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The general form of non-linear single-phase pressure diffusion @model with the finite number of wells is given by: 

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anchorPZ

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\phi \cdot c_t \cdot \partial_t p

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+

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\nabla  {\

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bf u}  
+ c \cdot ( {\bf u} \, \nabla p)
= \sum_k q

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_k(t) \cdot \delta({\bf r}-{\bf r}_k)

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anchor
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uu
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{\
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bf 
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u} = - M \cdot 
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(
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 \nabla 
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p 
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- \
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rho \, {\bf g}
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)
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anchorqGamma
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{\rm F}_{\Gamma}(p, {\bf 
...
u}
...
) = 0
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where
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Table. 1. Notations and Definitions
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body--uriencoded--p(t, %7B\bf r%7D)
reservoir pressure
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bodyt
time
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body--uriencoded--\
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rho(%7B\bf r%7D,p)
fluid density 
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body
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объем элементарной ячейки
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--uriencoded--%7B\bf r %7D
position vector
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body
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--uriencoded--\phi(%7B\bf r%7D, p)
effective porosity 
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body
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--uriencoded--%7B\bf r %7D_k
position vector of the 
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body
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k
-th source
...
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body
...
--uriencoded--c_t(%7B\bf r%7D,p)
total compressibility 
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body
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\delta ( \bf r )
Dirac delta function
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bodyM = k / \mu
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phase mobility
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body\
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nabla
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gradient operator
...
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body
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k
formation permeability to a given fluid
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body--uriencoded--
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%7B \bf g %7D
gravity vector
...
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body
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\mu
dynamic viscosity of a given  fluid
...
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body--uriencoded--
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%7B \bf u %7D
fluid velocity under Darcy flow 
...
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body
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q_k(t)
sandface flowrates of the 
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body
...
k
-th well
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body
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\Gamma
reservoir boundary
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body--uriencoded--
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%7B\rm F%7D_%7B\Gamma%7D(p, %7B\bf u%7D)
reservoir boundary flow condition through the reservoir boundary 
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body\
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Gamma
, which is usually the aquifer or gas cap


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titleDerivation
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Derivation of Single-phase pressure diffusion @model


The alternative form is to write down equations 
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 and 
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anchoruu
 in reservoir volume outside wellbore and match the solution to the fluid flux through the well-reservoir contact
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body--uriencoded--\%7B (\Sigma_x, \, \Sigma_%7Bx+\delta x%7D), \, (\Sigma_y, \, \Sigma_%7By+\delta y%7D), \, (\Sigma_z, \, \Sigma_%7Bz+\delta z%7D), \%7D
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:
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anchor
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PZ1
alignmentleft
\
...
phi 
...
\
...
cdot c_t \
...
cdot 
...
\
...
anchor1
alignmentleft
...
partial_t p + \nabla  {\bf u}  
+ c \cdot ( {\bf u} \, \nabla p)
= 0
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anchor
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uu1
alignmentleft
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{
...
\
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bf 
...
u}
...
 = 
...
Let fluid density be 
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body\rho(x,y,z)
 being distributed through porous media with porosity 
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body\phi(x,y,z)
.
This elementary volume 
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body\delta V
 is holding the mass: 
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body\delta m = \rho(x,y,z) \cdot \delta V
.
...
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body\Sigma
...
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body\delta A
...
- M \cdot ( \nabla p - \rho \, {\bf g})
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anchor
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qk
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\
...
int_{\Sigma_k} \, {\bf 
...
u} \,  d {\bf \
...
where
...
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body--uriencoded--%7B\bf \delta A%7D = \delta A \cdot %7B\bf n%7D
...
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body--uriencoded--%7B\bf n%7D
...
normal vector to  elementary area  
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body\delta A
Sigma} = q_k(t)
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anchorqGamma
...
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body--uriencoded--\displaystyle \frac%7Bdm%7D%7Bdt%7D \Big%7C_%7B\delta A%7D = %7B\bf j%7D \, %7B\bf \delta A%7D 
The total mass balance of the elementary volume 
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body\delta V
 is going to be:
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alignmentleft
...
{
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\rm F}_{\
...
Gamma}(p, 
...
{\
...
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body--uriencoded--\displaystyle \frac%7Bdm%7D%7Bdt%7D = \sum_%7B\alpha%7D j_%7B\alpha%7DA_%7B\alpha%7D = j_x%7C_%7Bx%7D\cdot A_%7Byz%7D - j_x%7C_%7Bx+\delta x%7D\cdot A_%7Byz%7D + ...
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Consider the mass flow rate balance  along the 
Рассмотрим приращение массы в элементарном кубе объема 
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body\delta V
. Предполагаем, что в самой ячейке нет источников, знак минус появляется за счет того, что нормали к противоположным граням кубика противонаправлены.
...
bf u}) = 0
where
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body\Sigma_k
well-reservoir contact of the 
 LaTeX Math Inline
bodyk
-th well
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body--uriencoded--
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d %7B\
...
bf \Sigma%7D
normal vector of differential area on the well-reservoir contact, pointing inside wellbore
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bodyq_k(t)
sandface flowrates at the 
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bodyk
-th well (could be injecting to or producing from the reservoir )

