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I was just looking at the geothermal house heating in Iceland, and came to know that in the winter the water is warm, hence cools the house, but how is it hot, not about us feeling it.. My question is not why we feel it warm, but why is it warm? I saw videos where they explain me how the temperature difference makes us feel it.. But my question is not about that, can you please explain me how the water under the ground itself is hot? This is the animation of it.. Please explain it to me.
So there are two different things here.
The first is your animation from “Colorado Geothermal Drilling”. This is a long thin hole that goes maybe 100 meters into the ground with a U-shaped tube in it, and water circulates through that tube. So I take issue with, say, Agnius’s diagram above because you can't even see this depth on the diagram, literally one four-thousandth of the distance to the first mark where the lithosphere ends.
The way this works is much simpler than that... It is just that the thick rocks of Earth’s crust, filled with liquid water and such, are dense, they take a loooot of heat energy just to change a little in temperature. The atmosphere is not so dense, and so you get these big temperature swings in the air from summer to winter. And the very top layer of the ground feels some of that too, but as you dig deeper you get further into the rock and the groundwater and all this “thermal inertia” that keeps a constant pleasant temperature.
It's the same basic reason why people store wine in the basement, or why bears go into caves for winter, or why many animals burrow underground rather than migrating. The deeper you go the more predictable it is, “At this depth it is 10°C (50°F) and it stays that temperature all year round.” And when you pipe water down and back up, no matter what temperature of the water goes down with, it exchanges heat with the vast amounts of rock and comes back up at that fixed stable temperature of the rock.
The easiest thing you could do with the water is, just pump it through your walls as-is. It would keep your house at around that same temperature, the 10°C is warmer than the freezing temperatures outside during the winter, but cooler than the sweltering temperatures of summer. Your entire house become this one big cool wine-basement. In practice people prefer a slightly warmer temperature though, and more control, so they use a little electricity to power what's called a heat pump. I won't get too deep into those work because you didn't ask, but the basic idea is that in winter heat wants to flow out of this 10°C water into the freezing air, maybe if we help it flow out then we can get more heat per unit of electricity, than if we just dump that electrical straight into heat energy (say with a space heater). This means that some very cold water will be coming out of the heat pump after we are done with it, but we send that water down into the Earth and it comes back at 10°C again so we don't have to worry about it.
So that is how to use the constant temperature 100m under the ground to heat a house in winter and cool it in summer.
So, to recap, that previous idea is something you can do just about anywhere, drill 100 m into the ground and you will find that the temperature is not affected by the weather in the atmosphere, and this can exchange enough heat to cool or heat a typical house. But the Earth is so big that you can do some much more extreme things, if you can find a good location. Instead of heating one house with some electricity requirements to run a heat pump, you can just lay out a grid underneath a city block, of pipes that conduct very hot water. You turn hot water into a utility, which all of the buildings on the block can hook up to, they can run it through radiators to heat their house in the winter and they can use a heat exchanger to heat the potable water, so they can have hot water out of the tap without a water heater. As a bonus, if you run these pipes underneath sidewalks or roads, the residual heat leaking out of them can sometimes melt any snow that falls on those roads, so you also don't need to pay for snow plows. Doesn't do air conditioning, but if you live in Iceland that's less of an issue for you.
So how are we going to get this hot water grid? A constant wine cellar 10°C is no use. Instead, we have to use the fact that Earth's crust is basically a thin eggshell and the liquid yolk inside is lava. (Well, it's technically “magma” until it punctures the surface but same thing, molten rock.)
We want to keep the magma at a safe distance, but use its incredible heat to heat our water. And it turns out we can have both these things at once! Water naturally wants to fill up the cracks between rocks underground, so if you can find some sedimentary rock 1-2 km underground, it will just have a bunch of hot (~80°C, ~180°F) water flowing around inside of it.
