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Fracture mechanics

Dealing with and exploiting fracturing of rock has been part of mining engineering for hundreds of years, but the analysis of fracture of rock or other materials has only developed into an engineering discipline since the mid 1940s . In petroleum engineering, fracture mechanics theories have been used for more than 50 years. Rock fracture mechanics is about understanding what will happen to the rocks in the subsurface when subjected to fracture stress. Much of what is used in hydraulic fracturing theory and design was developed by other engineering disciplines many years ago. However, rock formatons cannot often be treated as isotropic and homogeneous. For example, their porous and fluid filled nature can require that poroelastic theory be used for some problems. There are a number of important parameters to consider in the fracturing of rock. Some of these are fracture toughness, in situ stress, Poisson’s ratio, and Young’s modulus.

In-situ stresses

Underground formations are confined and under stress. Fig. 1 illustrates the local stress state at depth for an element of formation. The stresses can be divided into three principal stresses. In Fig. 1:

• σ1 is the vertical stress
• σ2 is the minimum horizontal stress
• σ3 is the maximum horizontal stress

These stresses are normally compressive, anisotropic, and nonhomogeneous, which means that the compressive stresses on the rock are not equal and vary in magnitude on the basis of direction. The magnitude and direction of the principal stresses are important because they control the pressure required to create and propagate a fracture, the shape and vertical extent of the fracture, the direction of the fracture, and the stresses trying to crush and/or embed the propping agent during production.

A hydraulic fracture will propagate perpendicular to the minimum principal stress. For a vertical fracture, the minimum horizontal stress can be estimated with ....................(1)

where

σmin = the minimum horizontal stress,

ν = Poisson ’ s ratio, σ1 = overburden stress, α = Biot’ s constant, pp = reservoir fluid pressure or pore pressure, and σext = tectonic stress. Poisson ’ s ratio can be estimated from acoustic log data or from correlations based on lithology. Table 1 presents typical ranges for Poisson’ s ratio. The overburden stress can be computed with density log data. Normally, the value for overburden stress is approximately 1 psi/ft of depth. The reservoir pressure must be measured or estimated. Biot’

s constant is usually 1.0, but can be less than 1.0 on occasion.

Poroelastic theory is often used to estimate the minimum horizontal stress. Eq. 1 combines poroelastic theory with a term that accounts for any tectonic forces that are acting on a formation. The first term on the right side of Eq. 1 is a linear elastic term that converts the effective vertical stress on the rock grains into an effective horizontal stress on the rock grains. The second term in Eq. 1 represents the stress generated by the fluid pressure in the pore space. The third term is the tectonic stress, which could be zero in tectonically relaxed areas, but can be important in tectonically active areas.

In tectonically active areas, the effects of tectonic activity must be included in the analyses of the total stresses. To measure the tectonic stresses, injection tests are conducted to measure the minimum horizontal stress. The measured stress is then compared with the stress calculated by the poroelastic equation to determine the value of the tectonic stress.

Basic rock mechanics

In addition to the in-situ or minimum horizontal stress, other rock mechanical properties are important when designing a hydraulic fracture. Poisson ’ s ratio is defined as "the ratio of lateral expansion to longitudinal contraction for a rock under a uniaxial stress condition." The value of Poisson’ s ratio is used in Eq. 1 to convert the effective vertical stress component into an effective horizontal stress component. The effective stress is defined as the total stress minus the pore pressure. The theory used to compute fracture dimensions is based on linear elasticity. When applying this theory, the modulus of the formation is an important parameter. Young’ s modulus is defined as "the ratio of stress to strain for uniaxial stress." The modulus of a material is a measure of the stiffness of the material. If the modulus is large, the material is stiff. In hydraulic fracturing, a stiff rock results in more narrow fractures. If the modulus is low, the fractures are wider. The modulus of a rock is a function of the lithology, porosity, fluid type, and other variables. Table 1 illustrates typical ranges for modulus as a function of lithology.

Fracture orientation

A hydraulic fracture will propagate perpendicular to the least principle stress (see Fig. 1). In some shallow formations, the least principal stress is the overburden stress; thus, the hydraulic fracture will be horizontal. Horizontal fractures have been documented. In reservoirs deeper than approximately 1,000 ft, the least principal stress will likely be horizontal; thus, the hydraulic fracture will be vertical. The azimuth orientation of the vertical fracture will depend on the azimuth of the minimum and maximum horizontal stresses. Lacy and Smith provided a detailed discussion of fracture azimuth in Ref. 7.

Injection tests

The only reliable technique for measuring in-situ stress is by pumping fluid into a reservoir, creating a fracture, and measuring the pressure at which the fracture closes. The well tests used to measure the minimum principal stress are:

• In-situ stress tests
• Step-rate/flowback tests
• Minifracture tests
• Step-down tests

For most fracture treatments, minifracture tests and step-down tests are pumped ahead of the main fracture treatment. As such, accurate data are normally available to calibrate and interpret the pressures measured during a fracture treatment. In-situ stress tests and step-rate/flowback tests are not run on every well; however, it is common to run such tests in new fields or new reservoirs to help develop the correlations required to optimize fracture treatments for subsequent wells.

