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})

where

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 A} = q_k(t)

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