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 $\begin{array}{l}\Delta p_f\end{array}$ to get opened against the rock burden.

When injection bottomhole pressure $\begin{array}{l}p_{wf}\end{array}$ is below fracture opening value $\begin{array}{l}p_{wf} < \Delta p_f\end{array}$ then water is going to the main pay only (Reservoir Layer #1) and flow radially around the well.

When injection bottomhole pressure $\begin{array}{l}p_{wf}\end{array}$ is above fracture opening value $\begin{array}{l}p_{wf} > \Delta p_f\end{array}$ 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

(1)	$\begin{array}{l}\displaystyle q = q_1 + q_2\end{array}$

(2)	$\begin{array}{l}\displaystyle p_{wf} = p_e + q/J\end{array}$

(3)	$\begin{array}{l}\displaystyle J = J_1 + J_2\end{array}$

(4)	$\begin{array}{l}\displaystyle p_e = \Delta p_f + \frac{J_1 \cdot p_1 + J_2 \cdot (p_2- \delta p_2)}{J_1 + J_2}\end{array}$

(5)	$\begin{array}{l}\displaystyle p_e = \frac{J_1 \cdot p_1 + J_2 \cdot p_c}{J_1 + J_2}\end{array}$

(6)	$\begin{array}{l}\displaystyle p_c = \left(1 + \frac{J_1}{J_2} \right) \Delta p_f + p_2 - \delta p_2\end{array}$

where

Well
	$\begin{array}{l}q\end{array}$	total subsurface flowrate of the well
	$\begin{array}{l}J\end{array}$	total well productivity Index
	$\begin{array}{l}p_e\end{array}$	apparent formation pressure of dual-layer formation
	$\begin{array}{l}h\end{array}$	true vertical height between the layers tops
	$\begin{array}{l}\rho\end{array}$	wellbore fuid density
	$\begin{array}{l}g\end{array}$	gravity constant
	$\begin{array}{l}\Delta p_f\end{array}$	fracture opening pressure
Layer #1
	$\begin{array}{l}p_{wf}\end{array}$	bottom-hole pr4essure at Layer #1 top
	$\begin{array}{l}q_1\end{array}$	total subsurface flowrate of the Layer #1
	$\begin{array}{l}p_1\end{array}$	formation pressure of the Layer #1
	$\begin{array}{l}J_1\end{array}$	productivity Index of the Layer #1
Layer #2
	$\begin{array}{l}p_{wf2} = p_{wf} + \rho \, g\, h\end{array}$	bottom-hole pr4essure at Layer #2 top
	$\begin{array}{l}q_2\end{array}$	total subsurface flowrate of the Layer #2
	$\begin{array}{l}p_2\end{array}$	formation pressure of the Layer #2
	$\begin{array}{l}J_2\end{array}$	productivity Index of the Layer #2

Derivation

(7)	$\begin{array}{l}\displaystyle p_{wf, 1} = p_{wf} = \Delta p_f + p_1 + q_1/J_1\end{array}$

(8)	$\begin{array}{l}\displaystyle p_{wf,2} = p_{wf} + \delta p_2 = \Delta p_f + p_2 + q_2/J_2\end{array}$

This leads to

(9)	$\begin{array}{l}\displaystyle q_1 = J_1 \cdot (p_{wf} - p_1 - \Delta p_f)\end{array}$

(10)	$\begin{array}{l}\displaystyle q_2 = J_2 \cdot (p_{wf,2} - p_2 - \Delta p_f) = J_2 \cdot (p_{wf} - (p_2 + \Delta p_f-\delta p_2) )\end{array}$

and

(11)	$\begin{array}{l}\displaystyle q = q_1 + q_2 = q_1 = J_1 \cdot (p_{wf} - (p_1 + \Delta p_f))+ J_2 \cdot (p_{wf} - (p_2-\delta p_2 + \Delta p_f) )\end{array}$

(12)	$\begin{array}{l}\displaystyle q = (J_1+J_2)\cdot p_{wf} - J_1 \cdot (p_1 + \Delta p_f) + J_2 \cdot ((p_2-\delta p_2 + \Delta p_f) )\end{array}$

or

(13)	$\begin{array}{l}\displaystyle q = J \cdot (p_{wf}-p_e), \ {\rm where} \ J = J_1 + J_2 \ {\rm and} \ p_e = J^{-1} \cdot (J_1 \cdot (p_1 + \Delta p_f) + J_2 \cdot (p_2-\delta p_2 + \Delta p_f))\end{array}$

or

(14)	$\begin{array}{l}\displaystyle p_e = \Delta p_f + J^{-1} \cdot (J_1 \cdot p_1 + J_2 \cdot (p_2-\delta p_2))\end{array}$

Page tree

See Also

Page tree

Dual-layer IPR with dynamic fracture

See Also