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)

The table below shows a list of popular well and reservoir pressure diffusion models which in some cases maybe simulated with analytical methods to speed up the calculations.

Wellbore storage model	Well model	Reservoir model	Boundary model
Constant	Skin-factor	Homogeneous	Unconstrained
Fair	Vertical well	Dual-porosity		Infinite
Rate-dependant	Fractured vertical well	Dual-permeability	Fully constrained
	Limited entry well	Anisotropic reservoir		Circle No Flow / Circle Constant P_i
	Horizontal well	Multi-layer reservoir		Rectangular No Flow / Rectangular Constant P_i
	Slanted well	Linear-composite	Partially constrained
	Multifrac horizontal well	Radial-composite		Single fault
				Parallel faults
				Intersecting Faults

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See also