We start with
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| anchor | rho_dif |
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| page | Derivation of Single-phase pressure diffusion @model |
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outside with reservoir pressure diffusion outside wellbore:
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| anchor | rho_dif |
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| alignment | left |
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| \frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = |
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\rho \cdot \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) | | LaTeX Math Block |
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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--d %7B\bf \Sigma%7D |
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| normal vector of differential area on the well-reservoir contact, pointing inside wellbore |
and use 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
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to arrive at:
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| \rho \, \phi \, c_t \cdot \frac{\partial |
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()}{\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
Let's assume Darcy flow and with constant permeability
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| body | --uriencoded--\displaystyle \frac%7Bdk%7D%7Bdp%7D = 0 |
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and ignore gravity forces:
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{\bf u} = \frac{k}{\mu} \nabla \, p |
so that diffusion equation becomes:
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| \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{\rho}{\mu} \, \nabla \, p) = 0 |
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| \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf |
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p\nabla } \, 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)} |
where
and assuming the fluid temperature
does not change over time and space during the modelling period:
rhophi \, 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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| \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf \nabla } \, p \cdot d {\bf A} = q_k(t) |
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or
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| \phi \, c_t \, \mu \cdot \frac{\partial \Psi}{\partial t} + \nabla \, ( k \cdot \nabla \, \Psi) = 0 |
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| \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf \nabla } \, p \cdot d {\bf A} = q_k(t) |
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where
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| body | --uriencoded--\displaystyle \Psi(p) =2 \, \int_0%5ep \frac%7Bp \, dp%7D%7B\mu(p) \, Z(p)%7D |
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| Pseudo-Pressure |
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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| body | --uriencoded--p_%7BBHP%7D(t) |
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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 |
See also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Pseudo-linear pressure diffusion @model
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