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2 SPE 163034

(2001) suggested that some geological and operational parameters such as well trajectory and rheological properties of drilling fluids can affect wellbore ballooning. Sanfillipo et al. (1997) approached this phenomenon by using diffusivity equation; they considered a non-deformable fracture with a constant aperture and a Newtonian drilling fluid. Verga et al. (2000) developed a model by using diffusivity equation. Using a non-Newtonian fluid as their drilling mud, they emphasized on the importance of the fluid rheology on the result of their model. Lavrov and Tronvoll (2004, 2005, and 2006) also worked on this phenomenon and published three publications. Two of their publications were based on circular fractures and one of them modelled fracture ballooning through a rectangular fracture; however, exponential pressure-aperture deformation law was not considered along with yield-power-law rheology at one publication. Majidi et al. (2008a) developed some theoretical frameworks about fracture ballooning. They developed a new model based on yield-power-law rheology and radial flow through a constant aperture fracture with an unlimited extension. They showed that when the pressure gradients drop below a threshold level, drilling fluid cannot penetrate into the fracture. They showed that the greater the yield value the sooner the loss will stop. In another study Majidi et al. (2008b) considered a horizontal disk-shaped fracture located in an impermeable rock. They assumed that the fracture is initially saturated with liquid under the static formation pressurep ; all properties of the drilling fluid were assumed to be identical to the fluid that was already in the fracture and both were described by yield-power-law rheology. The fracture wall deformation was described by a linear aperture-pressure deformation law. Majidi et al. (2008c) finally developed a new model by considering formation fluid as a Newtonian fluid and drilling fluid as a non-Newtonian fluid flowing through an infinitely long disk-shaped fracture; but this time the fracture is not deformable and matrix rock was also assumed to be so tight that the leak-off through the fracture wall can be neglected. Ozdemirtas et al. (2007) considered the planar, horizontal, square-shaped fracture in a non-permeable formation. The linear deformation law was assumed for the fracture, and the drilling and formation fluid were both considered as a Newtonian fluid. The emphasis was on the fracture roughness effect.

To the best of the author’s knowledge the latest published paper pertaining to fracture ballooning was developed by Pordel Shahri et al. (2011) in which they analyzed the effect of different pressure-aperture deformation laws on the mud loss rate and cumulative loss.

In this work a deformable non-propagating circular fracture with non-permeable walls is considered. Formation fluid has the same rheology as the drilling fluid, and the fracture wall deforms while pressurizing/depressurizing under the exponential pressure-aperture deformation law. Yield-power-law rheology is considered for formation and drilling fluids. Smooth fracture surface is assumed for the single fracture. The aim of this study is to investigate the sensitivity of fracture ballooning to some parameters which are mainly associated with the mechanical properties of the fracture.

2. Mathematical Modeling

2.1. Mathematical Description A horizontal disk-shaped fracture with a radius of r is considered. The fracture is initially filled with a liquid under the static formation pressure P . Neither fracture plugging by mud particles nor fracture wall permeability are considered. The formation fluid and the drilling fluid are assumed to have the same properties; both are considered incompressible fluid with yield-power-law rheology. A constant initial aperture is considered along the fracture which implies that the fracture surfaces are smooth. Fracture aperture in a given point is assumed to be an exponential function of the fluid pressure within the fracture in that point. Leak-off through the fracture walls is neglected (which is not an unreasonable assumption in carbonate reservoir). At t 0 a vertical well of radius r hits the circular fracture in the center and thus the pressure at r r increases from P to the borehole pressure,P , during a short time interval ,t , and the drilling fluid propagates through the single circular fracture. In this study t is considered equal to zero for most calculations.

2.2. Fluid Rheology In this study drilling and formation fluids are modelled by yield-power-law rheology. The equation which relates shear stress to shear rate for fluids with yield-power-law rheology is as follows:

τ τ k γ 1 Where τ is the yield-power stress, k is consistency factor, and m is flowbehaviour index of the fluid.

2.3. Deformation Law for Fracture Wall

The main mechanism under which fracture ballooning occurs is the deformation of fracture walls. Two empirical correlations can be used to model the relation between fracture aperture and pressure inside the fracture: a linear deformation law, and an exponential deformation law. In this study exponential deformation law is considered; this empirical law was proposed by Bruel et al. (1994) describing the fracture aperture at a given

SPE 163034 3

point changes as an exponential function of fluid pressure at that point:

w w e 2

Where w is the fracture aperture in a particular point, p is the fracture pressure in that point, σ is the normal stress, w is the fracture aperture when p σ , and β is an empirical coefficient (compressibility of the fracture). Effect of some of these parameters on fracture ballooning will be analyzed at the end of this section.

2.4. Governing Equation Making use of conservation of mass and Reynolds’s lubrication theory for radial flow of an incompressible fluid in a deformable fracture in absence of leak-off through fracture walls, the following equation is derived:

∂ wv∂r

1rwv

∂w∂t

0 3

Where r is the distance from borehole center in the fracture plane z 0 and v is the average radial velocity of the fluid in the fracture.

