Motivation

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


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

Outputs

Subsurface temperature distribution

Inputs

	Time lapse after the temperature step from up to
	Spatial coordinate along the transversal direction to constant temperature plane
	TVDss of the top of the lateral flow unit
	True vertical thickness of the the lateral flow unit
	Boundary temperature at
	Thermal diffusivity of the surroundings
	Geothermal Temperature Profile

Equations

Driving equation

Initial conditions

Boundary conditions

\frac{\partial T_e}{\partial t} = a_e^2 \, \Delta T_e = a_e^2 \, \frac{\partial^2 T_e}{\partial z^2}

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

T_e(t, z_f \leq z \leq z_f + h_f) = T_f = {\rm const}

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

Solution

\mbox{if} \, z < z_f \; \Longrightarrow \;T_e(t,z) = T_f + (T_G(z) - T_f) \cdot \mbox{erf} \left( \frac{z_f-z}{\sqrt{4 a_e t}} \right)

\mbox{if} \, z_f \leq z \leq z_f + h_f  \; \Longrightarrow \; T_e(t,z) = T_f

\mbox{if} \, z > z_f + h_f  \; \Longrightarrow \; T_e(t,z) = T_f + (T_G(z) - T_f) \cdot \mbox{erf} \left( \frac{z-z_f-h_f}{\sqrt{4 a_e t}} \right)

where

Error function

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).