...
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body--uriencoded--\displaystyle \frac%7B\partial%7D%7B\partial t%7D(\rho \Phi) = - \nabla \cdot \vec j
...
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body--uriencoded--\displaystyle \frac%7B\partial%7D%7B\partial t%7D(\rho \Phi) + \nabla \cdot \vec j = 0
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body--uriencoded--\displaystyle \frac%7B\partial%7D%7B\partial t%7D(\rho \Phi) + \nabla \cdot (\rho \vec u) = 0
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Вспоминаем определение (4) поля 
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body--uriencoded--\vec%7Bj%7D
 
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body--uriencoded--\displaystyle \vec u = -\frac%7Bk%7D%7B\mu%7D \vec \nabla p
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body--uriencoded--\displaystyle \frac%7B\partial%7D%7B\partial t%7D(\rho \Phi) - \nabla \cdot \left( \rho \frac%7Bk%7D%7B\mu%7D \vec \nabla p \right) = 0
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Здесь и далее работаем в приближении
  1. процесс изотермический
  2. плотность флюида и пористость породы не зависят от времени явно
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body--uriencoded--\displaystyle \frac%7B\partial%7D%7B\partial t%7D (\rho \Phi) = \frac%7B\partial%7D%7B\partial p%7D (\rho \Phi)_T \frac%7B\partial p%7D%7B\partial t%7D = (\dot %7B\Phi%7D \rho + \dot %7B\rho%7D \Phi)\frac%7B\partial p%7D%7B\partial t%7D = \rho \Phi (c_%7Br%7D + c_%7Bf%7D)\frac%7B\partial p%7D%7B\partial t%7D
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Распишем временную производную в ур-нии (7)
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body--uriencoded--\displaystyle \nabla \cdot \left( \rho \frac%7Bk%7D%7B\mu%7D \vec%7B\nabla%7D p \right) =\frac%7Bk%7D%7B\mu%7D \vec%7B\nabla%7D\rho \cdot \vec%7B\nabla%7Dp + \rho \cdot \nabla \cdot \left( \frac%7Bk%7D%7B\mu%7D \vec%7B\nabla%7Dp \right)
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body--uriencoded--\displaystyle \nabla \cdot \left(\rho \frac%7Bk%7D%7B\mu%7D \vec%7B\nabla%7D p \right) =\dot%7B\rho%7D\frac%7Bk%7D%7B\mu%7D (\vec%7B\nabla%7Dp)%5e2 + \rho \cdot \nabla \cdot \left( \frac%7Bk%7D%7B\mu%7D \vec%7B\nabla%7Dp \right)
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body--uriencoded--\displaystyle \rho \Phi c_%7Bt%7D \frac%7B\partial p%7D%7B\partial t%7D =\rho \left(\nabla \cdot \left( \frac%7Bk%7D%7B\mu%7D \vec%7B\nabla%7Dp \right) + c_%7Bf%7D\frac%7Bk%7D%7B\mu%7D (\vec%7B\nabla%7Dp)%5e2 \right)
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Перепишем ур-ние (7), используя конечные соотношения в (10) и (8), и определения для 
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body--uriencoded--c_%7Bt%7D
 (6) и 
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body--uriencoded--c_%7Bf%7D
 (7)
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body--uriencoded--\displaystyle \Phi(p) c_%7Bt%7D(p) \frac%7B\partial p%7D%7B\partial t%7D =\nabla \cdot \left( \frac%7Bk(p)%7D%7B\mu (p)%7D \vec%7B\nabla%7Dp \right) + c_%7Bf%7D(p)\frac%7Bk(p)%7D%7B\mu (p)%7D (\vec%7B\nabla%7Dp)%5e2
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Классическая запись уравнения диффузии в приближении изотермического процесса и независимости от времени плотности флюида и пористости породы.
Правая часть уравнения представляет собой сумму двух частей. Первая отвечает за пространственное распределение давления, вторая же содержит множителем сжимаемость флюида.
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-linearLinear and Non-linear. 
These models are built using Numerical, Analytical or Hybrid pressure diffusion solvers.
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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.
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Fractured vertical well
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See also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion
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Multifrac horizontal well
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See also
Pressure diffusion / Pressure Diffusion @model
[ Aquifer Drive Models ] [ Gas Cap Drive Models ]
[ Linear single-phase pressure diffusion @model ]