The only problem is, you don't want it to be like oil where eventually you pump out all the hot water and have to go find new hot water elsewhere... So you insist on a closed system. This means you do not clean the water, people do not pour it straight out of their faucets... They always always always pump the hot water through a heat exchanger and return it to a parallel set of water pipes for waste heating-water, and you collect this cooler 40°C-or-whatever water and pour it back into that sedimentary rock, a decent distance away. (The actual plant, the building where the hot water emerges from the ground and the cool water goes back in, can be the same building for both, if the boreholes into the Earth are drilled diagonally rather than straight down.)
So it really is as simple as, they find a pocket of really hot water underground (what at the surface would be hot springs) and run it through some pipes at the surface level and then put it back into the ground a certain distance away. Probably the reason that Iceland pioneers this work, other than the fact that it's cold there, is that it's a volcanic region and the magma is closer to Earth's surface, so the drilling is probably cheaper.
The answer is that the ground has thermal inertia and it takes time for the variation in surface temperature to reach (attenuated) the deeper layers. This results in a phase shift between the seasonal changes in temperature at the surface and the changes in temperature down in the ground. After a certain depth, the soil reaches an average temperature that is colder than the average hot summer temperature, and hotter than the average cold winter temperature.
Solution for a generic periodic boundary condition
If you want to see how to derive the functional dependence of the temperature with depth, at a given time of the year, I suggest you read Sommerfeld's “Partial Differential Equations in Physics”, pp. 68-71. Sommerfeld starts from the simplified heat equation in the single spatial variable x (the distance from the surface of an idealized infinitely large flat Earth):
$$ \frac{\partial^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t} $$
(where $\alpha = \kappa / \rho c$ is the thermal diffusivity of the medium, $\kappa$ is its thermal conductivity, $\rho$ is its density, and $c$ is its specific heat at constant pressure), and finds a series solution T(x,t) that is compatible with a generic periodic evolution of the surface temperature expressed by the following Fourier series:
$$T(0,t) = \sum _{n=-\infty }^{\infty } e^{i n \omega t} c_n$$
where $\omega = 2\pi / P$ is the angular frequency of the fundamental with period $P$. The solution can be expressed in terms of the magnitude $\left|c_n\right|$ and the phase $\varphi_n$ of the complex coefficients of the the above series ($c_n = \left|c_n\right| e^{i \varphi _n}$ for the nth harmonic component of the boundary condition):
$$T(x,t) = T_{\text{ave}} + 2 \sum _{n=1}^{\infty } \left|c_n\right| e^{- k_n x} \cos \left( n \omega t -k_n x + \varphi _n\right)$$
where:
$T_{ave}$ is the average temperature (at an ideally infinite depth, after equilibrium is reached);
$k_n$ is the attenuation constant that depends on the soil's thermal conductivity but, most importantly, increases as the square root of the frequency $n \omega = 2\pi n / P$ of the different harmonic component of the boundary condition;
$$k_n = \sqrt{ \frac{|n| \omega}{2 \alpha} } = \sqrt{ \frac{|n| \pi}{\alpha P} } = \sqrt{ \frac{|n| \pi}{P} \frac{\rho c_p}{\kappa } } $$
Since $k_n$ depends on n in a nonlinear manner, the solution will show dispersion and the variuous harmonic components of the surface temporal temperature evolution will propagate through the ground with different velocities. Moreover, since $k_n$ also determines the exponential attenuation with depth, the faster the temperature changes at the surface the more it will be attenuated with depth: daily variations affects only the first centimeters of soil, yearly variations can be detected for 15-20 meters, while on the hundreds of meters scale the variations in the temperature of the ground reflects the evolution of climate on a scale of hundreds or thousands of years.
Simplified solution for a single harmonic
To simplify things, we can limit the analysis to a single harmonic component of the surface Fourier series by hypothesizing that during a sidereal revolution, the variation in temperature linked to the tilt of the Earth with respect to the plane of the orbit can be approximated by a single sinusoid with a period of one year.
