Difference between revisions of "6/π stimulated well potential"
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(→Math & Physics) |
(→Math & Physics) |
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From Diffusivity Equation: | From Diffusivity Equation: | ||
− | :<math>\frac{d^2P}{dx^2}=\frac{\phi c \mu}{k} \frac{dP}{dt}</math> | + | :<math>\frac{d^2P}{dx^2}=\frac{\phi c \mu}{k} \frac{dP}{dt}</math> ( 1 ) |
From [[Darcy's law]]: | From [[Darcy's law]]: | ||
Line 30: | Line 30: | ||
:<math>c=-\frac{1}{V} \frac{dV}{dP}</math> | :<math>c=-\frac{1}{V} \frac{dV}{dP}</math> | ||
− | :<math> \frac{dP}{dt} = -\frac{q}{2 c y_e h \phi} \frac{2}{x_e}</math> | + | :<math> \frac{dP}{dt} = -\frac{q}{2 c y_e h \phi} \frac{2}{x_e}</math> ( 2 ) |
+ | |||
+ | ( 2 ) - > ( 1 ) : | ||
+ | |||
+ | :<math>\frac{d^2P}{dx^2}=- \frac{q \mu}{2 k y_e h} \frac{2}{x_e}</math> | ||
==Diff eq== | ==Diff eq== |
Revision as of 09:47, 12 September 2018
Brief
6/π is the maximum possible stimulation potential for pseudo steady state linear flow in a square well spacing.
Math & Physics
Pseudo steady state flow boundary conditions:
From Diffusivity Equation:
( 1 )
From Darcy's law:
From Material Balance:
( 2 )
( 2 ) - > ( 1 ) :
Diff eq
From Mass conservation:
( 1 )
From Darcy's law:
( 2 )
( 2 ) →( 1 ):
( 3 )
( 4 )
( 5 )
( 5 ) -> ( 4 ):
( 6 )
( 7 )
Assumption that viscosity is constant cancels out first term in left hand side of (7):
( 8 )
( 9 )
( 9 ) -> ( 8 ):
( 10 )
Term in (10) is second order of magnitude low and can be cancelled out, which yields:
( 11 )
See also
optiFrac
fracDesign
Production Potential
Nomenclature
= cross-sectional area, cm2
= thickness, m
= permeability, d
= pressure, atm
= initial pressure, atm
= average pressure, atm
= flow rate, cm3/sec
= length, m
= drinage area length, m
= drinage area width, m
Greek symbols
= oil viscosity, cp