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

\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)

{\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})

\int_{\Gamma} \, {\bf u} \,  d {\bf \Sigma} = q_\Gamma(t)

where

	reservoir pressure		time
	fluid density		position vector
	effective porosity		position vector of the -th source
	total compressibility		Dirac delta function
	phase mobility		gradient operator
	formation permeability to a given fluid		gravity vector
	dynamic viscosity of a given  fluid		fluid velocity under Darcy flow
	sandface flowrates of the -th source		reservoir boundary
	flow through the reservoir boundary , which is  aquifer or gas cap

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

\phi \cdot c_t \cdot \partial_t p + \nabla  {\bf u}  
+ c \cdot ( {\bf u} \, \nabla p)
= 0

{\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})

\int_{\Sigma_k} \, {\bf u} \,  d {\bf \Sigma} = q_k(t)

\int_{\Gamma} \, {\bf u} \,  d {\bf \Sigma} = q_\Gamma(t)

where

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.

See also

Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model