Average radial velocity for yield-power-law drilling fluid at a distance r from the borehole axis can be derived using the momentum conservation equation in cylindrical coordinate.

vm

2m 1w

2 k

∂p∂r

2m 1m 1

2τw

4

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m2m 1

1

2 k

∂∂r

w∂p∂r

2m 1m 1

2τ

w

m2m 1

1

2 kw

1r∂p∂r

2m 1m 1

2τ

w∂w∂t

5

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m2m 1

w

2 k

∂∂r

e∂p∂r

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6

2.5. Mud Loss/Gain Flow Rate Mud loss flow rate at radius r is obtained by multiplication of the area (2πrw) and the average velocity at that radius:

Q Av 2πrwv 7 Using wellbore radius and entry fracture aperture, the mud flow rate at the wellbore at each time step is calculated from the following equation:

Q 2πr w v 8

2.6. Cumulative Loss Cumulative loss is the total amount of mud loss within the fracture from the initial time, t 0, to the final time, t t .

V Q dt 9

2.7. Boundary Conditions Fracture walls are considered impermeable and thus the fracture tip acts like a no-flow boundary.

∂p∂r

0 at r r 10

Vertical borehole intersects the circular fracture at the center and pressure at the borehole wall r r increases in a short period of time, t , from P to P . In this study a zero value for t is considered for most calculations. However, for simulating kick situation and comparing it with loss situation when borehole hits the fracture for the first time, and also investigating the effect of t on the amount and rate of loss another boundary condition is considered.

p pp p t

t0 t t

p p t t 11

This boundary condition is useful for modelling kick condition in fractured formations. Effect of t will be analyzed in sensitivity analysis section.

2.8. Initial Condition Initially, pressure within the fracture is uniform and equal to formation pressure P .

4 SPE 163034

3. Sensitivity Analysis

3.1. Pressure distribution Fig. 1 illustrates pressure distribution within the fracture at different times for the model simulated with default values listed in Table A.1. The figure clearly shows how the pressure within the fracture approaches the borehole pressure as time elapses.

3.2. Effect of β is an empirical coefficient associated with the fracture normal compressibility (7). It depends on the structure and the morphology of the fracture surfaces. According to Eq. 2 , it is expected that opening and closing of the fracture take place easier for higher β values, than they do for lower ones. This is depicted in Fig. 2a. The figure shows that the changes in fracture aperture due to changes in pressure are larger for a fracture with higher compressibility coefficient than a fracture with lower compressibility coefficient. This will result in more conductivity of the fracture and thus more mud loss flow rate (Fig. 2b) and cumulative mud loss (Fig. 2c).

3.3. Effect of Effect of fracture radius is illustrated in Figs.3a and 3b. The figures show that fracture radius does not affect the initial part of the mud loss curves, while it significantly affects the end of these curves. The figures also reveal that the larger the radius, the larger the cumulative loss and mud loss flow rate.

3.4. Effect of Effect of wellbore radius, , on the mud loss rate is shown in Fig. 4a and Fig. 4b. values influence the beginning of the loss rate curve but as time proceeds the mud loss rate curves for all values converge to zero. According to Eq. 8, a fracture with larger wellbore radius will undergo larger initial mud invasion. This will, in turn, result in faster pressurizing of the fracture and thus a sharper decline in the mud loss rate. Thus after a while, the mud loss rate in the fracture with larger wellbore radius drops below the fracture with smaller wellbore radius as shown in Fig. 4a.

3.5. Effect of σ is the normal stress acting perpendicular to the fracture plane and it virtually shows how much internal pressure (i.e., fluid pressure within in the fracture) the fracture can withstand without the risk of fracture propagation. Considering a non-propagating fracture implies that fluid pressure inside the fracture must not exceed σ . Effect of σ on fracture aperture is demonstrated in Fig.5a. The figure demonstrates that the fracture aperture gets larger as normal stress gets smaller. Effect of σ on mud loss flow rate is shown in Fig.5b.

Fractures with smaller σ have larger apertures (Fig.5a) and thus undergo larger initial mud invasion and faster loss rate decline, as shown in Fig.5b. Cumulative losses are also affected by σ ; the smaller the normal stress, the larger the cumulative loss, as shown in Fig.5c. This can also be explained by the larger fracture aperture which occurs in smaller normal stress values.

3.6. Effect of w is the maximum attainable fracture width i.e. the fracture width when the fluid pressure within the deformable non-propagating fracture is equal to the normal stress σ . According to Eq. (2), larger w yields larger fracture apertures. It should be noted that this larger aperture is not due to an elevated formation fluid pressure. Similar to the effect of wellbore radius, a fracture with larger w initially undergoes a larger mud invasion, but then due to the sharper decline in mud loss rate, mud loss curve of a fracture with higher w drops below the mud loss curve of a fracture with smaller w (Fig. 6a). Effect of w on the cumulative loss is also demonstrated in Fig. 6b.

