Motivation

Subsurface Temperature Profile around Lateral Flow makes adjustments to Geothermal Temperature Profile to account for the lateral reservoir flow with a temperature (see Fig. 1 and Fig. 2).


Fig. 1. Sample Subsurface Temperature Profile around a lateral flows with temperature	Fig. 2. Sample Subsurface Temperature Profile around two lateral flows with temperature and

Outputs

Temperature distribution

Inputs

	Time lapse after the temperature step from up to
	Spatial coordinate along the transversal direction to constant temperature plane
	Boundary temperature at
	Thermal diffusivity of the surroundings
	Geothermal Temperature Profile

Equations

Driving equation

Initial conditions

Boundary conditions

\frac{\partial T_b}{\partial t} = a^2 \Delta T_b = a^2\frac{\partial^2 T_b}{\partial z^2}

T_b(t=0, z) = T_G(z)

T_b(t, z=0) = T_f = {\rm const}

T_b(t, z \rightarrow \infty) = T_G(z)

Solution

T_b(t,z) = T_f + (T_G(z) - T_f) \cdot \frac{2}{\sqrt{\pi}} \int_0^{z/\sqrt{4at}} e^{-\xi^2} d\xi

Reference

Subsurface Temperature Profile around Lateral Flow @model.pptx

Heat flow equation for Semispace Linear Conduction:

\frac{\partial T}{\partial t} = a^2 \Delta T = a^2\frac{\partial^2 T}{\partial z^2}

Initial Conditions

T(t=0, z) = T_G(z)

Boundary conditions

T(t, z=0) = T_f = {\rm const}, \quad T(t, z \rightarrow \infty) = T_G(z)

The exact solution is given by following formula:

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

A fair approximation at late times () is given by expanding the integral:

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]

where

\zeta = \frac{z}{4 a t}

The final solution for temperature above the flowing unit is represented by RHK pipe flow solution where T_G is replaced with T_b from .

For the intervals between two injection units the one needs to account for the SLC contribution from upper flowing unit and from lower flowing unit which can be done using the superposition.

First, let's rewrite in terms of temperature gain:

dT(t, z) = T(t,z) - T_G(z)= -  (T_G(z) - T_f) \cdot    \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta}  + \frac{3}{4 \zeta^3} \bigg)

Now one can write down the temperature disturbance from the overlying flowing unit A1:

dT_{b,over}(t, z) = T_{b,up}(t,z) - T_G(z)= -  (T_G(z) - T_{f, A1}) \cdot    \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta}  + \frac{3}{4 \zeta^3} \bigg)

and from the underlying flowing unit A2:

dT_{b,under}(t, z) = T_{b,up}(t,z) - T_G(z)= -  (T_G(z) - T_{f, A2}) \cdot    \frac{\exp(-\zeta^2)}{\sqrt{\pi} \zeta} \bigg( 1- \frac{1}{2 \zeta}  + \frac{3}{4 \zeta^3} \bigg)

The background temperature disturbance between the flowing units will be:

T_b(t, z) = T_G(z) + dT_{b,over}(t, z) + dT_{b,under}(t, z)

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).