Motivation
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Reservoir pressure dynamics away from wellbore and and boundaries is representative of two very important sensitive to the two specific complex reservoir properties: transmissibility
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pressure diffusivity .
In case the reservoir flow has been created by a well (vertical or horizontal) it will trend to form a radial flow away away from boundaries and well itself itself.
In this case a pressure drop and well flowrate can be roughly related to each other by means of a simple analytical homogeneous reservoir flow model with wellbore and and boundary effects neglected.
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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= c_r + c, LaTeX Math Inline |
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body | {c_r, LaTeX Math Inline |
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body | c, pore and fluid
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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 |
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t^2r^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} = |
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Computational Model
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LaTeX Math Block |
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| p(t,r) = p_i |
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-+ \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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Late-time response=-+ \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 Added | 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
[ Radial Flow Pressure @model ] [ 1DR pressure diffusion of low-compressibility fluid ] [ Exponential Integral ]