It's coldest in the middle of the winter, then it gradually becomes warmer, it's hottest in the middle of the summer and so on...
For this yearly variation in temperature, Sommerfeld finds that at a depth of about 4 meters in average soil, the phase lag is 180 degrees and the attenuation is 1/16.
A more recent treatment was given by G. C. Berresford ("Differential Equations and Root Cellars", UMAP Journal 2 (1981), pp. 53-75) who focused on the annual variation alone, deriving the following simplified expression for the normalized sinusoidal temperature change:
$$ T(x,t) = \cos(2 \pi t - k x) \exp(-k x) $$
where x is the depth in meters, t is the time in years and k is the attenuation/propagation constant that, for the annual periodic variation in 'average soil', has the value 0.743224 (m^-1).
The harmonic seasonal temperature perturbation at the surface propagates in the terrain at a speed of 8.45 meters per year, and the first 'phase inversion' happens at a depth of 4.23 meters, a result in good agreement with Sommerfeld's approximate computation. This inversion of minima and maxima is the reason why cellars are particularly lukewarm in the winter and cool in the summer.
In the winter, the ground at 'cellar depth' is 'optimally hot' because it is 'remembering' or better 'belatedly experiencing' the hot of the summer (albeit quite a bit 'faded' due to the exponential attenuation). Vice-versa, in the summer the cellar goes through the (attenuated) cold peak of the previous winter.
The deeper you go the more you reach an average temperature which, despite being constant, still happens to be warmer than the cold winter temperatures, and cooler than the hot summer ones. At a depth of about 20 meters, the seasonal oscillations in surface temperature are no longer appreciable and the temperature of the soil is roughly the yearly average of the temperature at the surface.
What about the heat from below?
Regarding the role of geothermal heating from the inner core of the Earth, Wikipedia's page on the geothermal gradient states that:
The top of the geothermal gradient is influenced by atmospheric temperature. The uppermost layers of the solid planet are at the temperature produced by the local weather, decaying to approximately the annual mean-average temperature (MATT) at a shallow depth of about 10-20 metres depending on the type of ground, rock etc;[8][9][10][11][12] it is this depth which is used for many ground-source heat pumps.[13] The top hundreds of meters reflect past climate change;[14] descending further, warmth increases steadily as interior heat sources begin to dominate.
Therefore I would say that in the first 100 meters, the warmer/cooler temps are solely determined by the way the heat from the sun propagates into the ground.
Heat pumps don't actually require the source of thermal energy to be warm. You can run an air sourced heat pump to heat your house in the winter using energy extracted from the freezing cold air outside. Heat won't flow spontaneously from cold to hot, but you can make energy flow from cold to hot by doing work. It's exactly the same principle of operation as in a refrigerator.
While it's not a requirement that the ground be warmer than the ambient air for the heat pump to work, the ground will tend to be warmer than the air in the winter. The more shallow layers of soil thermally insulate the deeper layers, and thermal energy that slowly makes its way down into the soil over the warmer months takes time (months) to flow back up into the air over the winter.
Why is the ground warm during winter
Because in principle our Earth is a heat generator (like a "wood stove"). Check Geothermal gradient:
(Image credits, here.)
and you'll see that Earth inner core has temperature about $5000~^\circ \text{C}$, so basically answer is that it warms inner-ground of surface. In case surface is heated more than natural geothermal gradient predicts (by sun, fire place, etc), then surface becomes an additional heat absorber, because heat always flows from warmer to colder places. Hence this is answer to second part of your question.
BTW, greatest geothermal gradient is within first $100~km$ of depth which is about
$$ \left( \frac {\partial T}{\partial z} \right)_{100~km} \approx 20 ~^{\circ} \text{C/km}.$$ Hot geysers in the Iceland are due to the local bigger geothermal gradient, which happens because of higher volcanic activity and other heat exchange mechanisms in that area.
