Consider the Cartesian coordinates in 3D space:
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| body | --uriencoded--ℝ%5e3 \Big %7C_%7B \%7Bx, y, z \%7D %7D |
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and its infinitesimal volumetric element: | LaTeX Math Inline |
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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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with volume | LaTeX Math Inline |
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| body | \delta V = \delta x \, \delta y \, \delta z |
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bounded by six faces: | LaTeX Math Inline |
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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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which have the same area along corresponding axisWe start with the general from of | LaTeX Math Block Reference |
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| page | Single-phase pressure diffusion @model |
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partial_t p + \nabla {\bf u}
+ c \cdot ( {\bf u} \, \nabla p)
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\cdot ( \nabla p - \rho \, {\bf g}) |
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int_{\Gamma} \, {\bf u} \, d {\bf \Sigma} = q_\Gamma(t) |
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where
Consider the volumetric element is filled with porous media with porosity saturated by fluid with density .
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\frac{dm}{dt} \Big|_{\delta \Sigma} = {\bf j} \, {\bf \delta A} |
where
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\delta A%7D = \delta A \cdot %7B\bf n%7Dvector area n%7Dnormal vector to elementary area delta A j%7D = \rho \cdot %7B\bf u%7Dmass flux vector u%7Dfluid flow velocity | ...
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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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the rate of the mass variation which happens inside the volumetric element
Dividing the
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by the volume :| LaTeX Math Block |
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\frac{dm}{dt \, \delta V} \Big|_{\delta \Omega} = \frac {\partial (\rho \, \phi)}{\partial t} = \frac{j_x|_x - j_x|_{x+\delta x}}{\delta x} + \frac{j_y|_y - j_y|_{y+\delta y}}{\delta y} + \frac{j_z|_z - j_z|_{z+\delta z}}{\delta z} + \frac{\delta \dot m_q}{\delta V} |
or in differential form:
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\frac{\partial (\rho \phi)}{\partial t} = - \nabla \, {\bf j} + \frac{\delta \dot m_q}{\delta V} |
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\frac{\partial (\rho \phi)}{\partial t} + \nabla \, {\bf j} = \frac{\delta \dot m_q}{\delta V} |
The mass rate generated/consumed by a finite number of well-reservoir contacts can be expressed as:
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\frac{\delta \dot m_q}{\delta V} = \sum_k \rho_k \cdot q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
where
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volumetric flowrate of the source/stock at the reservoir point
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| body | --uriencoded--%7B\bf r%7D_k |
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| body | --uriencoded--\rho_k (t) = \rho(p(t, %7B\rm r%7D_k)) |
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fluid density at the reservoir point
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Let's neglect the non-linear term | LaTeX Math Inline |
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| body | --uriencoded--c \cdot ( %7B\bf u%7D \, \nabla p) |
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for low compressibility fluid which is equivalent to assumption of nearly constant fluid density: | LaTeX Math Inline |
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| body | \rho(p) = \rho = \rm const |
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Together with constant pore compressibility this will lead to constant total compressibility | LaTeX Math Inline |
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| body | c_t = c_\phi + c \approx \rm const |
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Assuming that permeability and fluid viscosity do not depend on pressure and | LaTeX Math Inline |
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| body | \mu(p) = \mu = \rm const |
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one arrives to the differential equation with constant coefficients:
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| \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u}
= \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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| body | --uriencoded--%7B\bf j%7D = \rho \cdot %7B\bf u%7D |
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and
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| \int_{\Sigma_k} \, {\bf u} |
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where
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| body | --uriencoded--\displaystyle c_r = \frac%7B1%7D%7B\phi%7D \, \frac%7B\partial \phi%7D%7B\partial p%7D |
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| body | --uriencoded--\displaystyle c = \frac%7B1%7D%7B\rho%7D \, \frac%7B\partial \rho%7D%7B\partial p%7D |
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Substituting the
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and | LaTeX Math Block Reference |
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in | LaTeX Math Block Reference |
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and cancelling the fluid density one arrives to:...
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See also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Single-phase pressure diffusion @model
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