Possible inconsistency on radial distance of inclined finite line sources

[metab0t]

As mentioned in #50, the radial distance between inclined FLS is difficult to evaluate because they also depend on the depth.

I notice that the work of Lazzarotto (A methodology for the calculation of response functions for geothermal fields with arbitrarily oriented boreholes - Part I) does not mention the special case when the radial distance between heads of two inclined FLS are smaller than the radius of borehole: $\Delta x^2 + \Delta y^2 \leq r_b^2$.

In pygfunction, the radial distance is clamped to $r_b$ when it is too small for both vertical and inclined FLS.

The problem is that this will introduce inconsistency for two exactly same FLS when they are tilted.

Here we consider two short adjacent FLS.
When they are vertical, their radial distance is fixed to $r_b$ because $\Delta x = \Delta y = 0$.
If they are rotated a big angle, $\Delta x^2 + \Delta y^2 > r_b^2$ holds and $r_b$ will not impact their coupling factors.

Now it gives unrealistic result where the coupling between two FLS depends on their titled angles.

This is the small example to reproduce:

import numpy as np
import matplotlib.pyplot as plt

import pygfunction as gt

time = 86400.0 * 30
alpha = 1e-6

rb = np.array([0.1])

x = np.array([0.0])
y = np.array([0.0])

D = np.array([5.0])
H = np.array([0.5])

orientation = np.radians([0.0])

tilts = np.radians(np.linspace(0.0, 90.0, 90)).tolist()

inclined_factors = np.empty((len(tilts),))

for i, tilt in enumerate(tilts):
    x1 = x
    y1 = y
    D1 = D
    H1 = H

    x2 = x1 + H * np.sin(tilt)
    y2 = y1
    D2 = D1 + H * np.cos(tilt)
    H2 = H
    inclined_factors[i] = gt.heat_transfer.finite_line_source_inclined_vectorized(
        time,
        alpha,
        rb,
        x1,
        y1,
        H1,
        D1,
        tilt,
        orientation,
        x2,
        y2,
        H2,
        D2,
        tilt,
        orientation,
        reaSource=True,
        imgSource=False,
        approximation=False,
    )

fig, ax = plt.subplots(figsize=(5,3))

ax.plot(np.degrees(tilts), inclined_factors)

ax.set_xlabel('Tilt Angle (degrees)')
ax.set_ylabel('Heat Transfer')
ax.set_title('Heat Transfer for Different Tilt Angles')

The result:

[MassimoCimmino]

Thanks @metab0t.

I was able to reproduce the issue. The problem seems to be related to the evaluation of the horizontal distance in the integrand of the inclined FLS solution (note: it is squared in the following lines):

pygfunction/pygfunction/heat_transfer.py

Line 1221 in 443d866

rr = np.maximum(dx**2 + dy**2, rb1**2)

pygfunction/pygfunction/heat_transfer.py

Line 1237 in 443d866

rr = np.maximum(dx**2 + dy**2, rb1**2)

A possible fix would require a change of coordinates to evaluate the radial and axial distances with respect to the line source. Only the radial distance should be corrected (instead of the horizontal distance).

[metab0t]

Thanks for your confirmation.

I think the FLS coupling between the segments from the same borehole should be patched specifically by adding an extra $r_b^2$ in their distance.

Another issue is the mirror source. Should we add the $r_b^2$ when computing distance between the mirror FLS and real FLS?

[MassimoCimmino]

Adding an extra $r_b^2$ is what we implemented in a recent paper.

The current implementation does not differentiate (in the way the FLS is evaluated) between FLS to the same borehole and FLS to another borehole. It could be implemented along with the refactoring of the heat_transfer (#211) and solver (#323, #324) modules, for example by adding a to_self: bool parameter to the finite_line_source.

As a fix to this specific issue, evaluating the radial distance to the line should lead to the wanted behavior since only points along the borehole should have a radial distance of zero.

I tend to approach the mirror source as a different borehole. In most cases, $r_b << D$ and the correction of the radial distance makes little practical difference.

[metab0t]

I have read your latest publication this week and it is very inspiring.

I agree that adding to_self would solve this issue by adding $r_b^2$ to the square of radial distance when to_self=True.