Among the parameters in a geothermal design, the effective borehole thermal resistance, $R_b^{\ast}$, is one of the few under the designer's direct control. It expresses the resistance to heat transfer between the circulating fluid and the borehole wall, rolling into a single value the thermal properties of the pipe, the grout and the rock or soil surrounding the borehole, together with the number, position and geometry of the pipes, the borehole length and the circulation flow rate. Lowering $R_b^{\ast}$ shortens the required borehole field and reduces the initial construction cost of the system, in exchange for a modest, one-time investment. It is, accordingly, one of the more rewarding parameters in ground-source design — and unlike the ground itself, it is something the designer specifies rather than inherits.
Composition of effective borehole resistance
$R_b^{\ast}$ is not a single physical quantity but a chain of resistances, and the breakdown indicates where to act. Between the fluid and the borehole wall, three terms act in series:
- $R_f$, the convective resistance between the fluid and the pipe wall, set by the flow regime, the fluid properties and the pipe diameter;
- $R_p$, the pipe resistance, set by the pipe material and wall thickness;
- $R_g$, the grout resistance, set by the grout conductivity and the position of the pipes within the borehole.
Together these give the equivalent borehole resistance at a single depth,
$$ R_b = R_f + R_p + R_g $$
The value that actually governs design is the effective resistance, which adds one further term:
$$ R_b^{\ast} = R_b + R_\eta $$
where $R_\eta$ is the advective term. It captures the thermal short-circuiting between the up and down legs of the loop — heat that passes straight from the warm leg to the cold leg instead of into the ground — and it depends on the circulation flow rate, and so on how long the fluid resides in the pipes. A slower flow leaves the fluid in the borehole longer, letting more heat leak between the legs, so $R_\eta$ grows. On the contrary, a faster flow shortens the residence time and shrinks $R_b^{\ast}$ (Hellström 1991).
The distinction is not cosmetic: $R_b^{\ast}$ is always the larger of the two — by a few percent in a shallow borehole at high flow, but by several times $R_b$ in a deep borehole run at low flow, where the short-circuiting term comes to dominate. Predicting both before drilling is therefore worth doing well: a poor estimate of the resistance under- or over-sizes the field, and the error is paid for either way — in wasted capital or in excess energy — for the life of the system.
Many techniques exist to evaluate $R_b$ and $R_b^{\ast}$ from the borehole geometry and the material conductivities — shape-factor formulas, equivalent-diameter approximations and empirical correlations — and they do not all agree. The most thorough test of them is the systematic comparison by Javed and Spitler (2017), who benchmarked ten published methods for a grouted single U-tube against a rigorous reference solution across the full range of borehole diameters, pipe spacings, and grout and ground conductivities met in practice. Most of the simplified formulas were accurate only over the narrow conditions their authors had in mind and drifted well beyond it — common shape-factor and equivalent-diameter methods overestimate the grout resistance by 15–50% in unfavourable cases — whereas the closed-form first-order multipole expressions stayed within about 2% for the borehole resistance, and 6% for the internal short-circuiting term, over the entire range. It is the method to prefer wherever accuracy matters. The same approach has since been given explicit formulas for the double U-pipe, for which — unlike the much-studied single U-tube — almost no other methods exist (Claesson and Javed 2019).
The internal geometry of the borehole is where the largest controllable gains are found. Figure 1 shows these four contributions and how each shifts with flow rate.
The choice of borehole configuration
Before the grout and the flow rate are settled, the configuration of the heat exchanger itself sets the baseline. A single U-loop is the simplest and least expensive but offers the highest resistance. A double U-loop — with the inlet and outlet pipes either adjacent or diagonally opposed — adds pipe surface and lowers the resistance. A coaxial configuration, with concentric inner and outer pipes, lowers it further still, and a standing column well — an open borehole with no U-tube or grout in the heat path — has typically a low borehole resistance. Each step is a trade-off between a one-time cost and a permanent reduction in field length; the configurations, and their cost and installation trade-offs, are compared in a companion article on ground heat exchanger types.
Pipe spacing: the gain easily lost
The resistance model assumes the pipes are held symmetrically apart within the borehole at maximum shank spacing. Practice is less tidy. Once grouting begins, the pipes tend to drift together and rest against one another unless spacers are installed systematically along the full borehole depth.
A set of pipes bundled together can readily double the effective borehole resistance, erasing the entire benefit of an enhanced grout. Two pipes touching short-circuit heat directly from the warm leg to the cold leg — inflating the $R_g$ and $R_\eta$ terms the geometry was meant to keep small. This is the quiet failure mode of an otherwise sound design: the thermal work is completed, the premium grout is specified, and the field then under-performs because the pipes were never held apart. Specifying spacers in the contract documents, and verifying their installation on site, is as much a part of the thermal design as the calculation itself.
