...
We start with reservoir pressure diffusion outside wellbore:
| LaTeX Math Block |
|---|
| anchor | rho_dif |
|---|
| alignment | left |
|---|
| \frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \sum_k m_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
| |
qkint{} \, \rho) \ \, d {\bf A}m_{\Gamma}(t)where
| |
| LaTeX Math Inline |
|---|
| body | --uriencoded--d %7B\bf \Sigma%7D |
|---|
|
| normal vector of differential area on the well-reservoir contact, pointing inside wellbore |
| mass rate at -th well | LaTeX Math Inline |
|---|
| body | m_k(t) = \rho(p) \cdot q_k(t) |
|---|
|
|
| |
| |
...
| LaTeX Math Block |
|---|
| anchor | rhophi |
|---|
| alignment | left |
|---|
|
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:
| LaTeX Math Block |
|---|
| anchor | pre_filnal |
|---|
| alignment | left |
|---|
| \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \rho \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
| |
qkint{} \ \, d {\bf A} = q_{\Gamma}(t)where
...
| LaTeX Math Block |
|---|
| anchor | pre_filnal |
|---|
| alignment | left |
|---|
| \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla {\bf u}
+ c \cdot ( {\bf u} \, \nabla p) = \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
| | LaTeX Math Block |
|---|
| anchor | qkqGamma |
|---|
| alignment | left |
|---|
| {\intrm F}_{{\Gamma}} \(p, {\bf u} \, d {\bf A} = q_{\Gamma}(t)) = 0 |
|
See also
...
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Single-phase pressure diffusion @model
...
