Motivation

Reservoir pressure dynamics away from wellbore and boundaries is representative of two very important complex reservoir properties: transmissibility $\begin{array}{l}\sigma\end{array}$ and pressure diffusivity $\begin{array}{l}\chi\end{array}$ .

These can be roughly estimated with a homogeneous reservoir model where wellbore and boundaries effects can be neglected.

Inputs & Outputs

Detailing

$\begin{array}{l}\sigma = \frac{k \, h}{\mu}\end{array}$	transmissibility	$\begin{array}{l}\mu\end{array}$	dynamic fluid viscosity
$\begin{array}{l}\chi = \frac{k}{\mu} \, \frac{1}{\phi \, c_t}\end{array}$	pressure diffusivity	$\begin{array}{l}c_t = c_r + c\end{array}$	total compressibility
$\begin{array}{l}k\end{array}$	absolute permeability	$\begin{array}{l}{c_r}\end{array}$	pore compressibility
$\begin{array}{l}{\phi}\end{array}$	porosity	$\begin{array}{l}c\end{array}$	fluid compressibility

Zero wellbore radius $\begin{array}{l}r_w = 0\end{array}$

(1)	$\begin{array}{l}\displaystyle \frac{\partial p}{\partial t} = \chi \, \left[ \frac{\partial^2 p}{\partial t^2} + \frac{1}{r} \frac{\partial p}{\partial r} \right]\end{array}$

(2)	$\begin{array}{l}\displaystyle p(t=0,r) = p_i\end{array}$

(3)	$\begin{array}{l}\displaystyle p(t, r=\infty) = p_i\end{array}$

(4)	$\begin{array}{l}\displaystyle \left[ r \frac{\partial p}{\partial r} \right]_{r=0} = - \frac{q_t}{2 \pi \sigma}\end{array}$

(5)	$\begin{array}{l}\displaystyle p(t,r) = p_i - \frac{q_t}{4 \pi \sigma} {\rm Ei} \left(-\frac{r^2}{4 \chi t} \right)\end{array}$

(6)	$\begin{array}{l}\displaystyle p(t,r) = p_i - \frac{q_t}{4 \pi \sigma} \left[ \gamma + \ln \left(\frac{r^2}{4 \chi t} \right) \right] = p_i - \frac{q_t}{4 \pi \sigma} \ln \left(\frac{2.24585 \, t}{r^2} \right)\end{array}$