Dimensionless transient water flux response from aquifer to unit step change in net pay pressure:

W_{eD}(t, r_{aD}) =  \int_0^{t}  

\frac{\partial p_1( t_D, r_D)}{\partial r_D}  \Bigg|_{r_D=1} dt_D

where

solution of unit-pressure radial composite reservoir transient flow –

\frac{\partial p_1}{\partial t_D} =  \frac{\partial^2 p_1}{\partial r_D^2} + \frac{1}{r_D}\cdot \frac{\partial p_1}{\partial r_D}

p_1(t_D = 0, r_D)= 0

p_1(t_D, r_D=1) = 1

\frac{\partial p_1(t_D, r_D)}{\partial r_D} 
\Bigg|_{r_D=r_{aD}} = 0

or

 p_1(t_D, r_D = \infty) = 0

The Dimensionless Water Influx is a unique function of time for each dimensionless value of which represents the ratio of external aquifer size to net pay size :

r_{aD} = \frac{r_a}{r_e}

This function is readily tabulated for a wide range of variations.

There are also polynomial approximations.

See Also