3.7. Effect of t is the duration of pressure increase or decrease. During this time interval fracture pressure increases from P to the wellbore pressure P , or in a kick situation it is the time interval during which the fracture pressure decreases from P to the initial fracture pressure, P . Fig.7 shows the result of the simulation for a non-zero t value, where a vertical well hits a disk-shaped fracture at the center at t 0 and the pressure within the fracture is increased and then remained constant. The second half of this figure is used to model a kick situation in which a rapid influx of fracture fluid happens when the overbalance pressure due to the mud circulation is removed and thus a pressure drop occurs along the fracture fluid. Effect of t on modeling of a kick situation is demonstrated in Fig. 7a to Fig. 7c. According to this simulation, considering larger times for pressure drop/rise intervals affects the duration of mud loss or kick before stabilization.

4. Conclusions A mathematical model which describes the physics of borehole ballooning is developed. The mechanism under which opening/closing of fractures of a naturally fractured reservoir occur for a circular non-deformable single fracture is simulated. The following conclusions can be drawn from the sensitivity analysis that has been carried out:

Larger compressibility coefficient, β, results in larger mud loss rate and greater cumulative loss

SPE 163034 5

Increasing the fracture radius increases the cumulative loss Increasing the borehole radius increases the rate of the mud loss at initial times and the rate of mud loss rate decline Larger normal stress, σ , results in higher cumulative loss The maximum attainable fracture aperture,w ,can largely influence the rate of the mud loss and the cumulative loss t affects the duration of pressure stabilization. Increasing t increases the time under which mud loss event or influx of formation fluid takes place. This may also increases the time needed for the phenomenon to be observed on the surface.

A shortcoming of this model is that a smooth fracture surface is considered, a further development in fracture surface modelling would be attainable by introducing a surface based on fractal geometry. Further improvements of the model include: considering a permeable wall around the fracture, modelling the plugging phenomena by particles contained in drilling fluid, modelling formation of a mud cake, and finally developing a model for fracture networks which is more realistic than a single fracture model.

Nomenclature Symbol Parameter Unit

Empirical coefficient of fracture normal compressibility

[ ]

Shear rate 1/

Drilling mud yield stress [ ] Normal stress [ ]

Cross sectional area of fracture at radius

Consistency factor [ . ] Flow behaviour index Formation fluid pressure [ ] Borehole pressure [ ]

Mud loss rate [ ⁄ ]

Radial distance to well Wellbore radius [ ] Fracture radius [ ]

Time Final time [ ] Duration of pressure

increase/decrease [ ]

Cumulative loss [ ] Local average radial velocity

of the fluid in the fracture /

Average radial velocity of the fluid at

/

Local fracture aperture Initial fracture aperture

Maximum attainable fracture aperture Fracture aperture at the wellbore

References 1. Bruel, D., Cacas, M.C., Ledoux, E., and de

Marsily, G. (1994) “Modelling Storage Behaviour in a Fractured Rock Mass”, J. Hydrology 162, 267.

2. Gill, J.A. (1989) “How Borehole Ballooning Alters Drilling Responses”, Oil & Gas J., 87, 43.

3. Lavrov, A., and Tronvoll, J., (2004) “Modelling Mud Loss in Fractured Formations” Paper SPE 88700 presented at the 11thAbu Dhabi International Petroleum Exhibition and Conference held in Abu Dhabi, UAE.

4. Lavrov, A., and Tronvoll, J., (2005) “Mechanics of Borehole Ballooning in Naturally-Fractured Formations” Paper SPE 93747 presented at the 14th SPE Middle East Oil & Gas show and Conference held in Bahrain International Exhibition Centre, Bahrain.

5. Lavrov, A., and Tronvoll, J. (2006) “Numerical Analysis of Radial Flow in a Natural Fracture: Applications in Drilling Performance and Reservoir Characterization” Paper SPE 103564 presented at the Abu Dhabi International Petroleum Exhibition and Conference held in Abu Dhabi U.A.E.

6. Majidi, R., Miska, S.Z., Yu, M. and Thompson, L.G., (2008a) “Quantitative Analysis of Mud Losses in Naturally Fractured Reservoirs: The Effect of Rheology” Paper SPE 114130 presented at the 2008 SPE Western Regional and Pacific Section AAPG Joint Meeting held in Bakersfield, California, U.S.A.

7. Majidi, R., Miska, S.Z., Yu, M. and Thompson, L.G., (2008b) “Fracture Ballooning in Naturally Fractured Formations: Mechanism and Controlling Factors” Paper SPE 115526 presented at the 2008 SPE Annual Technical Conference and Exhibition held in Denver, Colorado, USA.

8. Majidi, R., Miska, S.Z., Yu, M. and Thompson, L.G., (2008c) “Modelling of Drilling Fluid losses in Naturally Fractured Formations” Paper SPE 114630 presented at the 2008 SPE Annual Technical Conference and Exhibition held in Denver, Colorado, USA.

9. Pordel Shahri, M., Zeyghami, M., and Majidi, R. (2011) “Investigation of Fracture Ballooning and Breathing in

6

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