*By Bernt S. Aadnoy, University of Stavanger, and Mesfin Belayneh, International Research Institute of Stavanger*

The fracture equation used in the oil and gas industry is derived from the Kirsch equation for the hoop stress. Due to its simplicity, it is almost exclusively used for prediction of fracture initiation pressures. However, it is not useful for analysis of load history.

An analytic study was undertaken to model load history leading to fracturing of the borehole. To use the model, initial conditions must be established, given by the virgin in-situ stress state and the pore pressure, followed by the load history and the temperature history. Imposing a volumetric strain balance, a new fracturing equation is developed. Because the borehole is loaded in the radial direction, causing tension in the tangential direction, a Poisson’s effect arises. In addition, the general solution includes effects of temperature history.

Example cases will show the improvement with the new model. The first case compares the new load-history fracture model with the Kirsch solution. The Poisson’s scaling factor in the new solution leads to a higher fracture pressure than the conventional solution. This may explain some of the discrepancy between models and field data. The second case investigates the thermal effects by comparing the fracture pressure for the drilling phase with a hot production phase and a cold water injection phase.

It is believed that by including the pressure and temperature load history, a better assessment of the fracture strength is obtained, leading to better predictions.

Introduction Basis for the model

Although the Kirch solution for stresses in a circular hole was published more than 100 years ago, it was not until 1980 that borehole mechanics started being applied to petroleum drilling. At that time, deviated wells were evolving, and due to the complexity, high inclination and increasing length of these, borehole stability was identified as a critical factor. Bradley (1979) is considered to be the person who introduced application of classical mechanics into the petroleum industry, by analysing borehole fracturing and collapse in deviated boreholes. Later, Aadnoy and Chenevert (1987) presented the mathematical framework and elaborated on the applications. These early works still form the basis for modern wellbore stability analyses.

Continuing work the past two decades have led to many contributions to the classical solutions. A full review will not be given here, but Fjaer et al (1992) serves as a good general reference. More recently, Aadnoy and Belayneh (2004) have shown that the boundary condition given by the drilling fluid can be better represented as an elasto-plastic barrier.

However, in this article we will apply the standard boundary condition by assuming that the fluid pressure on the borehole wall is given by a step function. The temperature effects are identified as having effect on the fracture pressure. Examples are given by Maury and Sauzay (1987) and Maury and Guenot (1995). More recent work by Gil et al (2006) considers the temperature issues from a poroelastic and “stress cage” perspective.

The variable in the solution is the borehole pressure, which causes the borehole to be loaded in the radial direction. However, the solution used today does not include the full Poisson’s effect, i.e., the effects in the tangential and axial direction of a radial loading. The objective of this article is to include this effect. In the following, we will present the resulting expression for the new model, which is derived in the Appendix.

Effects of Poisson’s ratio

During fracturing, a change in stresses occur at the borehole. The local stress field is affected in three dimensions. This implies a coupling between the stresses. This coupling is taken into account with the Poisson’s ratio.

The starting assumption is that there exists a principal stress state in the rock before the hole is drilled. If the borehole pressure is equal to the in-situ stress state, the near wellbore stress state is still principal (Aadnoy, 1996). Lowering or increasing the mud pressure from this stress level results in Poisson’s effect on the stresses. Assume a principal stress state consisting of σv, σh and σH. As shown in the appendix, the fracturing pressure from the linear elastic solution is then given by:

(1)

Equation 1 is similar to the so-called Kirsch solution that is commonly used in rock mechanics, except for the scaling factor in front. We will investigate this factor before proceeding. Let:

(2)

Figure 1 shows the magnitude of the Poisson’s effect. A typical value of Poisson’s ratio is 0.25. For this value, the scaling factor is C=0.605. This implies that a different fracturing pressure results if the Poisson’s effect is taken into account. The limiting value is C=1 at zero Poisson’s ratio. This is actually the result used by the oil industry today.

Effect of temperature on the fracturing pressure

Also derived in the appendix is the fracturing model, which includes temperature effects. If the borehole is heated or cooled, the fracturing pressure will change because of hoop stress change due to expansion or contraction.

The temperature effect on the fracturing equation can be expressed as:

(3)

Where K is a scaling factor that is given by the Poisson’s effect (see the Appendix).

