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Consider a system of net hydrocarbon pay and aquifer as  a radial composite reservoir with inner composite area being a Net Pay Area and outer composite area an Aquifer.

The transient pressure diffusion flow in this system is going to honour the following equation:

\frac{\partial p_a}{\partial t} = \chi \cdot \left[ \frac{\partial^2 p_a}{\partial r^2} + \frac{1}{r}\cdot \frac{\partial p_a}{\partial r} \right]

p_a(t = 0, r)= p(0)

p_a(t, r=r_e) = p(t)

\frac{\partial p_a}{\partial r} 
\bigg|_{(t, r=r_a)} = 0

Consider dimensionless solution 

\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

 p_1(t_D, r_D = \infty) = 0

which represents a specific function of dimensionless time

Now consider a convolution integral a pressure convolution:

p_a(t, r) = p(0) + \int_0^t p_1 \left(\frac{(t-\tau) \cdot \chi}{r_e^2}, \frac{r}{r_e} \right) \dot p(\tau) d\tau

\dot p(\tau) = \frac{d p}{d \tau}

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One can easily check that

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