The general form of non-linear single-phase pressure diffusion model is given by:

(1)	$\begin{array}{l}\displaystyle \beta({\bf r},p) \, \frac{\partial p}{\partial t} = \nabla \Big( M({\bf r},p, \nabla p) \cdot \nabla p \Big)\end{array}$

with non-linear dependence of fluid mobility $\begin{array}{l}M\end{array}$ on reservoir pressure $\begin{array}{l}p\end{array}$ and spatial pressure gradient $\begin{array}{l}\nabla p\end{array}$ :

(2)	$\begin{array}{l}\displaystyle M = k_{air}({\bf r}) \, M_r(p, \nabla p)\end{array}$

and non-linear dependence of compressivity $\begin{array}{l}\beta\end{array}$ and compressibility $\begin{array}{l}c_t\end{array}$ on reservoir pressure $\begin{array}{l}p\end{array}$ :

(3)	$\begin{array}{l}\displaystyle \beta = c_t({\bf r},p) \cdot \phi({\bf r},p)\end{array}$

(4)	$\begin{array}{l}\displaystyle c_t({\bf r},p) = c_r({\bf r},p) + \sum_\alpha s_\alpha({\bf r}) c_\alpha(p)\end{array}$

where

$\begin{array}{l}M(p, \nabla p)\end{array}$	Fluid mobility as function of reservoir pressure $\begin{array}{l}p\end{array}$ and spatial pressure gradient $\begin{array}{l}\nabla p\end{array}$
$\begin{array}{l}M_r(p, \nabla p)\end{array}$	Relative mobility as function of reservoir pressure $\begin{array}{l}p\end{array}$ and spatial pressure gradient $\begin{array}{l}\nabla p\end{array}$
$\begin{array}{l}\beta(p)\end{array}$	Compressivity as function of reservoir pressure $\begin{array}{l}p\end{array}$
$\begin{array}{l}c_t({\bf r},p)\end{array}$	Total compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$ and location $\begin{array}{l}\bf r\end{array}$
$\begin{array}{l}c_r({\bf r},p)\end{array}$	Rock compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$ and location $\begin{array}{l}\bf r\end{array}$
$\begin{array}{l}c_\alpha(p)\end{array}$	$\begin{array}{l}\alpha\end{array}$ -phase compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$ for $\begin{array}{l}\alpha = \{ w, \, o, \, g \}\end{array}$
$\begin{array}{l}s_\alpha({\bf r})\end{array}$	$\begin{array}{l}\alpha\end{array}$ -phase reservoir saturation for $\begin{array}{l}\alpha = \{ w, \, o, \, g \}\end{array}$
$\begin{array}{l}\phi_e({\bf r}, p)\end{array}$	Effective porosity as function of reservoir pressure $\begin{array}{l}p\end{array}$ and location $\begin{array}{l}\bf r\end{array}$
$\begin{array}{l}k_{air}({\bf r})\end{array}$	Formation permeability at initial formation pressure $\begin{array}{l}p_0\end{array}$ as function of location $\begin{array}{l}\bf r\end{array}$
$\begin{array}{l}\mu(p_0)\end{array}$	Dynamic fluid viscosity at initial formation pressure $\begin{array}{l}p_0\end{array}$
$\begin{array}{l}\xi (p, \|\nabla p\|)\end{array}$	Some function of reservoir pressure $\begin{array}{l}p\end{array}$ and spatial pressure gradient $\begin{array}{l}\nabla p\end{array}$ with the following asymptotic behaviour: $\begin{array}{l}\xi (p \rightarrow p_0, \|\nabla p\| \rightarrow 0) \rightarrow 1\end{array}$

The same account for non-linearity can be applied for non-linear multi-phase pressure diffusion when Pressure Diffusion Model Validity Scope is met and multi-phase pressure dynamics can be modeled as effective single-phase pressure dynamics.

Below is the list of popular physical phenomena and their mathematical models which can be covered by (1) model.