In-situ stress tests

An in-situ stress test can be either an injection-falloff test or an injection-flowback test. The in-situ stress test is conducted with small volumes of fluid (a few barrels) and injected at a low injection rate (tens of gal/min), normally with straddle packers to minimize wellbore storage effects, into a small number of perforations (1 to 2 ft). The objective is to pump a thin fluid (water or nitrogen) at a rate just sufficient to create a small fracture. Once the fracture is open, the pumps are shut down, and the pressure is recorded and analyzed to determine when the fracture closes. Thus, the term "fracture-closure pressure" is synonymous with minimum in-situ stress and minimum horizontal stress. When the pressure in the fracture is greater than the fracture-closure pressure, the fracture is open. When the pressure in the fracture is less than the fracture-closure pressure, the fracture is closed. Fig. 2 illustrates a typical wellbore configuration for conducting an in-situ stress test. Fig. 3 shows typical data that are measured. Multiple tests are conducted to ensure repeatability. The data from any one of the injection-falloff tests can be analyzed to determine when the fracture closes. Fig. 4 illustrates how one such test can be analyzed to determine in-situ stress.

Minifracture tests

Minifracture tests are run to reconfirm the value of in-situ stress in the pay zone and to estimate the fluid-loss properties of the fracture fluid. A minifracture test is run with fluid similar to the fracture fluid that will be used in the main treatment. Several hundred barrels of fracturing fluid are pumped at fracturing rates. The purpose of the injection is to create a fracture that will be of similar height to the one created during the main fracture treatment. After the minifracture has been created, the pumps are shut down, and the pressure decline is monitored. The pressure decline can be used to estimate the fracture-closure pressure and the total fluid leakoff coefficient. Data from minifracture treatments can be used to alter the design of the main fracture treatment, if required.

Step-down tests

For any injection-falloff test to be conducted successfully, a clean connection between the wellbore and the created fracture is needed. The main objective of an in-situ stress test and the minifracture test is to determine the pressure in the fracture when the fracture is open and the pressure when the fracture is closed. If there is excess pressure drop near the wellbore because of poor connectivity between the wellbore and the fracture, the interpretation of in-situ stress test data can be difficult. In naturally fractured or highly cleated formations, multiple fractures that follow tortuous paths are often created during injection tests. When these tortuous paths are created, the pressure drop in the "near-wellbore" region can be very high, which complicates the analyses of the pressure falloff data. To determine the cause of near-wellbore pressure drop, step-down tests are run.

A step-down test is pumped just before the minifracture treatment. A step-down test is pumped at fracturing rates with linear fluids, the friction pressures of which are well known. The pressure at the bottom of the hole during the injection is a function of the net pressure in the fracture and the near-wellbore pressure drop. To measure the near-wellbore pressure drop, the net pressure in the fracture needs to be relatively constant during the step-down portion of the test. To do this, the step-down test is started by injecting into the well for 10 to 15 minutes. Experience has shown that, in most cases, the net pressure is relatively stable after approximately 10 to 15 minutes of injection. The injection rate is then "reduced in steps" to a rate of zero. The injection rate at each step should be held constant for approximately 1 minute so the stabilized injection pressure can be measured. The injection rate should be stepped from the maximum value to zero, in three to five steps, in less than 5 minutes. The objective of the step-down test is to measure the near-wellbore pressure drop as a function of injection rate. If the net pressure in the fracture is relatively stable, then the change in bottomhole injection pressure as the injection rate is reduced will be a function of the near-wellbore pressure drop.

The key to analyzing a step-down test is that the two main causes of near-wellbore pressure drop can be distinguished easily as the data are analyzed. When the pressure drop near the wellbore is caused by perforation friction, the near-wellbore pressure drop will be a function of the injection rate squared, as Eq. 2 shows. ....................(2)

If the near-wellbore pressure drop is caused by tortuosity, then the near-wellbore pressure drop will be a function of the injection rate raised to a power of one-half (0.5), as Eq. 3 shows. ....................(3)

A graph of the value of near-wellbore pressure drop vs. injection rate will provide a clear indication of what is causing the near-wellbore pressure drop. Fig. 5 illustrates that the graph of pressure drop vs. injection rate will be concave upward when the pressure drop is dominated by tortuosity and will be concave downward when the pressure drop is dominated by perforation friction.

Net pressure

The reason for computing values of in-situ stress and conducting stress tests, minifracture tests, and step-down tests is to compute the net pressure in the fracture. The net pressure is the difference between the actual pressure in the fracture and the minimum in-situ stress, σmin. ....................(4)

The net pressure is generated by both tip effects and the pressure drop down the fracture caused by viscous fluid flow. Fig. 6 illustrates the net pressure profile down a typical fracture. In many formations, the pressure drop down the fracture is dominated by the pressure increases near the tip of the fracture as propagation occurs. The net pressure profile controls both the fracture height and fracture width distribution along the fracture length.

The value of net pressure is important because the engineer needs to know for which value to design the main fracture treatment, to perform onsite analyses of the fracturing pressures, and to perform postfracture analyses of the fracturing pressures. One of the best methods to analyze a fracture treatment is to use a fracture propagation model to analyze the net pressures measured during a fracture treatment.

Nomenclature

 a = constant (solved for) dp = proppant diameter, L pf = actual pressure in the fracture, m/Lt2 pn = net pressure, m/Lt2 pp = pore pressure (reservoir pressure), m/Lt2 ppfr = perforation friction, psi Q = injection rate, L3/t ν = Poisson’ s ratio σmin = minimum horizontal stress (in-situ stress), m/Lt2 σ1 = vertical (overburden) stress, m/Lt2 α = discharge coefficient, usually 0.9