Hydraulic fracturing is a technique for simulating wells completed in low-permeability reservoirs. The process involves the pressurization of an isolated perforated section of the wellbore with a viscous fluid until the induced stresses exceed the formation strength, which causes a failure and thus creates the fracture. Proppants are then pumped into the newly created fracture with viscous fracturing fluid as a carrier. Once initiated, the fracture propagates as additional fracturing fluid is injected. Following the release of the fracturing pressure, the proppants hold the fracture open and provide a conductive channel through which the reservoir fluids flow to the wellbore. Only vertical fractures are considered here.
The effectiveness of a hydraulic-fracturing treatment is determined by the propped fracture conductivity and the geometry of the created fracture, more specifically by the propped fracture height and the areal extent. The areal extent, by definition, includes both the penetration length and the fracture width. Pad volume and proppant concentration also play an important role in the fracture-treatment design because they determine final propped fracture penetration and conductivity. Insufficient pad volume causes premature screenout because of early depletion of pad fluid, whereas excess pad volume injection often results in short propped penetration. The propped fracture conductivity depends on the type of proppants (mesh size, shape), extent of permeability damage caused by fracture fluid residue, and other factors, e.g., fluid viscosity, fluid-loss characteristics, and pumping schedule-that determine fracture width. The intrusion of a fracture from the pay zone into the formations lying above and below is another factor that affects the effectiveness of the fracture treatment. If the hydraulic fracture is not contained within the producing formation and propagates in the vertical direction, the postfracturing production response may be lower than anticipated because substantial amounts of fracture fluid and proppants are used to fracture the unproductive formations. The containment of a fracture within the producing zone is even more important if the adjacent zones are waterbearing.
Hydraulic-fracturing simulators are used to design the treatment volume, proppant size and type, and pumping schedules to obtain the desired fracture geometry and conductivities. The pumping schedules are designed by running these simulators on a trial-and-error basis until a desired propped geometry is obtained that will ensure maximum proppant coverage at the end of pumping. With these propped fracture geometry and conductivity data, an economic estimate is usually made for the treatment for a given fracture length and an optimal design is selected for the maximum return of the well.
A realistic model must consider such basic phenomena of fracturing as
rock deformation, characterized by the variation of mechanical properties of the rock, vertical in-situ stress distribution, pore pressure distribution, and stress contrasts in the layered rock media;
fluid flow through tubular goods, fracture, and perforations;
rock deformation with fluid flow interaction;
fracture propagation that takes into account the reservoir or pore pressure in the vicinity of the propagating fracture; and
proppant distribution and placement in the fracture, influenced by fracturing fluid leakoff characteristics.
Five principal mathematical relationships that dictate the opening and propagation of hydraulic fractures include:
the mass conservation equation governing a total mass balance for a proppant-laden fluid-i.e., the total volume of slurry injected minus the volume of the slurry in the fracture, and the volume of fluid lost to the formation by leakoff and spurt loss must be equal to zero;
fracture-width-opening equations that relate the crack-opening thickness to the fracture length, height, geometry, and to the fluid-pressure distribution along the fracture, assuming that the rock formation deforms according to the laws of linear elasticity;
momentum equations describing the fluid-flow behavior of theologically complex fracturing fluids in response to the pressure gradients in the fracture;
continuity equations governing the exchange of fluid between the fracture and its surroundings; and
fracture propagation criteria that determine the direction of the fracture propagation in a given time.
Two-Dimensional (2D) Models. Hydraulic-fracture propagation modeling began initially with these two basic approaches.
The fracture zone deforms independently of the upper and lower layers, allowing free slippage on these layers (i.e., the plane-strain solution is considered in a horizontal plane). The fracture is assumed to have a constant height with a vertically constant width (but the width decreases with distance from the wellbore) as shown in Fig. la, characterized by a decreasing net wellbore pressure with time when injection is at a constant rate. This model, commonly known as the Geertsma-de Klerk model, would approximate a fracture with a horizontal penetration much less than the vertical penetration. The idea of the Geertsma-de Klerk model was originally proposed by Khristianovich and Zheltov. Therefore, it is sometimes referred to in the literature as the Khristianovich-Zheltov-Geertsma-de Klerk (KZ-GDK) model.
A constant-height model assumes that the cross section of the fracture lies in the vertical plane, perpendicular to the axis of the fracture. The fracture is limited to a given zone and the plane-strain condition is assumed in vertical planes only.