Thermally induced fracturing (TIF) during water injection is a well-established phenomenon. TIF modeling implies solving equations simultaneously that conventional petroleum engineering applications deal with separately. Combining these equations leads to very complex computer programs. This has led to the need for a simple model, which we present in this paper. Coupling analytical expressions representing each of these phenomena, rather than the basic physical equations, has led to a computer program that can be run on a modern desk-top computer. This program has successfully matched the daily wellhead pressure and injection rate during a period of 3 to 5 years for injection wells in complex sandstone/dolomite reservoirs. The model can be used for injection-well monitoring as well as in a predictive mode when planning new water-injection projects. The algorithm is sufficiently simple to be implemented in a conventional reservoir simulator.


The concept of a constant productivity index, extrapolated below bubble point by use of Vogel's curve, is one of the fundamental tools of petroleum engineering. One of its most interesting features is that it depends on reservoir and reservoir properties alone. It is independent of downstream wellbore equipment and surface facilities.

One would like to be able to use a similar concept for water injection wells, but, unfortunately, calculating an injectivity index,

Equation 1

turns out to be much more complex. Water available for injection is often much colder than the reservoir, and numerous temperature-induced phenomena often having opposite effects occur within the first few days or weeks of injection.

From the beginning of injection, the bottomhole flowing temperature decreases and finally reaches a stabilized value depending on surface and reservoir temperature, injection rate, depth, and well completion. During that time, matrix flows will have a reducing effect on injectivity. This is because, in such conditions, the bottom-hole viscosity can often increase two- to four-fold. Also, when water displaces oil, there is a relative-permeability effect tied to the growth of the zone from which oil has been displaced.

At the same time, mechanical effects will tend to decrease injectivity inversely. The reservoir stress near the well is reduced when the reservoir is cooled, and fracturing will occur if the reservoir stress falls below bottomhole flowing pressure. This phenomena is called TIF.1–6 It leads to a continuous increase in injectivity when fracture develops. In fact, the final reduced stress is the result of a thermal reducing effect (thermoelasticity) and a fluid-pressure increasing effect (poroelasticity) at the injector. In general, however, the latter is much smaller.

As we have shown, the injectivity index cannot be calculated without taking into account the wellbore pressure and temperature performance. Injectivity therefore depends on the situation both upstream (wellbore equipment and surface facilities) and downstream (reservoir properties). Modeling water injectivity therefore leads to very large computer programs in which the complexities of both reservoir models and fracturing simulators are intermingled. The pioneering work of Hagoort7 and Perkins and Gonzalez8 on thermo-poroelasticity were followed by more refined models, such as that published by Dikken and Niko.9 More recently, Settari10,11 and Clifford12 presented three-dimensional (3D) fracturing calculations.

This paper presents a model that uses simple analytical formulas representing all these intermingled physical processes that influence the injectivity index. The model has been programmed on a PC and used to match the performance of wells injecting into a complex sandstone/dolomite reservoir in the Gulf of Guinea. Well behavior is modeled as a sequence of timesteps. The basic assumption is that steady-state equilibrium is reached at the end of each timestep. This is a good approximation for long-term well behavior.

We do not aim to simulate short-term phenomena such as those encountered during well tests; in our model, reservoir pressure transients are ignored, as are the mechanics of fracture propagation. Our model starts at the wellhead, with a given injection rate and wellhead temperature. The model calculates a wellhead pressure, which can be compared to measurements. The least known parameters are adjusted within their plausible range of values until a satisfactory match is obtained. The algorithm also has been programmed so that when wellhead pressure and temperature are given as data, the model calculates the injection rate. This calculation mode is of particular interest when planning waterfloods.

Part 1: The Model
Wellbore Temperature Profile.

The first task is to calculate bottomhole flowing temperature, ?wf, from surface temperature, injection rate, and wellbore equipment. A linear geothermal gradient is assumed. Bottomhole flowing temperature is calculated from wellhead temperature by dividing the tubing into 25 segments. We use the transient heat-exchange solution13 between each segment and the surrounding earth to calculate the quantity of heat that reaches the water in the tubing.

This solution assumes that the well rate is constant. To cope with rate-varying behavior, an effective injection time has been defined with the cumulative injection Wi and the current injection rate i :

Equation 2

As long as injection rate does not decrease too abruptly, this simple algorithm gives satisfactory results. The reason that such a simple algorithm works is that most of the heat exchange between an injection well and the surrounding earth takes place at depth, where the well geometry is simplest: a tubing and one casing. On the contrary, such a simple calculation is impossible on a production well because, in this case, the biggest temperature contrast and therefore most of the heat exchange are close to the surface where well geometry and its surroundings are most variable.

This algorithm does not give realistic results when the injection rate is reduced abruptly (for instance, when the well is shut in). A smoothing function has therefore been introduced to limit the change in ?wf during any one timestep.

Calculation Assuming Radial Injection.

We use the term radial injection when flow is radially outwards from the well; the alternative, when the reservoir is fractured by the water injection process, is called fractured injection.

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