Compressible fluids

Pressure diffusion equation is going to be:

(5)	$\begin{array}{l}\displaystyle c_t(p) \, \phi({\bf r}) \, \frac{\partial p}{\partial t} = \nabla ( \alpha(p) \nabla p )\end{array}$

where

  
	(6)
	c_t({\bf r},p)  = c_r({\bf r}) + \sum_\alpha s_\alpha({\bf r}) c_\alpha(p)

Total compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$ and location $\begin{array}{l}\bf r\end{array}$

$\begin{array}{l}c_r({\bf r})\end{array}$

Rock compressibility as function of location $\begin{array}{l}\bf r\end{array}$

$\begin{array}{l}c_\alpha(p)\end{array}$

$\begin{array}{l}\alpha\end{array}$ -phase compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$ for $\begin{array}{l}\alpha = \{ w, \, o, \, g \}\end{array}$

$\begin{array}{l}\displaystyle \alpha(p) = \frac{k}{\mu(p)}\end{array}$

Fluid mobility as function of reservoir pressure $\begin{array}{l}p\end{array}$

$\begin{array}{l}k({\bf r})\end{array}$

Formation permeability as function of location $\begin{array}{l}\bf r\end{array}$

$\begin{array}{l}\mu(p)\end{array}$

Dynamic fluid viscosity as function of reservoir pressure $\begin{array}{l}p\end{array}$

$\begin{array}{l}c_t(p)\end{array}$

Total compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$

$\begin{array}{l}B(p)\end{array}$

Formation Volume Factor as function of reservoir pressure $\begin{array}{l}p\end{array}$

Compressible rocks

Pressure diffusion equation is going to be:

(7)	$\begin{array}{l}\displaystyle c_t(p) \phi(p) \frac{\partial p}{\partial t} = \nabla ( \alpha(p) \nabla p)\end{array}$

where

$\begin{array}{l}\phi(p)\end{array}$	Dynamic fluid viscosity as function of reservoir pressure $\begin{array}{l}p\end{array}$
$\begin{array}{l}k(p)\end{array}$	Formation permeability as function of reservoir pressure $\begin{array}{l}p\end{array}$
$\begin{array}{l}c_f(p)\end{array}$	Total compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$

Dependance on pressure gradient

Pressure diffusion equation is going to be:

$\begin{array}{l}\displaystyle \с_t \phi_e \frac{\partial p}{\partial t} = \nabla ( \frac{k(\nabla p)}{\mu} \nabla p)\end{array}$

where

$\begin{array}{l}k(\nabla p)\end{array}$	Dynamic fluid viscosity as function of reservoir pressure $\begin{array}{l}p\end{array}$
$\begin{array}{l}k(p)\end{array}$	Formation permeability as function of reservoir pressure $\begin{array}{l}p\end{array}$
$\begin{array}{l}c_f(p)\end{array}$	Total compressibility as function of reservoir pressure $\begin{array}{l}p\end{array}$

$\begin{array}{l}c_r\end{array}$	formation compressibility
$\begin{array}{l}n_k = \frac{\ln (k/k_0)}{\ln(\phi/\phi_0)}\end{array}$	power degree of permeability-porosity correlation
$\begin{array}{l}k_0\end{array}$	formation permeability at reference pressure $\begin{array}{l}p_0\end{array}$
$\begin{array}{l}\phi_0\end{array}$	formation porosity at reference pressure $\begin{array}{l}p_0\end{array}$
$\begin{array}{l}p_0\end{array}$	reference pressure, usually picked up at initial formation pressure $\begin{array}{l}p_0 = p_i\end{array}$

The substantial reduction of formation pressure leads to shrinking of numerous pore throats and massive reduction in permeability deviating from exponential.

The substantial increase of formation pressure leads to microfracturing and massive increase in permeability deviating from exponential.

Зависимость проницаемости от депрессии

В сильно-сжимаемых коллекторах при больших депрессиях (или репрессиях) наблюдается падение проницаемости

(12)	$\begin{array}{l}\displaystyle k=k(\|\nabla p\|)\end{array}$

Как правило используется экспоненциальный закон

(13)	$\begin{array}{l}\displaystyle k=k_0 \, e^{-\beta_G \|\nabla p\|}\end{array}$

Так как максимальная депрессия формируется в ближайшей зоне пласта вокруг скважины, то именно там и проявляются нелинейные эффекты в первую очередь.

При практическом анализе это выглядит как скин-фактор, зависящий от депрессии (или от дебита) и часто моделируется в первом приближении как пересечение линейной модели с постоянной проницаемостью и динамическим скин-фактором.

