FLS Calculation Methods

pyCaRM precomputes a thermal response matrix using the Finite Line Source (FLS) analytical model. Two methods are available to evaluate the effective distance between boreholes, controlled by the fls_mode parameter of Simulation.

Overview

Each borehole is assigned a Voronoi cell of area \(A_i\), from which an equivalent radius is derived:

\[r_{eq,i} = \sqrt{\frac{A_i}{\pi}}\]

This radius represents the effective influence area of borehole \(i\) and is used to compute the distance entering the FLS integral. Two strategies are implemented.

sqrt method (default)

The effective distance between source borehole \(j\) and receiver borehole \(i\) is computed as:

\[d_{ij} = \sqrt{d_{Euclidean,ij}^2 + r_{eq,i}^2}\]

where \(r_{eq,i}\) is the equivalent radius of the receiver cell. This is a scalar approximation — a single distance value per borehole pair — and is the default method due to its low computational cost.

continuous method

The receiver borehole wall is discretized into \(N_\theta\) equally spaced points along its equivalent radius:

\[\mathbf{p}_k = (x_i + r_{eq,i}\cos\theta_k,\ y_i + r_{eq,i}\sin\theta_k), \quad \theta_k = \frac{2\pi k}{N_\theta},\ k = 0, \ldots, N_\theta - 1\]

The FLS response is evaluated at each point and averaged:

\[\bar{h}_{ij} = \frac{1}{N_\theta} \sum_{k=0}^{N_\theta - 1} h(d_{ijk})\]

where \(d_{ijk}\) is the distance from source \(j\) to point \(k\) on receiver \(i\). This method is more accurate but significantly more expensive.

Angular discretization sensitivity

The figure below shows the absolute difference between results at \(N_\theta\) and the reference solution at \(N_\theta = 64\), for a regular 9-borehole field over 3 years.

From \(N_\theta = 8\) onward, the error is below 0.01 °C across all boreholes. The default value is therefore set to \(N_\theta = 8\) as the best compromise between accuracy and computational cost.

Method comparison

The figures below compare the two methods against the continuous reference solution over a 3-year simulation on a 9-borehole field.

Absolute error vs continuous

The clamp strategy produces errors up to 10 °C for central boreholes and is not recommended. The sqrt method stays within approximately 1.7 °C across all boreholes and both regular and irregular layouts, making it suitable as the default method.

Spacing sensitivity

The error between sqrt and continuous decreases as borehole spacing increases. For spacings of 10 m and above, the maximum error is below 1 °C over 3 years. At tight spacings (3 m), the error can reach approximately 1.6 °C for central boreholes.

Choosing a method

fls_mode	Accuracy	Cost
"sqrt"	Max error ~1.7 °C over 3 years	Low — single distance per pair
"continuous"	Reference solution	High — scales with \(N_\theta \times n_{BHE}^2 \times n_{steps}\)

Use "sqrt" (default) for parametric studies and large fields. Use "continuous" when high accuracy is required and the field is small.