GLOSSARY > Dual-layer IPR > 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 = \frac{J_1 \cdot p_1 + J_2 \cdot (p_2- \delta p_2)}{J_1 + J_2}

where

Well
		total subsurface flowrate of the well
		total well productivity Index
		apparent formation pressure of dual-layer formation
Layer #1
		bottom-hole pressure 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
		wellbore pressure loss between the tips of two layers
		total subsurface flowrate of the Layer #2
		formation pressure of the Layer #2
		productivity Index of the Layer #2

In many practical cases one can safely assume:

\delta p_2 = \rho \, g \, h

where

	wellbore fuid density
	gravity constant
	true vertical height between -th layer and reference layer

The above equations are valid for both producers and injectors .

p_{wf, 1} = p_{wf} = p_1 - q_1/J_1

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

This leads to

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

q_2 = J_2 \cdot (p_2 - p_{wf,2}) = J_2 \cdot ((p_2-\delta p_2)- p_{wf})

and

q = q_1 + q_2 = q_1 = J_1 \cdot (p_1 - p_{wf})+ J_2 \cdot ((p_2-\delta p_2)- p_{wf})

q =  - (J_1+J_2)\cdot  p_{wf} + J_1 \cdot p_1 + J_2 \cdot (p_2-\delta p_2)

or

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

See Also