We start with
outside wellbore:
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where
\Sigma_k | well-reservoir contact of the k-th well |
d {\bf \Sigma} | normal vector of differential area on the well-reservoir contact, pointing inside wellbore |
and use
to arrive at:
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where
Let's assume Darcy flow with constant permeability
\displaystyle \frac{dk}{dp} = 0 and ignore gravity forces:
(5) | {\bf u} = \frac{k}{\mu} \nabla \, p |
so that diffusion equation becomes:
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Let's express the density via Z-factor:
(8) | \rho = \frac{M}{RT} \, \frac{p}{Z(p)} |
where
T | fluid temperature |
M | molar mass of a fluid |
R | gas constant |
and assuming the fluid temperature T does not change over time and space during the modelling period:
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or
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where
\displaystyle \Psi(p) =2 \, \int_0^p \frac{p \, dp}{\mu(p) \, Z(p)} | Pseudo-Pressure |
In some practical cases the complex
c_t \, \mu can be considered as constant in time which makes
(11) a linear differential equation.
But during the early transition times the pressure drop is usually high and the complex c_t \, \mu 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 p_{BHP}(t):
(13) | \tau(t) = \int_0^t \frac{dt}{\mu(p_{BHP}(t)) \, c_t(p_{BHP}(t))} |
to correct early-time transient behaviour.
See also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Pseudo-linear pressure diffusion @model