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

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(→‎Nomenclature)
(→‎Nomenclature)
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:<math> h </math> = thickness, m
 
:<math> h </math> = thickness, m
 
:<math> k</math> = permeability, d
 
:<math> k</math> = permeability, d
−
:<math> x </math> = length, m
 
−
:<math> x_e</math> = drinage area length, m
 
−
:<math> y_e</math> = drinage area width, m
 
 
:<math> P </math> = pressure, atm
 
:<math> P </math> = pressure, atm
 
:<math> P_i </math> = initial pressure, atm
 
:<math> P_i </math> = initial pressure, atm
 
:<math> \bar P</math> = average pressure, atm
 
:<math> \bar P</math> = average pressure, atm
 
:<math> q </math> = flow rate, cm<sup>3</sup>/sec
 
:<math> q </math> = flow rate, cm<sup>3</sup>/sec
 +
:<math> x </math> = length, m
 +
:<math> x_e</math> = drinage area length, m
 +
:<math> y_e</math> = drinage area width, m
  
 
===Greek symbols===
 
===Greek symbols===

Revision as of 11:30, 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}{\mu}\ \frac{dP}{dx}
A =y_e*h
dP=\frac{q \mu}{2ky_eh} dx

Integration gives: P-P_{wf}=\frac{q \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 \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}
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}

See also

optiFrac
fracDesign
Production Potential

Nomenclature

A = cross-sectional area, cm2
h = thickness, m
k = permeability, d
P = pressure, atm
P_i = initial pressure, atm
\bar P = average pressure, atm
q = flow rate, cm3/sec
x = length, m
x_e = drinage area length, m
y_e = drinage area width, m

Greek symbols

\mu = oil viscosity, cp
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