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CANADIAN APPLIEDMATHEMATICS QUARTERLYVolume 10, Number 3, Fall 2002
ELECTROMAGNETIC WELLBORE HEATING
Based on work from the Fourth Annual Industrial Problem SolvingWorkshop, June 2000.
C. SEAN BOHUN, BRUCE MCGEE AND DAVID ROSS
ABSTRACT. A model is proposed for the flow of fluids ina horizontal well that is electrically heated from an externalsource. By analysing the physical processes involved, a second
order nonlinear ODE is derived for the volume flux of the wellas a function of distance along the wellbore. Comparing theextracted average radial temperature and pressure profiles witha commercial CFD solver illustrates that this simplified modelcaptures much of the structure involved in the solution to thefull PDE solution.
1 Introduction. In this paper we derive a simple model that de-scribes the recovery of petroleum fluids from an oil reservoir by themethod of electromagnetic heating. By its very nature this problemmust deal with both the equations that describe the fluid flow as wellas the heat flow. In fact, one approach to this problem is to write outthe full system of coupled partial differential equations that relate thetemperature and the velocity flux and then to solve them numericallywith a computational fluid dynamics (CFD) program. This method hasbeen used in the past  and the results from a commercial CFD solverwill be used to test the accuracy of our simplified model in the absenceof experimental data.
In general, the oil in the wellbore is very viscous with the consequencethat the fluid moves slowly. As a result, the amount of oil collected ina given time is quite small. To increase the production rate of the well,
Accepted for publication on December 16, 2001.AMS subject classification: 80A20, 35Q35, 76R99.Keywords: electrical heating, incompressible viscous flow, heat flow, horizontal
well.Copyright cApplied Mathematics Institute, University of Alberta.
354 C. SEAN BOHUN, BRUCE MCGEE AND DAVID ROSS
the oils velocity needs to be increased, and one method of accomplish-ing this is by heating the fluid using an electromagnetic induction tool(EMIT). The simple principle behind the EMIT is that it heats the fluid,thereby decreasing its viscosity and increasing its velocity. This methodof increasing the production rate of a given wellbore is currently beingutilized with the generalization that for wells of several hundred metersin length, several EMIT regions are placed in the wellbore at intervalsof about one hundred meters. So that they are all supplied sufficientpower, these EMIT regions are connected by a cable surrounded by asteel housing.
The purpose of this paper is to carefully analyze each of the physi-cal processes in this system, and by making some basic assumptions toderive a simple set of equations that can be solved rapidly while still cap-turing the main features of the system modelled with the CFD code. Inthis process we find that under our assumptions, the flux of oil from thewellbore can be modelled with a single nonlinear second order boundaryvalue problem.
For comparison of the two models, the production rate at the pumpwas computed for each model in the unheated case. The difference be-tween them was found to be less than 5%. This is quite remarkableconsidering the relative complexity of the two models. When the well-bore is heated the deviation between the CFD package and our solutionincreases, but it does so in a manner consistent with the formation ofa thermal boundary layer at the wellbore casing. Since the commercialcode is time dependent and does not model the wellbore as an idealizedpipe, the comparison in this heated case required many hours of com-putation. As such only one iteration of the CFD solution was pursued.
One of the advantages of the simplified model is that it allows oneto search wide ranges of parameter space. With a large commercialpackage this procedure can be prohibitively expensive. We consider twoproblems along these lines. First, we determine the production rate atthe pump as a function of position in the wellbore and the amount ofpower applied. Second, the rule of thumb of placing the EMIT regionsat 100 m intervals in a long well is analyzed.
This paper is organized in the following way. Section 2 describes theoverall geometry of the problem and establishes the coordinates used todescribe the model. At this point the problem is broken into three sub-problems: (i) the radial flow of fluid in the reservoir, (ii) the horizontalflow of fluid in the wellbore and (iii) the generation of temperature fromthe heat sources in any EMIT regions and how this couples to parts (i)and (ii). Parts (i) and (ii) result in a second order ODE for the radially
ELECTROMAGNETIC WELLBORE HEATING 355
average oil flux determined at a fixed viscosity. From part (iii) it is foundthat the temperature of the fluid is inversely proportional to the velocity.Consequently, fluid that moves slowly past an EMIT region will absorbmore heat than the same amount of fluid that moves quickly past anEMIT. As a result, slowing the fluid velocity increases the temperatureand therefore decreases the viscosity. This viscosity is used in parts (i)and (ii) to close the system of equations.
