**Introduction**

In this paper we derive a simple model that describes the recovery of petroleum fluids from an oil reservoir by the method of electromagnetic heating. By its very nature this problem must deal with both the equations that describe the fluid flow as well as the heat flow. In fact, one approach to this problem is to write out the full system of coupled partial di erential equations that relate the temperature and the velocity flux and then to solve them numerically with a computational fluid dynamics (CFD) program. This method has been used in the past [5] and the results from a commercial CFD solver will be used to test the accuracy of our simpli ed model in the absence of experimental data.

In general, the oil in the wellbore is very viscous with the consequence that the fluid moves slowly. As a result, the amount of oil collected in a given time is quite small. To increase the production rate of the well, the oil's velocity needs to be increased and one method of accomplishing 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 method of increasing the production rate of a given wellbore is currently being utilized with the generalization that for wells of several hundred meters in length, several EMIT regions are placed in the wellbore at intervals of about one hundred meters. So that they are all supplied sucient power, these EMIT regions are connected by a cable surrounded by a steel housing.

The purpose of this paper is to carefully analyze each of the physical processes in this system and by making some basic assumptions, to derive a simple set of equations that can be solved rapidly while still capturing the main features of the system modelled with the CFD code. In this process we nd that under our assumptions, the flux of oil from the wellbore can be modelled with a single nonlinear second order boundary value problem.

As a comparison of the two models, the production rate at the pump was computed for the two models in the unheated case. The di erence between the two models was found to be less than 5%. This is quite remarkable considering the relative complexity of the two models. When the wellbore is heated the deviation between the CFD package and our solution increases but it does so in a manner consistent with the formation of a thermal boundary layer at the wellbore casing. Since the commercial code is time dependent, and does not model the wellbore as an idealized pipe the comparison in this heated case required many hours of computation. As such only one iteration of the CFD solution was pursued.

One of the advantages of the simpli ed model is that it allows one to search wide ranges of parameter space. With a large commercial package this procedure can be prohibitively expensive. We consider two problems along these lines. First, we determine the production rate at the pump as a function of position in the wellbore and the amount of power applied. Second, the rule of thumb of placing the EMIT regions at 100 m intervals in a long well is analyzed.

This paper is organized in the following way. Section 2 describes the overall geometry of the problem and establishes the coordinates used to describe the model. At this point the problem is broken into three subproblems: i) the radial flow of fluid in the reservoir, ii) the horizontal flow of fluid in the wellbore and iii) the generation of temperature from the heat sources in any EMIT regions and how this couples to part i) and ii). Parts i) and ii) result in a second order ODE for the radially average oil flux determined at a xed viscosity. From part iii) it is found that the temperature of the fluid is inversely proportional to the velocity. Consequently, fluid that moves slowly past an EMIT region will absorb more heat than the same amount of fluid that moves quickly past an EMIT. As a result, slowing the fluid velocity increases the temperature and 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 the axial 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 relationship for the velocity and the pressure from the Navier-Stokes equations is obtained by averaging over the radius of the wellbore. Under the assumptions made, the pressure is found to be related to the radius of the wellbore by a form of Poiseuille's law. Section 5 illustrates the analytical solution of the resulting model in the simple situation when no heat is applied to the oil.

Section 6 details the derivation of part iii), the temperature equations. This derivation is complicated by the fact that there are four radial regions of the radial problem to consider; EMIT, casing, reservoir and wellbore with the rst three forming the boundary conditions for the heat equation in the wellbore region. Furthermore, there are three axial regions: EMIT region, cable region, and a region where there is neither EMIT nor cable. Section 7 summarizes resulting nonlinear ODE obtained by pulling the results of Sections 3, 4 and 6 together.

In Section 8, we discuss the numerical results of the simpli ed model and how they compare to the results predicted by the CFD code. On comparison, we nd considerable qualitative agreement between the two models. These aspects are further discussed in the nal section of the paper.