Threshold pressure gradient (TPG)

A quantity (usually denoted as $\begin{array}{l}G\end{array}$ ) representing the minimum pressure gradient required to initiate the reservoir flow:

(14)	$\begin{array}{l}\begin{equation} \begin{cases} {\bf u}= - \frac{k}{\mu} ( \nabla p - G \, {\bf e}_{\nabla p} ), & \|\nabla p\| > G, \\ {\bf u}= 0, & \|\nabla p\| \leq G . \end{cases} \end{equation}\end{array}$

where $\begin{array}{l}{\bf e}_{\nabla p} = \frac{\nabla p}{|\nabla p|}\end{array}$ – unit vector along the pressure gradient.

At high flow velocities and pressure gradients the model is reducing to Darcy equation.

This model can be reformulated in terms of non-linear permeability model:

(15)	$\begin{array}{l}\displaystyle {\bf u}= - \frac{k(\|\nabla p\|)}{\mu} \nabla p\end{array}$

where $\begin{array}{l}k(|\nabla p|)\end{array}$ is defined as:

(16)	$\begin{array}{l}\begin{equation} \begin{cases} k(\|\nabla p\|) = k_0 \, ( 1 - \frac{G}{\|\nabla p\|} ), & \|\nabla p\| > G, \\ k(\|\nabla p\|) = 0, & \|\nabla p\| \leq G . \end{cases} \end{equation}\end{array}$

Зависимость проницаемости от скорости потока

При больших линейных скоростях течения сильно-сжимаемого флюида через пористую среду наблюдается отклонение от линейного закона течения Дарси, вызванное дополнительным сопротивлением от турбулентности и которое выражается в виде зависимости проницаемости коллектора от скорости флюида.

(17)	$\begin{array}{l}\displaystyle k=k(\|\bf u\|)\end{array}$

Обычно это явление проявляется только в небольшой окрестности скважины.

Есть разные модели этого явления и наиболее популярной на практике является модель Форхгеймера.

Forchheimer model

@wikipedia

The momentum balance equation relating a pressure gradient $\begin{array}{l}\nabla p\end{array}$ in porous medium with induced fluid flow (percolation) with velocity $\begin{array}{l}\bf u\end{array}$

(18)	$\begin{array}{l}\displaystyle - \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, \| {\bf u} \| \, {\bf u}\end{array}$

where

$\begin{array}{l}k\end{array}$	formation permeability
$\begin{array}{l}\mu\end{array}$	fluid viscosity
$\begin{array}{l}\beta\end{array}$	Forchheimer coefficient

Forchheimer coefficient depends on flow regime and formation permeability as:

(19)	$\begin{array}{l}\displaystyle \beta = \frac{C_E}{\sqrt{k}}\end{array}$

where $\begin{array}{l}C_E\end{array}$ is dimensionless quantity called Ergun constant accounting for inertial (kinetic) effects and depending on flow regime only.

$\begin{array}{l}C_E\end{array}$ is small for slow percolation (thus reducing Forchheimer equation to Darcy equation) and grows quickly with high flow velocities.

Forchheimer equation can be approximated by non-linear permeability model as:

(20)	$\begin{array}{l}\displaystyle {\bf u} = - \frac{k}{\mu} \, k_f \, \nabla p\end{array}$

(21)	$\begin{array}{l}\displaystyle k_f(\|\nabla p\|) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big]\end{array}$

(22)	$\begin{array}{l}\displaystyle w = 4 \, \left(\frac{k}{\mu} \right)^2 \, \beta \, \rho \, \|\nabla p\| \, < \, 1\end{array}$

Reference

Philipp Forchheimer (1886). "Über die Ergiebigkeit von Brunnen-Anlagen und Sickerschlitzen". Z. Architekt. Ing.-Ver. Hannover. 32: 539–563.

Page tree

Compressible fluids

Compressible rocks

Dependance on pressure gradient

See also

Зависимость кинетических коэффициентов от пластового давления

Зависимость проницаемости от депрессии

Threshold pressure gradient (TPG)

Зависимость проницаемости от скорости потока

See also

Reference

Page tree

Non-linear single-phase pressure diffusion for compressible rocks @model

Compressible fluids

Compressible rocks

Dependance on pressure gradient

See also

Зависимость кинетических коэффициентов от пластового давления

Зависимость проницаемости от депрессии

Threshold pressure gradient (TPG)

Зависимость проницаемости от скорости потока

Forchheimer model

See also

Reference