Part (i) is described in Section 3, where a relationship between theaxial changes in the fluid flux and the pressure in the wellbore is derived.The details of part (ii) can be found in Section 4 where a relationshipfor the velocity and the pressure from the Navier-Stokes equations isobtained by averaging over the radius of the wellbore. Under the as-sumptions made, the pressure is found to be related to the radius of thewellbore by a form of Poiseuilles law. Section 5 illustrates the analyticalsolution of the resulting model in the simple situation when no heat isapplied to the oil.
Section 6 details the derivation of part (iii), the temperature equa-tions. This derivation is complicated by the fact that there are fourradial regions of the radial problem to consider: EMIT, casing, reservoirand wellbore, with the first three forming the boundary conditions forthe heat equation in the wellbore region. Furthermore, there are threeaxial regions: EMIT region, cable region, and a region where there isneither EMIT nor cable. Section 7 summarizes resulting nonlinear ODEobtained by pulling the results of Sections 3, 4 and 6 together.
In Section 8, we discuss the numerical results of the simplified modeland how they compare to the results predicted by the CFD code. Oncomparison, we find considerable qualitative agreement between the twomodels. These aspects are further discussed in the final section of thepaper.
2 Geometry. Figure 1 depicts the overall geometry of the problem.A horizontal cylindrical well extends from z = 0 to z = L. Fluid flowsradially into the well from the surrounding media and is drawn out witha pump which is located at z = L where a fixed producing pressure ofPP is maintained.
At z = 0, where the end cap is situated, the motion of the flowis radially inward through the reservoir and the casing that lines thecomplete length of the wellbore (no horizontal flow at this point). Asz increases, the action of the pump comes into effect and imparts ahorizontal component to the fluid flow.
356 C. SEAN BOHUN, BRUCE MCGEE AND DAVID ROSS
FIGURE 1: Cross section of the overall geometry for the horizontalwellbore problem. The rate and direction of the oil flow is indicatedwith the arrows. At z = 0 there is an end cap and the horizontal flowis zero while at z = L there is a pump that maintains the producingpressure PP .
This figure shows only one EMIT region located at z = ze of length2Le but the analysis can be easily generalized to the case of N EMITregions. It is in these EMIT regions that the oil is heated. The casingin these regions acts as a single turn secondary winding of a transformerthereby heating the surrounding fluids. Power is supplied to the EMITsthrough a cable housing resulting in three different regions. Starting atthe end cap the wellbore is open with no impediment to the horizontalflow. This extends to the first EMIT. If there are other EMIT regionsthen they must also be joined with cable housing and eventually, afterthe last EMIT region, we have a cable housing region that extends tothe pump.
There are a number of physical constants associated with the fluidand heat flow within the wellbore. Since these are required to generate
ELECTROMAGNETIC WELLBORE HEATING 357
numerical solutions and to justify some of the assumptions, they havebeen collected in Table 1.
3 Axial velocity: Darcys Law. Once the horizontal well isdrilled, fluid seeps from the surrounding region into the wellbore. Havingreached the wellbore, the fluid is drawn out with a pump that maintainsa fixed pressure at one end of the well. The rate at which the fluidseeps into the wellbore is a function of the pressure differential and theviscosity of the fluid. Indeed, the flow rate (volume/time) of the fluidinto a segment of the wellbore of length z is given by the expression 
(1) q(z) =2k[PR P (z)]o ln(Rd/Rc)
where k is the permeability of the reservoir, PR is the reservoir pressureand P (z) is the pressure inside the wellbore at the axial position z. Aswell, o is the viscosity at the ambient temperature Ta and Rd/Rc is theratio of the drainage radius to the outer radius of the casing.
Since we are assuming that we are at a steady state, we make theassumption that the radially flowing fluid remains unheated until itreaches the outer radius of the casing at which point it instantly be-comes heated to the average temperature of the fluid at that particularz position. Consequently the viscosity in expression (1) will remain aso even when the temperature of the wellbore is incre