Summary 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. Introduction 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, 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 : 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.

When comparing various wells on a given field in terms of sand production, it often happens that they display quite different behaviours: for example, deviated wells will produce sand whilst vertical ones won't. A major reason behind such an apparent paradox is the orientation of the well and its perforations in the in-situ stress field. Such a problem is usually studied by using complex non linear 3D Finite Element Models. In an attempt to make such an approach more flexible, this paper presents a simplified, semi quantitative analytical model able to evaluate sand production risks in cased holes with various deviations. This pseudo 3D model, based on linear elasticity, confirms that sand production risk for a given perforation strongly depends on well deviation and on the orientation of the perforation in the plane orthogonal to the wellbore. A field case illustrates how the model can hence provide an understanding for observations which could have otherwise remained unexplained. Finally, by introducing special failure criteria calibrated on hollow cylinder and cavity failure tests, an extension of the model is compared favourably to some 3D non linear numerical simulations in the case of a second field case. Because of its analytical nature, such an extension can easily be used in conjunction with logs in order to give a rather precise idea of the sand production risk on a newly drilled well. Introduction From an industrial point of view, accurate sand production prediction is an essential issue: a recent internal survey within Elf revealed that for 70 % of all wells drilled, the question of sand production is posed in one way or another before deciding upon the completion method. The Productivity Index (PI) of wells being itself very strongly affected by the completion method e.g., one can understand why such an issue has been considered as essential by the industry for more than two decades. In practice, the question of sand production is essentially raised when the development of a reservoir is planned. At this stage, it is usually decided whether the development wells for a given field will be equipped with sand control or if natural completion will be adopted. However, in many dubious cases, the available experience from exploration and appraisal wells - e.g. observations from tests - is not sufficient to allow for a complete assessment of the sand production risk and a conservative attitude is generally adopted - i.e. all wells will be equipped with sand control. P. 323^

The linear mechanics of porous rocks is summarized : basic equations – i.e. constitutive law and fluid mass balance – and most useful relationships are given. For reservoir engineering applications, the hydrostatic loading case is dealt with in detail. The conventional diffusivity equation -i.e. uncoupled pressure equation-is then derived. Experimental procedures are suggested and results obtained on outcrop and reservoir sandstones are given. The data indicate that the sandstones have a nonlinear behavior and that volumetric coefficients are functions of the difference between the confining pressure and the pore pressure. A model is then proposed based on Biot's semilinear theory. The reservoir rock behavior is shown to be in agreement with this model for the experiments performed. As an application effective stress is computed. Finally the role of pore pressure in damaging cores during the pulling out phase is investigated.

Production history, fluid pressure and uniaxial compressive strength are basic data to evaluate the risk of sand production. Evaluations of the compressive strength based on logs are illustrated for the example of a well on the Germigny-sousCoulombs structure. Then relationships between compressive strength and porosity are developed using a theoretical approach of grain contacts, the analysis of published rock mechanics data and mechanical measurements on plugs taken from well cores. It is shown that compaction factor can complement porosity in the analysis. A field example illustrates the relationship between a critical differential pressure and compaction factor or porosity.

Production history, fluid pressure and uniaxial compressive strength are basic data to evaluate the risk of sand production. Evaluations of the compressive strength based on logs are illustrated for the example of a well on the Germigny-sous-Coulombs structure. Then relationships between compressive strength and porosity are developed using a theoretical approach of grain contacts, the analysis of published rock mechanics data and mechanical measurements on plugs taken from well cores. It is shown that compaction factor can complement porosity in the analysis. A field example illustrates the relationship between a critical differential pressure and compaction factor or porosity. Introduction The general frame of this study is the production of oil and gas from poorly consolidated formations. It is well known that such a production can be hindered by the phenomenon of sand production. Various gravel-pack equipment can be used to prevent sand production but they are costly and generally harmful to well productivity. Therefore the decision to use such equipment has considerable economic impact and requires a clear-sighted evaluation of the risk of sand production. Tensile rupture and compressive rupture are proposed as possible mechanisms of sand failure. Tensile rupture is possible under two conditions:the fluid pressure gradient at the production face is larger than the gradient of the radial stress,the tangential effective stress does not exceed the level of compressive failure (less than ucs). As fluid pressure and tangential effective stress are linked through the equilibrium equation of the sand, conditions (a) and (b) impose an upper limitp max to the pressure difference Pd - Pw through the zone drained by a perforation. p max is proportional toucs and various values of the ratio p max/ ucs can be found depending on the drainage geometry (radial, spherical) and on the production history (influence of shut-in periods). Compressive rupture is also possible if, under symmetrical conditions: the pressure gradient remains smaller than the gradient of the radial stress, the effective tangential stress reaches theucs critical level. To these conditions corresponds a maximum of the total depletion ptd which depends linearly on ucs. The field observations apparently confirm the general trends of these theoretical analyses. For example satisfactory production conditions are obtained if the drawdown pressure Pdd is maintained under 0.5 ucs. Therefore using theoretical modelling and field observations two categories of data appear to be very important: pressure data characterizing the production history and the fluid flow as well as data characterizing the uniaxial compressive strength of the rock. P. 381^