Motivation
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Reservoir pressure dynamics away from wellbore and boundaries is representative of two very important complex reservoir properties: transmissibility
and
pressure diffusivity .
These can be roughly estimated with a homogeneous reservoir model where wellbore and boundaries effects can be neglected.
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Physical Model
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Mathematical Model
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| LaTeX Math Block |
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| \frac{\partial p}{\partial t} = \chi \, \left[ \frac{\partial^2 p}{\partial t^2} + \frac{1}{r} \frac{\partial p}{\partial r} \right] |
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| LaTeX Math Block |
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| p(t=0,r) = p_i |
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| LaTeX Math Block |
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| p(t, r=\infty) = p_i |
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| LaTeX Math Block |
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| \left[ r \frac{\partial p}{\partial r} \right]_{r=0} = - \frac{q_t}{2 \pi \sigma} |
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Computational Model
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| LaTeX Math Block |
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| p(t,r) = p_i - \frac{q_t}{4 \pi \sigma} {\rm Ei} \left(-\frac{r^2}{4 \chi t} \right) |
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Approximations
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| Late-time response |
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| LaTeX Math Block |
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| p(t,r) = p_i - \frac{q_t}{4 \pi \sigma} \left[
\gamma + \ln \left(\frac{r^2}{4 \chi t} \right) \right]
= p_i - \frac{q_t}{4 \pi \sigma} \ln \left(\frac{2.24585 \, t}{r^2} \right)
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See also
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Physics / Fluid Dynamics / Radial fluid flow / Line Source Solution
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