The fluid velocity $\begin{array}{l}u_L\end{array}$ across fracture area:

(1)	$\begin{array}{l}\displaystyle u_L = \frac{C_L}{\sqrt{t-t_0}}\end{array}$

where

$\begin{array}{l}C_L\end{array}$	Carter leak-off coefficient
$\begin{array}{l}t\end{array}$	time
$\begin{array}{l}t_0\end{array}$	exposure time (also called fill-in time) which require for fracture area to get exposed to a fluid flow

The Carter leak-off coefficient $\begin{array}{l}C_L\end{array}$ can be simulated numerically.

There are various analytical approximations, with the most popular being as:

(2)	$\begin{array}{l}\displaystyle C_L = \Delta P \, \sqrt{\frac{k \, \phi \, c_t}{\pi \, \mu}}\end{array}$

where

$\begin{array}{l}\Delta P = P_{wf} - P_e\end{array}$	drawdown pressure across fracture face area
$\begin{array}{l}P_{wf}\end{array}$	bottom-hole pressure across fracture face area
$\begin{array}{l}P_e\end{array}$	formation pressure around fracture
$\begin{array}{l}k\end{array}$	reservoir phase permeability to fracture fluid
$\begin{array}{l}\phi\end{array}$	reservoir porosity
$\begin{array}{l}\mu\end{array}$	fracture fluid viscosity
$\begin{array}{l}c_t = c_r + c_f\end{array}$	total reservoir compressibility

Volumetric leak-off rate $\begin{array}{l}q_L\end{array}$ is given by:

(3)	$\begin{array}{l}\displaystyle q_L = 2 \, h_f \, X_f \, \, u_L = \frac{2 \, h_f \, X_f \, C_L}{\sqrt{t-t_0}}\end{array}$

where

$\begin{array}{l}h_L\end{array}$	leak-off fracture height (usually $\begin{array}{l}h_L = h\end{array}$ , where $\begin{array}{l}h\end{array}$ is net reservoir thickness)
$\begin{array}{l}X_f\end{array}$	fracture half-length

The Carter's leak-off productivity index is given by:

(4)	$\begin{array}{l}\displaystyle J_L = \frac{q_L}{\Delta P}= 2 \, h_f \, X_f \, \sqrt{\frac{k \, \phi \, c_t }{\pi \, \mu \, (t-t_0)}}\end{array}$

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