The general form of non-linear single-phase pressure diffusion model is given by:

\beta({\bf r},p) \, \frac{\partial p}{\partial t} = \nabla \Big( M({\bf r},p, \nabla p) \cdot \nabla p \Big)

with non-linear dependence of fluid mobility on reservoir pressure and spatial pressure gradient :

M = k_a({\bf r}) \, M_r(p, \nabla p)

\beta = c_t({\bf r},p) \cdot \phi({\bf r},p)

c_t({\bf r},p)  = c_r({\bf r},p) + \sum_\alpha s_\alpha({\bf r}) c_\alpha(p)

where

	Fluid mobility as function of reservoir pressure and spatial pressure gradient
	Relative mobility as function of reservoir pressure and spatial pressure gradient
	Compressivity as function of reservoir pressure
	Total compressibility as function of reservoir pressure and location
	Rock compressibility as function of reservoir pressure and location
	-phase compressibility as function of reservoir pressure for
	-phase reservoir saturation for
	Effective porosity as function of reservoir pressure and location
	Absolute formation permeability at initial formation pressure as function of location
	Dynamic fluid viscosity at initial formation pressure
	Some function of reservoir pressure and spatial pressure gradient with the following asymptotic behaviour:

The same account for non-linearity can be applied for non-linear multi-phase pressure diffusion when Pressure Diffusion Model Validity Scope is met and multi-phase pressure dynamics can be modeled as effective single-phase pressure dynamics.

Below is the list of popular physical phenomena and their mathematical models which can be covered by model.

Compressible fluids

Pressure diffusion equation is going to be:

c_t(p) \, \phi({\bf r})  \, \frac{\partial p}{\partial t} = \nabla (M(p)  \nabla p )

where

c_t({\bf r},p)  = c_r({\bf r}) + \sum_\alpha s_\alpha({\bf r}) c_\alpha(p)

Rock compressibility as function of location

Formation permeability as function of location

See also