Modelling a Borehole Subjected to Fluid Pressure

Fluid pressure inside a borehole produces hydraulic fracture and damage zones in the vicinity of the borehole. These fractures result from stress concentrations around the borehole. The results of numerical simulation of a borehole subjected to fluid pressure using a micromechanical damage model [2] are presented in this paper. It is observed that tensional concentrated stresses are generated around the bottom of the borehole by the applied fluid pressure. Furthermore, fractures develop from the corners of the bottom of the borehole. Introduction In oil and gas production, a process known as hydraulic fracturing is often used. Hydraulic fracturing involves pumping a fluid under pressure into a reservoir. When the pressurized fluid enters the reservoir it produces stress concentration in the surrounding rock which may lead to rock fracture. The fracture creates a conductive path for the production to flow towards the reservoir. The fracture growth in the vicinity of a borehole is largely controlled by rock failure. In order to maximize the efficiency of hydraulic fracturing method, it is of great importance to investigate the fracturing process in rock under fluid pressure and to predict the pressure needed to cause the desired fracture. In this paper, we study mechanical response of rock around a vertical borehole drilled in rock under fluid pressure by using the micromechanical damage model proposed by Golshani et al. (2006). Special attention will be paid to the fracture process under compression in the surrounding rock in terms of microcracking which takes place in association with inelastic deformation. Overview of micromechanical damage model In this model, the rock matrix is regarded as an elastic solid with N groups of microcracks distributed at different orientations, and the i-th group is characterized by the microcracks orientation ) (i θ , the number density of the microcracks ) (i ρ , and the average microcracks length ) ( 2 i c . ) (i θ is the inclination angle of the unit vector ) (i n , normal to a microcrack, to the global axis 1 x (see figure 1). In the following discussion, “ ́” indicates quantities in the local coordinate − ′ i x axes. Figure 1: A single microcrack By assuming that microcrack growth occurs in tensile mode I [1, 2], the stress intensity factor I K for a single microcrack with respect to local axes ) 2 , 1 ( = ′ i xi is approximated by: t I c K σ π ′ − = (1) where t σ ′ is the tensile stress acting normal to the microcrack surface, and is expressed as: 22 22 ) ( S c f t ′ + ′ = ′ σ σ (2) It should be noted that the compressive stress is taken to be positive. The first term on the right hand side of Eq. (2) stems from the far field compression, hence it takes a positive (compression) in a common case. This means that the first term acts as an inhibiting factor for microcracking. The second term is the tensile stress, which is locally generated as a result of the inhomogeneity of rock and sliding movement on asperities. Following the suggestion by Costin [1], we assume that the local tensile stress increases proportionally to the deviatoric stress 22 S ′ , and that ) (c f is a proportionality coefficient depending only on half the microcrack length c. It is of particular importance to point out that the local tensile stress must decrease as the microcrack grows. Otherwise, the microcrack would propagate without any limit as soon as the stress intensity factor I K reaches the fracture toughness IC K . This unsatisfactory situation is easily avoided if the proportional coefficient ) (c f is inversely proportional to half the microcrack length: