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Outputs
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| Temperature distribution alon transversal direction to |
Inputs
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Inputs
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| Time lapse after the temperature step from up to |
| Spatial coordinate along the transversal direction to constant temperature plane |
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| Thermal diffusivity of the surroundings |
Equations
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| Driving equation | Initial conditions | Boundary conditions |
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| LaTeX Math Block |
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| \frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2} |
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| LaTeX Math Block |
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| T(t=0, z) = T_G(z) |
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| LaTeX Math Block |
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| T(t, z=0) = T_f = {\rm const}, \quad T(t, z \rightarrow \infty) = T_G(z) |
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Solution
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| LaTeX Math Block |
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| T(t,z) = T_f + (T_G(z) - T_f) \cdot \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi |
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Approximations
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| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \zeta = \frac%7Bz%7D%7B4 a t%7D \sim 0 |
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| LaTeX Math Block |
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| T(t,z) = T_f + (T_G(z) - T_f) \cdot \Bigg[ 1- \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta} + \frac{3}{4 \zeta^3} \bigg) \Bigg] |
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See also
Heat flow equation for Semispace Linear Conduction:
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Replacing the static value of
in RHK model with dynamic value of
one arrives to the final wellbore temperature model with account of heat exchange with surrounding rocks and cooling effects from flowing units (Semispace Linear Conduction).
See also
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Physics / Fluid Dynamics / Linear Fluid Flow
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