Ohmic Heating via Potential Difference Across a Geologic Unit I am a current GeoPhysics graduate student studying how high-voltage / high-current electricity can be used to heat up an underground reservoir. I have designed a basic experiment, where I will drill two vertical wells into a bedrock containing water / electrolyte, and  place electrodes (anode and cathode) into each well. I will then apply a voltage difference (alternating current) across the wells, to joule heat the rocks. Think electrolysis in an underground rock formation filled with electrically conductive fluid.
Here are some basic parameters:
Voltage Difference = 400,00 kV
Peak Current = 500 amps
Impedance = 800 ohms*m
Distance between wells = 30 ft.
I am struggling to figure out:
1.) What equations should I use to figure out how the electricity will attenuate/decrease with distance from the source to receiver? The goal is to determine how heat will be produced in the rock.
2.) What effect, if any, would the frequency of the current have on effectiveness? Should I be thinking about the applied potential difference as an electromagnetic wave?
Any advice/assistance would be much appreciated!
-P

 A: Disclaimer: I do not know anything about geology or whether DC pulses (anode/cathode) would be better than AC. But I would worry about electrolysis, especially with direct current. And about the extreme electrical inhomogeneity of bedrock with fractures.
You may have trouble calculating because the resistivity between contacts in a resistive bulk medium depends on the size of the electrode.
Current density is greatest near the electrodes, so I expect that that would be where most of the heat is developed (assuming the medium is not too inhomogeneous). 
In a homogeneous medium, ohmic heating is proportional to the square of the current density. Current density decreases with distance from the electrodes. When the electrodes are spheres with a radius that is small compared to the distance between them, the current density would approximately decrease like $1/r^2$ when close to the electrode. So that is not a desirable geometry, with $1/r^4$ Joule heating.
But in the figure, both electrodes seem to connect to fractures filled with electrolyte. Maybe such fractures can be modeled as two-dimensional sheet conductors with varying thickness. There are no equations for that, you could only try simulations. Or rely on experience.
