Page tree

We start with the general from of (Single-phase pressure diffusion @model:1):

(1) \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)
(2) {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})
(3) \int_{\Gamma} \, {\bf u} \, d {\bf \Sigma} = q_\Gamma(t)


p(t, {\bf r})

reservoir pressure



\rho({\bf r})

fluid density 

{\bf r }

position vector

\phi({\bf r})

effective porosity 

{\bf r }_k

position vector of the k-th source

c_t({\bf r})

total compressibility 

\delta ( \bf r )

Dirac delta function

M({\bf r})

reservoir fluid mobility M({\bf r}) = \frac{k({\bf r})}{\mu}


gradient operator

k({\bf r})

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


reservoir boundary


flow through the reservoir boundary \Gamma, which is  aquifer or gas cap

Let's neglect the non-linear term c \cdot ( {\bf u} \, \nabla p) for low compressibility fluid c \sim 0 which is equivalent to assumption of nearly constant fluid density: \rho(p) = \rho = \rm const.

Together with constant pore compressibility c_\phi = \rm const this will lead to constant total compressibility c_t = c_\phi + c \approx \rm const.

Assuming that permeability and fluid viscosity do not depend on pressure k(p) = k = \rm const and \mu(p) = \mu = \rm const one arrives to the differential equation with constant coefficients

(4) \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u} = \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)
(5) {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})
(6) \int_{\Gamma} \, {\bf u} \, d {\bf \Sigma} = q_\Gamma(t)

(7) \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u} = 0
(8) {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})
(9) \int_{\Sigma_k} \, {\bf u} \, d {\bf \Sigma} = q_k(t)
(10) \int_{\Gamma} \, {\bf u} \, d {\bf \Sigma} = q_\Gamma(t)

See also

Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Single-phase pressure diffusion @model

  • No labels