We start with reservoir pressure diffusion outside wellbore:
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| anchor | rho_dif |
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| \frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0 |
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| \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t) |
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where
Table. 1. Notations and Definitions
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| body | --uriencoded--\delta A_%7Byz%7D=\delta y\delta z |
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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 | --uriencoded--\vec%7Bj%7D = \rho \vec%7Bu%7D |
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, | LaTeX Math Inline |
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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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| body | --uriencoded--\displaystyle c_%7Bf%7D = \frac%7B1%7D%7B\rho%7D \frac%7B\partial \rho%7D%7B\partial p%7D |
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| body | --uriencoded--c_%7Bt%7D = c_%7Br%7D + c_%7Bf%7D |
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Then use the following equality:
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d(\rho \, \phi) = \rho \, d \phi + \phi \, d\rho = \rho \, \phi \, \left( \frac{d \phi }{\phi} + \frac{d \rho }{\rho} \right)
= \rho \, \phi \, \left( \frac{1}{\phi} \frac{d \phi}{dp} \, dp + \frac{1}{\rho} \frac{d \rho}{dp} \, dp \right)
= \rho \, \phi \, (c_{\phi} \, dp + c \, dp) = \rho \, \phi \, c_t \, dp |
to arrive at:
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| \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0 |
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| \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t) |
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where
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| body | --uriencoded--\delta \Omega = \%7B (x, x+\delta), (y, y+\delta y), (z, z+\delta z) \%7D \in ℝ%5e3 |
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| body | \delta V = \delta x \, \delta y \, \delta z |
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| body | --uriencoded--\%7B (\delta \Sigma_x, \, \delta \Sigma_%7Bx+\delta x%7D), \, (\delta \Sigma_y, \, \delta \Sigma_%7By+\delta y%7D), \, (\delta \Sigma_z, \, \delta \Sigma_%7Bz+\delta z%7D) \%7D |
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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 is filled with porous media with porosity saturated by fluid with density .
The pore volume is going to be | LaTeX Math Inline |
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| body | --uriencoded--\delta V_%7B\phi%7D = \phi \cdot \delta V |
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and the fluid mass contained in this volume is | LaTeX Math Inline |
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| body | --uriencoded--\delta m = \rho \cdot \delta V_%7B\phi%7D = \rho \cdot \phi \cdot \delta V |
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The mass flowrate through any face
with area is defined as:| LaTeX Math Block |
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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 |
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Let's assume Darcy flow with constant permeability
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| \displaystyle \frac%7Bdk%7D%7Bdp%7D = 0 |
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and ignore gravity forces:| LaTeX Math Block |
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{\bf u} = \frac{k}{\mu} \nabla \, p |
so that diffusion equation becomes
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normal vector to elementary area
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| body | --uriencoded--%7B\bf j%7D = \rho \cdot %7B\bf u%7D |
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| body | --uriencoded--%7B\bf u%7D |
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The total mass balance of the volumetric element
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| \rho \, \phi \, c_t \cdot \frac{ |
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\partial t} + \nabla \, ( k \cdot \frac{\rho}{\mu} \, \nabla \, p) = 0 |
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\, p \cdot d {\bf A} = q_k(t) |
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Let's express the density via Z-factor:
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\rho = \frac{M}{RT} \, \frac{p}{Z(p)}\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
| \delta \dot m_q | the rate of the mass variation which happens inside the volumetric element \delta \Omega | and assuming the fluid temperature Dividing the
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by the volume does not change over time and space during the modelling period:
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c_t \, \mu \cdot \frac{p}{\mu \, Z} \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{p}{\mu \, Z} \, \nabla \, p) = 0 |
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or in differential form:
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or
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The mass rate generated/consumed inside the volumetric element by a finite number of sources can be expressed as:
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which turns
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into:t} + \nabla \, ( k \cdot \ |
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where
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| body | --uriencoded--q(%7B\bf r%7D) |
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\nabla } \, p \cdot d {\bf |
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where
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| body | --uriencoded--\displaystyle \ |
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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 + ...Consider the mass flow rate balance along the Рассмотрим приращение массы в элементарном кубе объема . Предполагаем, что в самой ячейке нет источников, знак минус появляется за счет того, что нормали к противоположным граням кубика противонаправлены. | | 4 | | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \frac %7B\partial%7D%7B\partial t%7D%7B\rho \Phi%7D = \frac%7Bj_x%7C_x - j_x%7C_%7Bx+\delta x%7D%7D%7B\delta x%7D + \frac%7Bj_y%7C_y - j_y%7C_%7By+\delta y%7D%7D%7B\delta y%7D + \frac%7Bj_z%7C_z - j_z%7C_%7Bz+\delta z%7D%7D%7B\delta z%7D |
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| Разделим ур-ние (1) на объем ячейки |
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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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| Ур-ние (2) есть развернутая форма записи ур-ния (3) |
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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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| Классическое уравнение непрерывности |
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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 | LaTeX Math Inline |
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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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| Феноменологический закон Дарси, связывающий скорость потока с градиентом давления |
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In some practical cases the complex
can be considered as constant in time which makes | LaTeX Math Block Reference |
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a linear differential equation.But during the early transition times the pressure drop is usually high and the complex
can not be considered as constant in time which leads to distortion of pressure transient diagnostics at early times.In this case one can use Pseudo-Time, calculated by means of the bottom-hole pressure
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Здесь и далее работаем в приближении
- процесс изотермический
- 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 |
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(6) и | LaTeX Math Inline |
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| body | --uriencoded--c_%7Bf%7D |
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(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.
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\tau(t) = \int_0^t \frac{dt}{\mu (p_{BHP} ) \, c_t (p_{BHP}) } \, , \ \ p_{BHP} = p_{BHP}(t) |
to correct early-time transient behaviour which turn equation
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into:| LaTeX Math Block |
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\phi \cdot \frac{\partial \Psi}{\partial \tau} + \nabla \, ( k \cdot \nabla \, \Psi) = 0 |
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See also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / SinglePseudo-phase linear pressure diffusion @model
