Hydraulic fracturing is the process of transmitting pressure by liquid or gas to create cracks or to open the existing cracks in hydrocarbon bearing rocks underground. The purpose of hydraulic fracturing in an oil or gas reservoir is to enable the oil or gas to flow more easily from the formation to the wellbore. Mathematical modeling of the hydraulic fracturing process is usually performed to predict different aspects and phenomena through the process. For this reason, many models have been developed over the past several decades. All of these solutions are approximated as they require assumptions about either the fracture opening or the pressure field. Such assumptions are necessary because of the difficulty in treatment of the complex fracture geometry growing under different stress and well conditions. This paper surveys a review of the most commonly used and accurate existing hydraulic fracturing prediction.Keywords: Hydraulic fracturing; modeling; two-dimensional; three-dimensional.
Hydraulic fracturing is a widely used technique for the exploitation of oil, gas, and geothermal resources [1, 2]. The technique of hydraulic fracturing was introduced to the petroleum industry during the 1930s when Dow Chemical Company discovered that by applying a large enough down-hole fluid pressure, it was possible to deform and fracture the rock formation to have a more effective acid stimulation, and is now a standard operating procedure . By 1981, more than 800,000 hydro fracturing treatments had been performed and recorded. Nowadays, about 40% of all currently drilled wells are hydraulically fractured . In some cases, hydraulic fracturing is used to improve the connection between a well and a productive reservoir . In other applications, hydraulic fracturing is used to engineer the reservoir, creating or stimulating fractures in a low permeability matrix. Two well-known examples of the latter case are enhanced geothermal systems (EGS)  and gas shale stimulation .
Hydraulic fracturing occurs when fractures initiate and propagate as a result of pressure applied by a fluid inside the fractures. Fractures in the earth's crust are desired for a variety of reasons, including enhanced oil and gas recovery, re-injection of drilling or other environmentally sensitive wastes, measurement of in situ stresses, geothermal energy recovery, and enhanced well water production. The size of fractures can vary from a few meters to hundreds of meters . A typical hydraulic fracturing process in the petroleum industry is shown in Figure 1. The process is to work as the following steps :
It is generally assumed that the induced fracture has two wings, which extend in opposite directions from the well and is oriented, more or less, in a vertical plane. Other fracture configurations, such as horizontal fractures, are also reported to occur, but they constitute a relatively low percentage of situations documented [4, 8].
A successful hydraulic fracturing may increase the production up to 10s of times, making the technique economically attractive . Experience indicates that at a depth of below 600 meters, fractures are usually oriented vertically. At shallow depths, horizontal fractures have been reported . The fracture pattern, however, may not be the same for different types of soil and rock.
The geometry of the induced fracture is significantly affected by the rock’s mechanical properties, in-situ stresses, the rheological properties of the fracturing fluid and local heterogeneities such as natural fractures and weak bedding planes. For performing hydraulic fracturing in an isotropic and homogeneous medium, the in-situ stress state is the controlling factor on fracture development . Generally, because of reservoirs do not satisfy these ideal conditions, it should understand the effect of other factors on hydraulic fractures .
Modeling of fracture propagation by hydraulic induced fractures is of great interest in order to define the required amount of fluid, injection pressure, and proppant volume and to predict the effectiveness and feasibility of the treatment. Hydraulic fracturing is intrinsically a three dimensional nonlinear coupled problem, where fluid flow and diffusion into rock formation, fracture propagation, and inelastic rock deformation are mechanisms to be described by the model . A hydraulic fracture will grow in the direction normal to the smallest of the three principal stresses as it tends to open in the direction of least resistance. For most reservoir depths of interest in the petroleum industry, the smallest principal stress is in the horizontal plane, which restricts fractures to the vertical plane . Therefore, one may map fractures on the horizontal plane. However, an ideal model should include the three-dimensional aspects of the problem; the available techniques in the literature for three-dimensional analysis of fractures are computationally so expensive that current computers can only handle one or two fractures. Therefore to save the computational efforts and avoid further complexities, the investigation was limited to the two-dimensional analysis. However, two dimensional models will provide a framework for further developments to three dimensional analyses. Despite the development of sophisticated 3D fracture modeling tools, hydraulic fracture behavior is difficult to predict with any degree of confidence.
This article presents a review of the most commonly used and accurate existing hydraulic fracturing prediction models for reservoir rocks with emphasis on their qualitative and quantitative predictive capability, their scientific basis, and their limitations.
An easily static worked procedure for calculating the fluid displacement that accompanies hydraulic fracturing was introduced by Piggott and Elsworth . The solution applied to the particular case of a Perkins and Kern, Nordgren (PKN) [13, 14] hydraulic fracture subject to fracturing fluid loss to the formation at a rate that is equal to the rate of fluid injection. This is referred to as a static solution, since it assumes that the fluid particle is static with respect to the calculation of velocity .
A schematic illustration of a PKN hydraulic fracture is shown in Figure 2. As it can be seen from this figure, two symmetrical fracture segments propagate away from the wellbore with a constant height, H, to a maximum length of L=Lp at the end of fluid injection time of t= tp.
Other parameters that characterize fracture extension and fluid displacement are the diffusivity of the formation (D), porosity of the formation (n), the fracturing fluid leak-off coefficient (CL), and the rate of fracturing fluid injection (Q). The extension of a PKN hydraulic fracture subject to fluid loss at the rate of fluid injection is given by [14, 15]:
where the local rate of fluid loss to the formation through the two opposing fracture walls is defined by:
in this equation, τ is defined as time at which the formation is first exposed to fracturing fluid. Fluid flow through the formation occurs in the plane of x-y and fluid displacement is indexed by the displacement components in the directions of coordinate axes as:
where x0 and y0 and xf and yf are the initial and final positions of a reference fluid particle, respectively. ∆xj is the increment in fracture length. Then, the total displacement of the particle is given by:
Here, T denotes the transmissivity of the formation. In this solution it is assumed that the applied velocity to the particle is unchanged with respect to the motion of the particle due to sufficiently small displacement of the reference fluid particles. This approximation would results in the following equations:
In this equation, geometry is expressed in dimensionless form relative to the length of the fracture at the end of fluid injection xd =x/Lp and yd =y/Lp. In Eq. (5), integration implies the superposition of displacement increments due to fluid loss along the length of the fracture with
The major limitation of the static model is the assumption that the velocity of the fluid particle is everywhere equal to the velocity calculated at the initial location of the particle. From the basic theory of the two-dimensional static model, it is expected that the accuracy of this model will degrade as the displacement of the particle increases, and that a dynamic solution that explicitly represents the motion of the particle is required for the case of large displacement magnitudes that is considered extensively in section 2.2.