# Page History

## Key

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Consider a water injector with main pay in Reservoir Layer #1 and spontaneous fracture extending down to Reservoir Layer #2 (see Fig. 1).

Assume that fracture is not fixed and requires surplus pressure

LaTeX Math Inline
body \Delta p_f
to get opened against the rock burden.

When injection bottomhole pressure

LaTeX Math Inline
body --uriencoded-- p_%7Bwf%7D
is below fracture opening value
LaTeX Math Inline
body --uriencoded-- p_%7Bwf%7D < \Delta p_f
then water is going to the main pay only (Reservoir Layer #1) and flow radially around the well.

When injection bottomhole pressure

LaTeX Math Inline
body --uriencoded-- p_%7Bwf%7D
is above fracture opening value
LaTeX Math Inline
body --uriencoded-- p_%7Bwf%7D > \Delta p_f
then water is going to the fracture and then gets distributed between Reservoir Layer #1 and Reservoir Layer 2

 Fig. 1. Dual-layer well schematic

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anchor 23CBN left
q = q_1 + q_2

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anchor 23CBN left
p_{wf} = p_e + q/J

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anchor 23CBN left
J = J_1 + J_2

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anchor pe left
p_e = \Delta p_f + \frac{J_1 \cdot p_1 + J_2 \cdot (p_2- \delta p_2)}{J_1 + J_2}

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anchor pe left
p_e = \frac{J_1 \cdot p_1 + J_2 \cdot p_c}{J_1 + J_2}

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anchor pc left
p_c = \left(1 + \frac{J_1}{J_2} \right) \Delta p_f + p_2 - \delta p_2

where

Well

LaTeX Math Inline
body q

total subsurface flowrate of the well

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body J

total well productivity Index

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body p_e

apparent formation pressure of dual-layer formation

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body h

true vertical height between the layers tops

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body \rho

wellbore fuid density

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body g

gravity constant

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body \Delta p_f

fracture opening pressure
Layer #1

LaTeX Math Inline
body --uriencoded--p_%7Bwf%7D

bottom-hole pr4essure at Layer #1 top

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body q_1

total subsurface flowrate of the Layer #1

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body p_1

formation pressure of the Layer #1

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body J_1

productivity Index of the Layer #1
Layer #2

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body --uriencoded--p_%7Bwf2%7D = p_%7Bwf%7D + \rho \, g\, h

bottom-hole pr4essure at Layer #2 top

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body q_2

total subsurface flowrate of the Layer #2

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body p_2

formation pressure of the Layer #2

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body J_2

productivity Index of the Layer #2

Expand
title Derivation

Panel
borderColor wheat mintcream 7

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anchor 23CBN left
p_{wf, 1} = p_{wf} = \Delta p_f + p_1  + q_1/J_1

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anchor 23CBN left
p_{wf,2} = p_{wf} + \delta p_2 = \Delta p_f + p_2 + q_2/J_2

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anchor 23CBN left
q_1 = J_1 \cdot (p_{wf} - p_1 - \Delta p_f)

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anchor 23CBN left
q_2 = J_2 \cdot (p_{wf,2} - p_2 - \Delta p_f) = J_2 \cdot (p_{wf} - (p_2 + \Delta p_f-\delta p_2) )

and

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anchor 23CBN left
q = q_1 + q_2 = q_1 = J_1 \cdot (p_{wf} - (p_1 + \Delta p_f))+ J_2 \cdot (p_{wf} - (p_2-\delta p_2  + \Delta p_f) )

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anchor 23CBN left
q =  (J_1+J_2)\cdot  p_{wf} - J_1 \cdot (p_1 + \Delta p_f) + J_2 \cdot ((p_2-\delta p_2 + \Delta p_f) )

or

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anchor 23CBN left
q =  J \cdot (p_{wf}-p_e), \ {\rm where} \ J = J_1 + J_2 \ {\rm and} \ p_e = J^{-1} \cdot (J_1 \cdot (p_1 + \Delta p_f) + J_2 \cdot (p_2-\delta p_2 + \Delta p_f))

or

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anchor 23CBN left
p_e = \Delta p_f +  J^{-1} \cdot (J_1 \cdot p_1  + J_2 \cdot (p_2-\delta p_2))