The fluid velocity across fracture area:

u_L = \frac{C_L}{\sqrt{t-t_0}}

where

	Carter leak-off coefficient
	time
	exposure time (also called fill-in time) which require for fracture area to get exposed to a fluid flow

The Carter leak-off coefficient can be simulated numerically.

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

C_L = \Delta P \, \sqrt{\frac{k \, \phi \, c_t}{\pi \, \mu}}

where

	drawdown pressure across fracture face area
	bottom-hole pressure across fracture face area
	formation pressure around fracture
	reservoir phase permeability to fracture fluid
	reservoir porosity
	fracture fluid viscosity
	total reservoir compressibility

Volumetric leak-off rate is given by:

q_L = 2 \, h_f \, X_f \,  \, u_L = \frac{2 \, h_f \, X_f \, C_L}{\sqrt{t-t_0}}

where

	leak-off fracture height (usually , where is net reservoir thickness)
	fracture half-length

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

J_L = \frac{q_L}{\Delta P}=  2 \, h_f \, X_f \, \sqrt{\frac{k \, \phi \, c_t }{\pi \, \mu \, (t-t_0)}}

See Also