Grout: the largest single lever
Neat bentonite- or cement-based grout has a thermal conductivity of roughly 0.7–0.9 W/m·K. Thermally enhanced grouts, which incorporate conductive fillers such as high-conductivity silica sand, reach 1.7–3.3 W/m·K (Allan and Kavanaugh 1999). The effect on GHE length is substantial: modelling for a representative system predicts bore-length reductions of 22–37%, depending on soil conductivity and bore diameter, from the grout upgrade alone (Allan and Kavanaugh 1999).
The gains are not linear, however. Because the grout is only one term in the series, its conductivity matters most while $R_g$ dominates; as the grout is enhanced further, the resistance becomes increasingly limited by the pipe and the fluid film instead, and each additional increment of conductivity buys progressively less reduction (Allan and Kavanaugh 1999). The design target is a good grout, not the most conductive one obtainable at any price.
The economics of the grout upgrade usually favour it decisively: on a large installation the incremental cost of better grout is a small fraction of the drilling it avoids, so a field shortened by a quarter is a quarter fewer metres to drill, grout, connect and pay for — a gain that lasts the life of the system, for a single material decision taken before the rig arrives.
The wider point, though, is that the objective is the lowest total cost, not the lowest $R_b^{\ast}$ for its own sake — and the two need not coincide. Where the local economics run the other way, with cheap drilling or a resistance-reduction measure that is costly or slow to install, the rational design accepts a higher $R_b^{\ast}$ and a longer field. Grout is adopted almost everywhere precisely because it lowers the resistance for so little effort that it pays back on nearly any site, whereas a specialized configuration or enhanced pipe often costs enough that it is cheaper to simply drill deeper.
Flow regime: turbulent, but no more than necessary
A high circulation flow rate keeps the flow turbulent and the convective resistance $R_f$ low, but the relationship has sharply diminishing returns. Figure 1 plots $R_b^{\ast}$ against flow rate, and the trade-off is visible in it: at low flow, the curve rises steeply as the regime falls into the laminar transition — a condition to avoid at peak but that could be acceptable otherwise — while beyond the turbulent threshold the curve flattens, $R_f$ improving only marginally as pumping cost and energy climb steeply.
Two effects compound at low flow. Not only does $R_f$ rise as turbulence is lost, but the advective term $R_\eta$ also grows, for the reason given above — slower fluid short-circuits more heat between the legs. The effective resistance therefore climbs on both counts at once, and in a deep borehole at low flow it can reach several times its turbulent-regime value.
The design target is therefore the lowest flow that maintains turbulent conditions at peak, and no higher. Pursuing turbulence beyond that point purchases a marginal resistance gain at a steep and permanent pumping penalty. This trade-off, and its interaction with antifreeze viscosity, is examined in the pumping article.
Site execution
Two of the resistance levers are set not on the drawing but on the field, and both realise their value only if the installation matches the specification. They deserve equal attention, because a lapse in either permanently raises $R_b^{\ast}$ for the life of the system.
The first is the grout. Its thermal conductivity depends critically on the water-to-solids ratio and on thorough mixing; a poorly mixed or over-diluted grout can deliver a conductivity well below its specification, undoing the field length bought by an enhanced product. Site supervision during grouting is a worthwhile investment, particularly where the field was optimised around a high-performance grout specification — the alternative is to pay for a premium product and receive the performance of a standard one.
The spacers are the second, and they are a performance lever in exactly the same sense. A field sized around maximum shank spacing delivers that resistance only if the spacers are actually fitted — clipped on at the specified interval along the full borehole depth, not omitted below the first few metres or set too far apart to hold the pipes off one another as the grout is tremied in. Left to drift together, the pipes short-circuit the warm and cold legs and inflate $R_g$ and $R_\eta$ just as surely as an under-specified grout raises $R_g$. Verifying spacer placement on site therefore protects $R_b^{\ast}$ as directly as supervising the grout mix does; neither can be corrected once the borehole is backfilled.
Conclusion
Borehole resistance offers one of the best ratios of payoff to cost in geothermal design, but only if the thermal work survives contact with the drilling rig. Selecting an appropriate configuration, specifying enhanced grout where the numbers justify it, insisting on spacers and verifying them, holding the flow at the turbulent threshold rather than above it, and supervising the grouting together convert a design intention into realised performance — before a single borehole is drilled, and for every year the system runs thereafter.
References
- Allan, M. L., and S. P. Kavanaugh. 1999. Thermal conductivity of cementitious grouts and impact on heat exchanger length design for ground source heat pumps. HVAC&R Research 5 (2): 85–96.
- Claesson, J., and S. Javed. 2019. Explicit multipole formulas and thermal network models for calculating thermal resistances of double U-pipe borehole heat exchangers. Science and Technology for the Built Environment 25 (8): 980–992.
- Hellström, G. 1991. Ground Heat Storage: Thermal Analyses of Duct Storage Systems. PhD thesis, University of Lund, Sweden.
- Javed, S., and J. D. Spitler. 2017. Accuracy of borehole thermal resistance calculation methods for grouted single U-tube ground heat exchangers. Applied Energy 187: 790–806.
Part of the GSHP in Practice series.
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