Fracturing of a wellbore and 2D fracture models

Introduction

The hydraulic fracturing process has been employed to enhance the production of oil and gas from underground reservoirs for many decades. In the process, the frac-fluid is pumped at a high pressure into a selected section of wellbore. This fluid pressure creates one or more fractures extending into the rock medium that contains oil or gas. Since the fracturing operation is conducted at a great depth, the minimum compressive in situ stress is typically in horizontal direction, the hydraulically induced fracture is a vertical fracture.

The dimension and propagation characteristics of a hydraulic fracture are important information in design of fracturing operations. Knowing the properties of reservoir rock, frac-fluid, and the magnitude and direction of in situ stresses, one seeks an accurate prediction of the dimension (opening width, length, and height) of the hydraulically induced fracture for a given pumping rate and time. Many fracture models have been developed for this purpose. The initiation of a hydraulic fracture from a vertical wellbore and two-dimensional fracture models are discussed in the following sections.

Fracturing of a wellbore

Consider an uncased vertical wellbore (or an open hole) under the action of horizontal in situ stresses σmin and σmax as shown in Fig. 1-1.

Assume that the rock is an elastic medium and has a tensile failure stress σT. The breakdown pressure pb for introducing a fracture at the surface of borehole can be calculated by applying elasticity theory [1] to give

b=3σmin−σmax+σT

where σmin is the minimum in situ stress, σmax the maximum in situ stress, and σT the tensile failure stress of the rock.

The hydraulically induced fracture is a vertical fracture and the fracture plane is perpendicular to the minimum horizontal in situ stress σmin as shown. Note that the above equation is independent of hole size and the elastic moduli of rock medium. For a wellbore section at a depth of 10,000 ft, the typical values for the horizontal minimum and maximum in situ stresses are in the order of 5000-7000 psi, respectively. The rock has a tensile failure stress on the order of 500-1500 psi. Equation (1-1) clearly shows that the rock tensile failure stress σT has a small effect on the magnitude of breakdown pressure, and the hole breakdown pressure is mainly to overcome the compressive circumferential hoop stress produced by in situ stresses.

It is clear that the applied wellbore pressure first balances the reservoir pressure (or pore pressure), then overcomes the compressive circumferential hoop stress, causing a tensile stress on the hole surface. A fracture is initiated when this surface stress reaches the tensile failure stress of the rock medium.

The hydraulically induced fracture propagates from the wellbore into reservoir as pumping continues. A typical downhole pressure record (i.e., the pressure measured inside the hole near the opening of hydraulic fracture) is sketched in Fig. 1-2.

The hydraulically induced fracture propagates into the reservoir as pumping continues, and at the same time the frac-fluid leaks off from the fracture surface into the surrounding rock medium. It is important to observe that the opening of the fracture is maintained by the net pressure (fluid pressure minus the minimum in situ stress), while the fluid leak-off rate from the fracture surface is caused by the differential between fluid pressure and reservoir pressure.

Referring to Fig. 1-2 again, the maximum pressure is the initial breakdown pressure pb. The pressure drops, but not always in the field, when a fracture is initiated at the borehole surface. The near constant portion of the pressure curve is the propagation pressure pprog. This is the pressure that causes the propagation of hydraulic fracture into the reservoir. When pumping stops, the pressure drops instantly to a lower value, due to the vanishing frictional pressure loss in the pipe, perforation entrance and near-borehole area, and then continues to decrease slowly to the reservoir pressure due to fluid leaking off from the fracture and borehole as shown in the figure. The transition point is called the shut-in pressure psi (or the instantaneous shut-in pressure, ISIP). However, fluid continues to leak off from fracture surface and the fracture opening width continues to decrease. The fluid pressure inside the fracture eventually reaches to an equilibrium with the minimum in situ stress and at this point the hydraulic fracture closes. The fracture closure pressure, which can be determined from the pressure decline analysis, is taken as a measure of the minimum in situ stress. Although the ISIP is somewhat higher than the fracture closure pressure, the ISIP can be easily identified from the measured pressure-time curve. Field engineers often use ISIP to estimate the magnitude of the minimum horizontal in situ stress. Unfortunately, the situation is somewhat more complicated in field conditions. The underlying control factors for this pressure drop are discussed by McLennan and Roegiers [2].

Equation (1-1) is derived from the assumption that the rock is an elastic medium. However, most reservoir rocks are porous medium through which fluid can flow. The pressure difference between fracture and reservoir causes the fluid to flow from the fracture into reservoir, that is, fluid leak off. The experimental study carried out by Haimson and Fairhurst [3,4] and Medlin and Masse [5] have demonstrated that the porosity and pore fluid have an influence on the hole breakdown pressure. By applying the poroelasticity theory, Schmitt and Zoback [6] have modified Eq. (1-1) to the form as follows:

For a formation impermeable to frac-fluid,

b=3σmin−σmax+σT−βpb

For a formation permeable to frac-fluid,

b=3σmin−σmax+σT−αpp1−2v1−v1+β−α1−2v1−v

where pp is the pore pressure; β the pore pressure factor in tensile failure criterion, 1 ≥ β ≥ 0; v the Poisson’s ratio of dry rock; and =1−bulkmodulusofdryrockbulkmodulusofskeletonmaterial,1≥α>0. Parameter α is known as the Biot’s poroelastic parameter which approaches the upper limit of 1.0 for a compliant rock and less for a stiff low-porosity rock. Schmitt and Zoback [6] have demonstrated that Eqs. (1-2) and (1-3) give a better agreement with experimental data.

The above equations clearly show that the effect of rock porosity and pore pressure is to lower the hole breakdown pressure. They also suggest that the breakdown pressure of the hole is dependent on the filtercake-forming capability of the fluid.

Most wellbores that need fracturing are cased wellbores. To fracture a cased wellbore, the wellbore is first perforated with shaped charges to form a series of perforated holes spiraling along the wellbore surface as shown in Fig. 1-3.

The perforations are typically made at spacings of 4-6 shots per foot and at a phase angle of 60° or 120° as shown in the figure. When the wellbore is pressurized, the perforated holes in (or near) the direction of maximum horizontal in situ stress (σmax) will be fractured first.

The breakdown pressure can be calculated from Eq. (1-1) by replacing the maximum horizontal in situ stress σmax with the vertical stress σVert. The mini-fractures initiated from the perforations may or may not link up to form a large hydraulic fracture perpendicular to the minimum in situ stress along the direction of the wellbore axis. In practice, it is desirable for the mini-fractures to link up forming a large fracture along the wellbore. The linking up of mini-fractures will be discussed in Chapter Five.

Constant height fracture models

Since the wellbore is often fractured at a great depth (> 5000 ft) where the minimum in situ stress is in the horizontal plane, the fracture is a vertical fracture whose plane is perpendicular to the minimum in situ...