The general form of non-linear single-phase pressure diffusion @model with the finite number of sources/sinks is given by:
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where
p(t, {\bf r}) | reservoir pressure | t | time |
\rho({\bf r},p) | fluid density | {\bf r } | position vector |
\phi({\bf r}, p) | effective porosity | {\bf r }_k | position vector of the k-th source |
c_t({\bf r},p) | total compressibility | \delta ( \bf r ) | Dirac delta function |
\nabla | gradient operator | ||
formation permeability to a given fluid | { \bf g } | gravity vector | |
dynamic viscosity of a given fluid | { \bf u } | fluid velocity under Darcy flow | |
sandface flowrates of the k-th source | \Gamma | reservoir boundary | |
q_\Gamma(t) | flow through the reservoir boundary \Gamma, which is aquifer or gas cap |
The alternative form is to write down equations (1) and (2) in reservoir volume outside wellbore and match the solution to the fluid flux through the well-reservoir contact:
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where
\Sigma_k | well-reservoir contact of the k-th well |
\Gamma | |
d {\bf \Sigma} | normal vector of differential area on the well-reservoir contact, pointing inside wellbore |
q_k(t) | sandface flowrates at the k-th well (could be injecting to or producing from the reservoir ) |
q_\Gamma(t) | flow through the reservoir boundary \Gamma, which is aquifer or gas cap |
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.
Wellbore storage model | Well model | Reservoir model | Boundary model |
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Constant | Skin-factor | Homogeneous | Infinite |
Fair | Vertical well | Dual-porosity | Circle No Flow |
Rate-dependant | Dual-permeability | Circle Constant Pi | |
Limited entry well | Anisotropic reservoir | Single fault | |
Horizontal well | Multi-layer reservoir | Parallel faults | |
Slanted well | Linear-composite | Intersecting Faults | |
Radial-composite |
See also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model