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

Revision as of 10:06, 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}

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

Nomenclature

A

= cross-sectional area

k

= permeability

x

= length

P

= pressure

q

= flow rate

Greek symbols

\mu

= oil viscosity

@@ Line 35: / Line 35: @@
 ==Nomenclature==
-:<math> A </math> = cross-sectional area, cm<sup>2</sup>
+:<math> A </math> = cross-sectional area
-:<math> k</math> = permeability, d
+:<math> k</math> = permeability
-:<math> L </math> = length, cm
+:<math> x </math> = length
-:<math> P </math> = pressure, atm
+:<math> P </math> = pressure
-:<math> q </math> = flow rate, cm<sup>3</sup>/sec
+:<math> q </math> = flow rate
 ===Greek symbols===
-:<math> \mu </math> = [[Darcy's law]] oil viscosity, cp
+:<math> \mu </math> = oil viscosity
 [[Category:Technology]]

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