If the flux we want to suck out of the ground in winter is much bigger
than these natural fluxes then we know that our sucking is going to signif-
icantly alter ground temperatures, and may thus not be feasible. For this
calculation, I’ll assume the ground just below the surface is held, by the
combined influence of sun, air, cloud, and night sky, at a temperature that
varies slowly up and down during the year (figure E.16).

### Response to external temperature variations

Working out how the temperature inside the ground responds, and what
the flux in or out is, requires some advanced mathematics, which I’ve
cordoned off in box E.19 (p306).

The payoff from this calculation is a rather beautiful diagram (figure
E.17) that shows how the temperature varies in time at each depth.
This diagram shows the answer for any material in terms of the character-
istic length-scale
z0 (equation (E.7)), which depends on the conductivity κ
and heat capacity CV of the material, and on the frequency ω of the external
temperature variations. (We can choose to look at either daily and
yearly variations using the same theory.) At a depth of 2z0, the variations
in temperature are one seventh of those at the surface, and lag them by
about one third of a cycle (figure E.17). At a depth of 3z0, the variations
in temperature are one twentieth of those at the surface, and lag them by
half a cycle.

For the case of daily variations and solid granite, the characteristic
length-scale is z0 = 0.16 m. (So 32 cm of rock is the thickness you need
to ride out external daily temperature oscillations.) For yearly variations
and solid granite, the characteristic length-scale is z0 = 3 m.

Let’s focus on annual variations and discuss a few other materials.
Characteristic length-scales for various materials are in the third column
of table E.18. For damp sandy soils or concrete, the characteristic length-
scale z0 is similar to that of granite – about 2.6 m. In dry or peaty soils, the
length-scale z0 is shorter – about 1.3 m. That’s perhaps good news because
it means you don’t have to dig so deep to find ground with a stable tem-
perature. But it’s also coupled with some bad news: the natural fluxes are
smaller in dry soils.

The natural flux varies during the year and has a peak value (equation
(E.9)) that is smaller, the smaller the conductivity.

For the case of solid granite, the peak flux is 8 W/m2. For dry soils,
the peak flux ranges from 0.7 W/m2 to 2.3 W/m2. For damp soils, the peak
flux ranges from 3 W/m2 to 8 W/m2.

What does this mean? I suggest we take a flux in the middle of these
numbers, 5 W/m2, as a useful benchmark, giving guidance about what
sort of power we could expect to extract, per unit area, with a ground-
source heat pump. If we suck a flux significantly smaller than 5 W/m2,
the perturbation we introduce to the natural flows will be small. If on the Figure E.17. Temperature (in °C) versus depth and time. The depths are given in units of the characteristic depth z0, which for granite and annual variations is 3 m.

At “depth 2” (6 m), the temperature is always about 11 or 12 °C. At “depth 1” (3 m), it wobbles between 8 and 15 °C. 