Motivation
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Reservoir pressure dynamics away from wellbore and boundaries is sensitive to the two specific complex reservoir properties: transmissibility
and
pressure diffusivity .
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Since the well radius is neglected the well is modeled as a vertical 0-thickness line, sourcing the fluid from a reservoir, giving a model a specific name Line Source Solution.
Inputs & Outputs
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Physical Model
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Mathematical Model
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| Motion equation | Initial condition | Boundary conditions |
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| LaTeX Math Block |
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| \frac{\partial p}{\partial t} = \chi \, \left[ \frac{\partial^2 p}{\partial r^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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| LaTeX Math Inline |
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| body | \displaystyle t \gg \frac{r^2}{4\chi} |
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| LaTeX Math Block |
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| p(t,r) \sim 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 \, \chi t}{r^2} \right)
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Diagnostic Plots
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 Image Modified
| Pressure Drop | |
| LaTeX Math Block |
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| \delta p = p_i - p_{wf}(t) = \frac{q_t}{4 \pi \sigma} {\rm Ei} \left(-\frac{r^2}{4 \chi t} \right) |
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| LaTeX Math Block |
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| p' = t \frac{d (\delta p)}{dt} = \frac{q_t}{2\pi\sigma} \exp \left( - \frac{r^2}{4\chi t} \right) |
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| LaTeX Math Block |
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| \delta p \sim \ln t + {\rm const}, \ t \gg \frac{r^2}{4\chi} |
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| LaTeX Math Block |
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| p' \sim \rm const, \ t \gg \frac{r^2}{4\chi} |
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Fig. 1. PTA Diagnostic Plot for LSS pressure response for the 0.1 md reservoir in a close line source vicinity (0.1 m), which is about a typical wellbore size. One can easily see that with wellbore effects neglected even for a very low permeability reservoir the IARF regime is getting formed very early at 0.01 hr (36 s). |
See also
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Physics / Fluid Dynamics / Radial fluid flow / Line Source Solution
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