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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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\phi \cdot c_t \cdot \partial_t p

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+

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\nabla  {\

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bf

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u}

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+ 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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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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{\rm F}_{\Gamma}(p, {\bf 
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u}
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)
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titleDerivation
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Table. 1. Notations and Definitions
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 = 0
where
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body--uriencoded--
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body\delta V = \delta x \delta y \delta z
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объем элементарной ячейки
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body\rho
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body\phi
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bodyp
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bodym
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bodyk
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body\mu
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body\sigma
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p(t, %7B\bf r%7D)
reservoir pressure
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bodyt
time
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body--uriencoded--\
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body--uriencoded--\displaystyle j_%7B\delta A%7D = \frac%7B\delta m%7D%7B\delta t \delta A%7D
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body--uriencoded--\vec %7Bu%7D
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body--uriencoded--\displaystyle c_%7Br%7D = \frac%7B1%7D%7B\Phi%7D\frac%7B\partial \Phi%7D%7B\partial p%7D
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rho(%7B\bf r%7D,p)
fluid density 
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body--uriencoded--
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%7B\bf r %7D
position vector
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body--uriencoded--
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body--uriencoded--ℝ%5e3 \Big %7C_%7B \%7Bx, y, z \%7D %7D
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\phi(%7B\bf r%7D, p)
effective porosity 
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body--uriencoded--
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%7B
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\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--
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c_t(%7B\bf r%7D,p)
total compressibility 
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body\delta ( \bf r )
Dirac delta function
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bodyM = k / \mu
phase mobility
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body\nabla
gradient operator
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bodyk
formation permeability to a given fluid
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\delta A(\delta \Sigma_x) = \delta A(\Sigma_{x+\delta x }) = \delta A_{yz} = \delta y \cdot \delta z
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\delta A(\delta \Sigma_y) = \delta  A(\Sigma_{y+\delta y }) = \delta A_{xz} = \delta x \cdot \delta z
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\delta A(\delta \Sigma_z) = \delta A(\Sigma_{z+\delta z }) = \delta A_{xy} = \delta x \cdot \delta y
Consider the volumetric element 
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body\delta \Omega
 is filled with porous media with porosity 
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body\phi(x,y,z)
 saturated by fluid with density 
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body\rho(x,y,z)
.
...
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body--uriencoded--
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%7B \
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bf g %7D
gravity vector
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body\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
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\frac{dm}{dt} \Big|_{\delta \Sigma} = {\bf j} \, {\bf \delta A} 
where
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body--uriencoded--%7B\bf \delta A%7D = \delta A \cdot %7B\bf n%7D
k
-th well
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body
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\Gamma
reservoir boundary
...
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body--uriencoded--%7B\
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rm F%7D_%7B\Gamma%7D(p, %7B\bf u%7D)
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reservoir boundary flow condition through the reservoir boundary 
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body\
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anchormdot
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\frac{dm}{dt} \Big|_{\delta \Omega} =  \sum_{\alpha} j_{\alpha}A_{\alpha} + \delta \dot m_q
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\frac{dm}{dt} \Big|_{\delta \Omega} =  
j_x|_{x}\cdot \delta A_{yz} - j_x|_{x+\delta x}\cdot \delta A_{yz} +j_y|_{y}\cdot \delta A_{xz} - j_y|_{y+\delta y}\cdot \delta A_{xz} +
j_z|_{z}\cdot \delta A_{xy} - j_z|_{z+\delta z}\cdot \delta A_{xy}  + \delta \dot m_q
where
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body\delta \dot m_q
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the rate of the mass variation which happens inside the volumetric element 
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body\delta \Omega
   
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


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The alternative form is to write down equations 
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 and 
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body\delta V
uu
 in reservoir volume outside wellbore and match the solution to the fluid flux through the well-reservoir contact:
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\
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phi \
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or in differential form:
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\frac{\partial (\rho \phi)}{\partial t} = - \nabla \cdot {\bf j} + \frac{\delta \dot m_q}{\delta V}
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\frac{\partial (\rho \phi)}{\partial t} + \nabla \cdot {\bf j} =  \frac{\delta \dot m_q}{\delta V} 
The mass rate generated/consumed inside the volumetric element 
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body\delta \Omega
 by a finite number of sources can be expressed as:
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\frac{\delta \dot m_q}{\delta V} = \sum_k q({\bf r}) \cdot \delta({\bf r}-{\bf r}_k)
where
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body--uriencoded--q(%7B\bf r%7D)
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cdot c_t \cdot \partial_t p + \nabla  {\bf u}  
+ c \cdot ( {\bf u} \, \nabla p)
= 0
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anchoruu1
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{\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})
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anchorqk
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\int_{\Sigma_k} \, {\bf u} \,  d {\bf \Sigma} = q_k(t)
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anchorqGamma
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{\rm F}_{\Gamma}(p, {\bf u}) = 0
where
which turns 
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anchorprelast
 into:
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\frac{\partial (\rho \phi)}{\partial t} + \nabla \cdot ({\bf j}) = \sum_k q({\bf r}) \cdot \delta({\bf r}-{\bf r}_k)
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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.
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body\Sigma_k
well-reservoir contact of the 
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bodyk
-th well
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body--uriencoded--
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d %7B\
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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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Multifrac horizontal well
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See also
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Pressure diffusionPhysics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model
[ Aquifer Drive Models ] [ Gas Cap Drive Models ]
[ Linear single-phase pressure diffusion @model ]