Motivation

Reservoir pressure dynamics away from wellbore and boundaries is representative of two very important complex reservoir properties: transmissibility  and pressure diffusivity .

In case the reservoir flow has been created by a well (vertical or horizontal) it will trend to form a radial flow away from boundaries and well itself.

In this case a pressure drop and well flowrate can be roughly related to each other by means of a simple analytical homogeneous reservoir flow model with wellbore and boundary effects neglected.

Inputs & Outputs

Physical Model

Mathematical Model

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

p(t=0,r) = p_i

p(t, r=\infty) = p_i

\left[ r \frac{\partial p}{\partial r} \right]_{r=0} = - \frac{q_t}{2 \pi \sigma}

Computational Model

p(t,r) = p_i - \frac{q_t}{4 \pi \sigma} {\rm Ei} \left(-\frac{r^2}{4 \chi t} \right)

Approximations

p(t,r) = p_i - \frac{q_t}{4 \pi \sigma} \left[  
\gamma + \ln \left(\frac{r^2}{4 \chi t} \right) \right] 

= p_i - \frac{q_t}{4 \pi \sigma} \ln \left(\frac{2.24585 \, t}{r^2} \right)

See also

Physics / Fluid Dynamics / Radial fluid flow / Line Source Solution

[ Radial Flow Pressure @model ] [ 1DR pressure diffusion of low-compressibility fluid ]