Let's start with a few caveats - heat flow is the vital issue here and on Earth heat flow has proven to be very variable and is hard to measure accurately. What follows is based on a scientific paper that I prepared with Jeff kargel as co-author but it was rejected by the reviewers, so It is not accepted wisdom.

I nonetheless believe that the logic is sound, but you should use your own judgement here.

The data:-

Mars has an average surface temperature of -60 C. At the equator it averages (year round) -45 to -50 C. The poles freeze CO2 in the winter and it only partially boils off in the summer so they are frozen at -125 C.

Mars is smaller than Earth, and less dense.

In Antarctica, the average surface temperature is around -40 to -50 C and the ice is up to 4 km thick.

Average temperature gradients on Earth are ~ 30 C /km

The analysis:-

Let's focus on the depth to the zero centigrade isotherm. Of course, one may argue that strong acid or salt solutions may be mobile at much lower temperatures and hence shallower depths, but if they are too strong it becomes hard to imagine life as we know it, Jim.

Comparing Antarctica to the equator of Mars, we can see that if Mars had similar rocks as Earth (it does, to first approximation) and similar heat flow, then one would need to drill 4 km+ to find pure liquid water.

Where does heat flow come from? It is radiogenic heat from U, Th, K40. All slowly decaying. The heat escapes from hot interior by mantle convection, volcanic fluids, hot springs, but mostly by conduction through rocks (="heat flow")

IF Mars has identical composition to Earth, and the same fraction of radiogenic nucleides, and the same differentiation and convection history then:-

1) it would have1/10 the mass, so 1/10 the heat generation
2) it would have1/2 the radius so 1/4 the surface area

ergo:- it would have 4/10 the heat flow.

(Heat flow scales approx as gravity for terrestrial planets. Mars = 0.38 g)

Now, let's look at that density difference. it isn't just pressure and compaction - Mars is genuinely made of lighter stuff on average than Earth. When we look at calculations of what the distribution of elements is in the Solar Nebula, we expect more "metals" in close and more oxygen and other lighter elements further out. The pattern is not perfect, but Mercury certainly has a huge metallic core and Mars has almost none.

of the radiogeneic elements we expect less U and Th, but more K. Nett result, less radiogenic elements. - about 20% less

The differentiation history of Mars appears different from Earth. It differentiated early and strongly. (Earth may have done the same, then remixed with strong convection). Anyway, Mars has more heat-producing elements near the surface so can cool better. - about 10% less on heat flow due to less storage of historic heat from past times.

Final answer - Mars heat flow 25-30% that of Earth.

Thus we expect average temperature gradients on Mars to be 25-30% of the ~30 C /km on Earth. - i.e. 7-10 C /km.

Now let's go to Mars and look see how deep we need to go to find zero centigrade...

Equator: surface = -45 C. Require 4.5 to 6.5 km depth
Poles: surface = -125 C. require 12.5 to 18 km

Given this, it is remarkable that there was even a hint a disappointment when the ESA imaged the base of the polecap at 1.8 km depth with no hint of liquid water.

Unfortunately, there are some persistent and wholly wrong ideas floating in the literature as the result of the writings of Fanale, Clifford, Baker etc... These authors, bless their souls, claimed that liquid water was possible at depths of a few hundred metres to maybe a km or two.

I can go through the long argument of why they are wrong, but basically they required a thick dry, dusty, ice-free surface layer as an insulating blanket, plus heatflows of at least terrestrial scale. Odyssey evidence for extensive near-surface permafrost killed the dust blanket model and I am firmly of the opinion that their heat flow models are fallacious.

That's enough for a non-technical forum, anyway. Discussions happily entered into but do your background reading first or I will demolish your flimsy arguments. :-)