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 to get opened against the rock burden.

When injection bottomhole pressure is below fracture opening value then water is going to the main pay only (Reservoir Layer #1) and flow radially around the well.

When injection bottomhole pressure is above fracture opening value then water is going to the fracture and then gets distributed between Reservoir Layer #1 and Reservoir Layer 2

$GLOSSARY > Dual-layer IPR with dynamic fracture > image2020-4-19_14-50-24.png$

Fig. 1. Dual-layer well schematic

q = q_1 + q_2

p_{wf} = p_e + q/J

J = J_1 + J_2

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

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

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

where

Well
		total subsurface flowrate of the well
		total well productivity Index
		apparent formation pressure of dual-layer formation
		true vertical height between the layers tops
		wellbore fuid density
		gravity constant
		fracture opening pressure
Layer #1
		bottom-hole pr4essure at Layer #1 top
		total subsurface flowrate of the Layer #1
		formation pressure of the Layer #1
		productivity Index of the Layer #1
Layer #2
		bottom-hole pr4essure at Layer #2 top
		total subsurface flowrate of the Layer #2
		formation pressure of the Layer #2
		productivity Index of the Layer #2

p_{wf, 1} = p_{wf} = \Delta p_f + p_1  + q_1/J_1

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

This leads to

q_1 = J_1 \cdot (p_{wf} - p_1 - \Delta p_f)

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

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) )

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

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

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

See Also