...

Although the actual flow may not have an axial symmetry around the well-reservoir contact or reservoir inhomogeneities (like boundary and faults and composite areas) but still:

• the dominant part of wellbore and reservoir pressure variation is usually radial-flow or linear-flow and the two represent the basis for Pressure diffusion analysis

...

• 
  
  
• in most practical cases the long-term correlation between the flowrate and  bottom-hole pressure response can be approximated by a radial flow 

Inputs & Outputs

...

...

\frac{\partial p}{\partial t}  = \chi \, \left( \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} \right)

p(t = 0, {\bf r}) = p_i

p(t, r \rightarrow \infty ) = p_i

\left[ r\frac{\partial p(t, r )}{\partial r} \bigg|right]_{r \rightarrow 0r_w} = \frac{q_t}{2 \sigmapi \, dsigma}

p(t,r) = p_i + \frac{q_t}{4 \pi \sigma} \,  F \bigg( - \frac{r^2}{4 \chi t} \bigg)

p_{wf}(t) = p_i + \frac{q_t}{4 \pi \sigma} \, \bigg[ - 2S +   F \bigg( - \frac{r_w^2}{4 \chi t} \bigg) \bigg]

...