The general form of non-linear single-phase pressure diffusion @model with the finite number of wells is given by:

(1)	$\begin{array}{l}\displaystyle \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u} + c \cdot ( {\bf u} \, \nabla p) = \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)\end{array}$

(2)	$\begin{array}{l}\displaystyle {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})\end{array}$

(3)	$\begin{array}{l}\displaystyle {\rm F}_{\Gamma}(p, {\bf u}) = 0\end{array}$

where

$\begin{array}{l}p(t, {\bf r})\end{array}$	reservoir pressure	$\begin{array}{l}t\end{array}$	time
$\begin{array}{l}\rho({\bf r},p)\end{array}$	fluid density	$\begin{array}{l}{\bf r }\end{array}$	position vector
$\begin{array}{l}\phi({\bf r}, p)\end{array}$	effective porosity	$\begin{array}{l}{\bf r }_k\end{array}$	position vector of the $\begin{array}{l}k\end{array}$ -th source
$\begin{array}{l}c_t({\bf r},p)\end{array}$	total compressibility	$\begin{array}{l}\delta ( \bf r )\end{array}$	Dirac delta function
$\begin{array}{l}M = k / \mu\end{array}$	phase mobility	$\begin{array}{l}\nabla\end{array}$	gradient operator
$\begin{array}{l}k\end{array}$	formation permeability to a given fluid	$\begin{array}{l}{ \bf g }\end{array}$	gravity vector
$\begin{array}{l}\mu\end{array}$	dynamic viscosity of a given fluid	$\begin{array}{l}{ \bf u }\end{array}$	fluid velocity under Darcy flow
$\begin{array}{l}q_k(t)\end{array}$	sandface flowrates of the $\begin{array}{l}k\end{array}$ -th well	$\begin{array}{l}\Gamma\end{array}$	reservoir boundary
$\begin{array}{l}{\rm F}_{\Gamma}(p, {\bf u})\end{array}$	reservoir boundary flow condition through the reservoir boundary $\begin{array}{l}\Gamma\end{array}$ , which is usually the aquifer or gas cap

Derivation

Derivation of Single-phase pressure diffusion @model

The alternative form is to write down equations (1) and (2) in reservoir volume outside wellbore and match the solution to the fluid flux through the well-reservoir contact:

(4)	$\begin{array}{l}\displaystyle \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u} + c \cdot ( {\bf u} \, \nabla p) = 0\end{array}$

(5)	$\begin{array}{l}\displaystyle {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})\end{array}$

(6)	$\begin{array}{l}\displaystyle \int_{\Sigma_k} \, {\bf u} \, d {\bf \Sigma} = q_k(t)\end{array}$

(7)	$\begin{array}{l}\displaystyle {\rm F}_{\Gamma}(p, {\bf u}) = 0\end{array}$

where

$\begin{array}{l}\Sigma_k\end{array}$	well-reservoir contact of the $\begin{array}{l}k\end{array}$ -th well
$\begin{array}{l}d {\bf \Sigma}\end{array}$	normal vector of differential area on the well-reservoir contact, pointing inside wellbore
$\begin{array}{l}q_k(t)\end{array}$	sandface flowrates at the $\begin{array}{l}k\end{array}$ -th well (could be injecting to or producing from the reservoir )

Physical models of pressure diffusion can be split into two categories: Newtonian and Rheological (non-Newtonian) based on the fluid stress model.

Mathematical models of pressure diffusion can be split into three categories: Linear, Pseudo-Linear and Non-linear.

These models are built using Numerical, Analytical or Hybrid pressure diffusion solvers.

Many popular 1DR solutions can be approximated by Radial Flow Pressure Diffusion @model which has a big methodological value.

The simplest analytical solutions for pressure diffusion are given by 1DL Linear-Drive Solution (LDS) and 1DR Line Source Solution (LSS).

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See also