/2} \left ( \frac{q | + | :<math> \bar P = \frac{ \int \limits_{0}^{x_e/2} \left ( \frac{q \mu}{2ky_eh} x + P_{wf} \right ) dx}{\int dx} = \left. \frac{q \mu}{2ky_eh} \frac{x}{2} \right|_{x=0}^{x=x_e/2} + P_{wf} = \frac{q \mu x_e}{8ky_eh} + P_{wf}</math> |
| − | :<math>J_D=\frac{q | + | :<math>J_D=\frac{q \mu}{2 \pi k h} \frac{1}{( \bar P - P_{wf})} =\frac{q \mu}{2 \pi k h} \frac{8ky_eh}{q \mu x_e} = \frac{4y_e}{\pi x_e}=\frac{4}{\pi}</math> |
==See also== | ==See also== | ||
Revision as of 11:28, 10 September 2018
Brief
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:
From Darcy's law:
Integration gives: 
Since average pressure is: 
See also
optiFrac
fracDesign
Production Potential
Nomenclature
= cross-sectional area, cm2
= thickness
= permeability
= length
= drinage area length
= drinage area width
= pressure
= initial pressure
= average pressure
= flow rate
Greek symbols
= oil viscosity
