The general form of non-linear single-phase pressure diffusion @model with the finite number of wells 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})

{\rm F}_{\Gamma}(p, {\bf u}) = 0

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 well		reservoir boundary
	reservoir boundary flow condition through the reservoir boundary , which is usually the aquifer or gas cap

Derivation of Single-phase 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}  
+ 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)

{\rm F}_{\Gamma}(p, {\bf u}) = 0

where

	well-reservoir contact of the -th well
	normal vector of differential area on the well-reservoir contact, pointing inside wellbore
	sandface flowrates at the -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).

See also