Estimation of Heat Distribution in Selective Electric Reservoir Heating
Authors: Harvey Herbert A. , Arnold M.D. 
Published: December 2007

We have been investigating the possibility of using an alternating electriccurrent to heat portions of an oil reservoir that normally are bypassed byinjected fluids. The process is accompanied by water injection, and itsexpected to increase oil recovery in a waterflood. In a pattern flood, theapplication of heat to parts of the pattern that normally are bypassed byinjected fluids usually will improve the mobility ratio at the displacementfront, since the viscosities of most crude oils are more temperature-sensitivethan the viscosity of water. The improvement in mobility ratio should cause thesweep pattern to be extended to include some of the flood pattern to beextended to include some of the flood pattern that otherwise would be bypassed.Thermal pattern that otherwise would be bypassed. Thermal expansion of oil inthe normally bypassed part of the flood pattern also might make somecontribution to oil recovery if the bypassed region can be effectivelyheated.

In studying the process it was necessary to estimate the distribution ofresistive heating near electrodes. placed in water injection wells. For thetechnique to placed in water injection wells. For the technique to beeffective, heating in the immediate vicinity of the electrodes should beminimized so that adequate electrical is available for heating the desiredportions of the reservoir. portions of the reservoir. The resistance of anelectric conductor is described by

RL p =, .................................(1) A

and the relationship between electric potential and resistance is

V = Ip ........................................(2)

Since the surface area of the sphere equals 4 pi r2, Eqs. 1 and 2 can becombined to show that the distribution of electric potential about a sphericalelectrode in a thick homogeneous medium is

.......................(3)

where ro is the radius of the electrode and re is an arbitrary distance fromthe electrode. Eq. 3 may be integrated to obtain

...............(4)

If we consider a spherical electrode w/ radius of 0.5 ft, the potential dropat a distance of 10 ft. from the electrode is,

where the units conversion factor C = 3.28 ft/m. Similarly, the potentialdrop at a distance of 250 ft from the electrode is.

Thus, 95% of the potential drop to a distance of 250 ft will occur within120 ft of the electrode. Since heat generation is directly proportional tovoltage drop, it is evident that this system would not be suitable for heatingbypassed portions of an oil reservoir. However, the difficulty suggested by Eq.4 is less pronounced in a system than spherical. Heating near pronounced in asystem than spherical. Heating near the electrode also can be reduced byinjecting water that is less resistive than formation water while heating is inprogress. To simplify the problem we will limit our consideration tohorizontal, homogeneous, isotropic reservoirs that are bounded above and belowby highly resistive beds. For this system, we will assume that the flows ofboth injected water and electricity are radial near the injection well. Theregion of r12 and a resistivity of R1. The other boundary of the model used torepresent the system is a cylinder with a radius of r2, which is half thedistance between adjacent electrode wells in a five-spot flood pattern. Theresistivity of the portion of the reservoir that has not be contacted byinjected water is R2. The system is illustrated by Fig. 1 Since the surfacearea of a cylinder equals 2 pi rh, Eqs. 1 and 2 can be combined for this systemto show that

.....................(5)

where Rw1 is the resistivity of injected water. However, the first integral, which represents voltage drop within the wellbore, is negligible compared withthe sum of the other two integrals. Therefore a satisfactory approximation ofEq. 5 for the process under study is

......(6)