The general form of 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}  
= \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)

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

where

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

Derivation of Linear pressure diffusion @model

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}  
= 0

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

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

where

	well-reservoir contact of the -th well
	normal vector of differential area on the well-reservoir contact, pointing inside wellbore

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.

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

See also