(4)

Initial conditions and history matching

The Kirsch equation has been used with no concern to load path because the previously defined effects have not been taken into account. However, to perform load history analysis, the initial conditions must be established.

**Initial conditions**

We assume that a principal stress state exists in the formation before the hole is drilled. If the direction of the well deviates from this direction, the stresses must be transformed in space. During fracturing, the Poisson’s effect is effective only for the stress magnitude deviating from the principal stress state.

Assuming a vertical hole with the following in-situ stress state: σH, σh and σv, where the last is largest. For this case, a fracture will arise in the direction of σH. The in-situ stress acting normal to this direction is σh , which defines the initial condition. The fracture pressure is the in-situ stress plus the loading above the in-situ stress, which includes the Poisson’s effect.

Isotropic stress loading

If there exists an isotropic loading around the borehole (equal normal stresses on the borehole wall), the loading is simple. The initial stress condition is simply equal to the in-situ stress that existed before the hole was drilled, σ. The loading towards fracturing is this reference plus the hoop stress, including the Poisson’s effect until the fracture pressure is reached. With reference to the appendix, the fracturing pressure becomes:

(5)

Anisotropic stress loading

For this case, the two normal stresses on the borehole wall have different magnitudes. Assuming a vertical hole, these two stresses may be defined as σH and σh, defined as the maximum and minimum horizontal stresses. Because the borehole is filled with a fluid, both of these cannot be the initial condition simultaneously. At the position of fracture initiation, the initial stress state is σH. Choosing this as the initial state, the fracturing equation becomes:

(6)

Initial temperature conditions

Several publications address temperature effects on the fracture pressure, such as Fjaer et al (1992) and Gil et al (2006). The general equation for the change in stress due to temperature is:

The Appendix derives a fracture equation where the Poisson’s effect caused by radial loading is included. This solution is similar to the solution above, except that the scaling term is different.

Assume that there is an initial temperature, which exists at the in-situ stress conditions, Tinit. Any change in temperature from this value may create changed hoop stress and, hence, changed fracture pressure.

The correction term becomes (see the Appendix):

(7)

The complete model for history matching

The general fracturing model for arbitrary wellbore orientation is similar to the equations above except that the in-situ stresses should be transformed in space, now referred to the x,y coordinate system. The general fracturing equation becomes:

(8)

Here, σx is the least normal stress acting on the borehole.

Evaluation of the new model

In the following, several examples will be presented to demonstrate the significance of including the Poisson’s effect and the temperature effect. It is also observed that the new equations are simple to use.

Case 1: Comparison with the Kirsch model

Assuming normal stresses equal to 1.4 sg and a pore pressure of 1.2 sg, the fracture pressure is for the classical Kirsch equation (temperature effects are not included):

Assuming a Poisson’s ratio of 0.25, which is typical for rocks, and using the new model, Equation 8, the fracture pressure becomes:

This example shows that the classical Kirsch equation severely underpredicts the fracturing pressure and that the Poisson’s effect is significant.

Case 2: Comparison of cold water injection and hot gas injection

Typical water-alternating gas (WAG) wells are often injected with cold water over a period of time. When the gas cyclus is applied, the temperature rises because gas heats up when it is pressurized through the gas compressors. We will use the data from Example 1 to investigate changes in fracture pressures during these two scenarios.

The virgin well temperature at reservoir level is 80ºC at 2,000 m depth. During cold water injection over months, the bottomhole temperature approaches 30ºC. When gas is injected at a later stage, a temperature of 120ºC results. The following data are required to analyze these two scenarios.

Table 1 gives the data for the case. We choose to solve the problem in units of specific gravity (sg). The elastic modulus of the sandstone rock at 2,000 m depth is then equivalent to:

E-modulus in sg =

For the first case, the wellbore is heated from 80ºC to 120ºC. The fracture gradient will then increase to:

The second case where cold water is injected over a prolonged period of time leads to cooling of the wellbore. This will increase the tensile hoop stress and lead to a reduced fracture initiation pressure, which becomes:

Summary

A new model for fracture initiation is presented in this article. It is based on the classical Kirsch equation, but it includes the Poisson’s effect that arises when the borehole is pressurized from one stress state to another. The model also includes thermal effects to account for temperature changes in the borehole. As opposed to the classical Kirsch model, the new model starts with the initial in-situ stress and the virgin in-situ temperature. The mechanical and thermal loading towards fracturing is therefore modeled from this initial state.

