Another minor note as I read the rest of the page:

> the resistor material itself is $1-$2/kilowatt. (...) There are only a few materials that are even acceptable as resistors.

This depends strongly on your target temperature and conditions. Nichrome may cost US$2/kW but galvanized barbed wire doesn't. It can kill you with the fumes, and in oxidizing conditions it won't last long above 200°, but zinc fumes inside a dirt pile won't hurt anybody, and you can probably maintain reducing conditions by mixing some humus into your dirt initially and keeping the pile dry. If process heat or climate control is your objective, you don't need temperatures high enough to oxidize iron rapidly even in an oxidizing environment.

(Incidentally, reducing conditions will destroy nichrome at the temperatures people use nichrome for. Maintaining oxidizing conditions inside your dirt pile over the years is going to be a lot more challenging, I suspect.)

I suspect keeping the pile dry is the reason for not just trenching in existing soil. Dryness is essential for maintaining temperature over 100°.

My intuition is that you could probably scale the seasonal-sensible-TES-in-dirt approach further down from the few megawatts he's talking about if you can use real insulation instead of dirt. But insulation is expensive at this scale, even things like perlite and vermiculite.

Spitballing, suppose we want to withdraw 100kW energy for 4 months (1TJ) and can deal with a time constant of 8 months, so leakage is ballpark 50kW. Suppose we have 1J/g/K sand and ΔT = 200K. Then we need 5300 tonnes of sand; at US$20/kg that's US$100k, a dollar a watt, cheaper than power from a coal plant but not really competitive. A larger ΔT can save you but might shorten the life of your barbed wire. But at those low temperatures you pretty much can use any old dirt, US$5300 of it by Vernon's embankment figure.

But that's about 2000m³, whose minimum insulated surface area would be as a 7.8m sphere with 800m² of surface area. 50kW of leakage over that area would be 70W/m², or 0.3W/m²/K, or 3K·m²/W.