Numerical examples show that if the Poisson’s effect is neglected, the fracture pressure is severely underpredicted. Examples of thermal effects are also presented with one case of hot gas injection followed by one case of cold water injection. Heating increases the fracturing pressure whereas cooling decreases the fracturing initiation pressure.

Appendix: Poisson’s Effect

The stresses around a wellbore are governed by Hooke’s law, equilibrium equations and compatibility equations. Figure A1 illustrates a borehole with radial and tangential stresses. In the analysis to follow, we use effective stresses for porous media, which are defined as total stresses minus the pore pressure.

Assuming plane strain condition (εz = 0), the strain in terms of effective stress and temperature is given as (Boresi and Lynn, 1974):

(A1)

(A2)

where E = Young’s modulus, ν is Poisson’s ratio and κ is the coefficient of linear thermal expansion.

**Well deformation**

During pressure loading of the borehole, volumetric deformation of the well takes place. The final volume is reached when the well pressure approaches the fracturing pressure. The effective differential pressure is the difference between the well fracturing pressure and the pore pressure ΔP = (Pwf – Po), which is equivalent to the effective radial stress. This pressure is related to the volumetric strain as:

(A3)

Vo is the volume of the well before deformation, and K is the bulk modulus.

Because the borehole is expanding, the Poisson’s ratio is given as:

(A4)

Assuming plane strain conditions and inserting Equation A2 and A4 into Equation A3:

(A5)

where

The result is the coupling between well pressure and the effective borehole stresses:

(A6)

From Aadnoy and Chenevert (1987), the tangential stress in the direction of a fracture is (assuming vertical well):

(A7)

For penetrating situations at the wellbore, ΔP = Pwf – Po = 0, and Equation A6 becomes:

(A8)

For the non-penetrating case, Pwf > Po

(A9)

For equal normal stresses on the borehole wall:

(A10)

It is observed that if Poisson’s ratio is set equal to zero (and in the absence of temperature effect), Equation A10 reduces to:

which is the solution currently in use in the petroleum industry.

Figure 1 presents the magnitude of the scaling factors as a function of Poisson’s ratio. Table A1 is provided to obtain the specific numerical values.

*IADC/SPE 114829, “A New Fracture Model That Includes Load History, Temperature and Poisson’s Effects,” was presented at the IADC/SPE Asia Pacific Drilling Technology Conference and Exhibition, held in Jakarta, Indonesia, 25-27 August 2008.*

*References:*

*Aadnoy, B.S. and Chenevert, M.E., 1987. Stability of highly inclined boreholes. SPE Drilling Engineering, Vol.2, No. 4, 364-374.*

*Aadnoy, B.S., 1996: Modern Well Design, A.A. Balkema, Rotterdam/Brookfield 1996, ISBN 90 5410 633 6.*

*Aadnoy, B.S. and Belayneh, M., 2004: Elasto-plastic fracturing model for wellbore stability using non-penetrating fluids. Journal of Petroleum Science and Engineering 45 (2004) 179-192.*

*Boresi, A.P. and Lynn, P.P., 1974: Elasticity in Engineering Mechanics, Prentice-Hall, Inc, Englewoods Cliffs, New Jersey, ISBN 0-13-247080-2.*

*Bradley, W.B., 1979: Failure of inclined holes.J. Energy Res. Tech., Trans. Aime, Vol. 102, 232-239.*

*Fjaer, E., Holt, R.M., Horsrud, P., Raaen, A.M. and Risnes, R, 1992: Petroleum related rock mechanics. Developments in Petroleum Science, 33, Elsevier, Amsterdam, the Netherlands. ISBN 0-444-88913-2.*

*Gil, I., Roegiers, J.C. and Moos, D., 2006: Wellbore cooling as a means to permanently increase fracture gradient. Paper SPE 103256 presented at the 2006 Ann. Tech. Conf. And Exhibition, San Antonio, 24-27 September 2006.*

*Maury, V.M. and Sauzay, J.M., 1987: Borehole instability; Case histories, rock mechanics approach and results. Paper SPE/IADC 16051 presented at the 1987 SPE/IADC Drilling Conference, New orleans, LA, March 15-18, 1987.*

*Maury, V. and Guenot, A., 1995: Practical advantages of mud cooling systems for drilling. SPE Drilling and Completions, March 1995, p. 42-48.*