**1. Introduction**

Fracture propagation modeling is an important part of reservoir geomechanics and must be considered in injection modeling of wells. Classical modeling of fracture geometry is well established and documented in literature of rock mechanics and stimulation [3]. Direct coupling of fracture propagation (fracture dynamics) and fluid flow is computationally very expensive [4, 5]. The modeling of "complex" fracturing [6] is also expensive. However, proper representation of dynamic propagation in which the fracture is directly coupled into a reservoir simulator is important for many applications. It has been shown that the some degree of coupled treatment of fracture mechanics, reservoir modeling and geomechanics is important for better understanding of the unconventional fracturing applications as well as for tight gas fracturing treatments such as waterfracs [7, 8]. While the fully coupled approach [5, 12] is not

© 2013 Islam and Settari; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

yet feasible for practical work, we have developed a simplified method which will be described briefly here. Such coupling, described next, can be achieved in a simplified fashion using only a reservoir simulator, which makes the modeling computationally efficient. The method has been developed and refined over two decades and proved to be successful in modeling a large variety of injection processes.

#### **2. Theory of fracture propagation modelling**

The method is based on modifications of transmissibility in the reservoir flow model. For injection scenarios transmissibilities are modified dynamically. To model dynamic fracturing process, a transmissibility multiplier function is assigned to a line (or plane) of grid blocks assumed for fracture propagation extending from the well. The multiplier function is a table that can be derived from simple 2-D analytical fracture models which approximate the actual fracture. In an uncoupled modeling, transmissibility multipliers are a function of fracture injection pressure, while in a coupled (geomechanical) system they are a function of a mini‐ mum effective stress. The multipliers are calculated based on the estimation of a 2D crack opening in a cross-section by Equation (1) [9], then calculating the fracture permeability by Equation (2) as a function of the net pressure in the fracture, and finally calculating the transmissibility multiplier (*Tr*) on the reservoir transmissibility of the block containing the fracture as described in Equation (3).

$$\text{Cov}\_f = \frac{\Delta P\_f \cdot H\_f}{\cdot} = \frac{4 \,\Delta P\_f \cdot H\_f \cdot \left\{1 \cdot \nu^2\right\}}{E} = \frac{4 \,\left\{P\_f \cdot P\_{f\text{sc}}\right\} H\_f \cdot \left\{1 \cdot \nu^2\right\}}{E} \tag{1}$$

$$\mathcal{K}\_f = \mathcal{R}\_{fa} \frac{w\_f^2}{12} \tag{2}$$

Equation (3) from the known *Tr* at any grid cell in fracture plane and in time. Typically a single function is used, but variations of confining stress can be modeled by the use of multiple functions with different *Pfoc* values. Finally, the created fracture volume can be accounted for

Injection Modeling and Shear Failure Predictions in Tight Gas Sands — A Coupled …

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Obviously, the method is approximate in several respects. The fracture opening is computed from PK geometry for a fixed height while in reality the height varies along the distance from the well. The fracture opening also varies in the vertical direction. In spite of these approxi‐ mations, the method provides a very realistic approximation to the results obtained by models based on fracture mechanics [4], and is capable of history matching complex injection sequen‐

The wells studied are located in the tight gas sands in Western Canadian sedimentary basin. They were fractured with different techniques – Well A using cross-linked gelled water fracs, and Well B with un-gelled water fracs (slick water fracs). In both cases, the entire well was fractured through an open hole as opposed to multi-stage fracturing (which is the more common technique). However, microseismic monitoring and other techniques have shown

Field bottomhole injection pressure (BHIP) for wells A and B is given in Figure 1 and 2 respectively. Well B which is deeper than Well A has a higher bottomhole injection pressure than well A. Breakdown pressure, maximum pressure required to initiate fracture in forma‐ tion, is also higher for well B. Complete set of reservoir, geomechanical and stimulation data has been provided in references [1, 2]. For modeling it was assumed that 15 fractures were created along the well with spacing of 50 m, as it was indicated by microseismic monitoring. The simulations were carried out both in an uncoupled mode and coupled mode (solving both fluid flow and geomechanics). In both we employed the technique described above for fracture modeling, and also the pressure or stress dependent matrix permeability changes (which model the permeability enhancement in the SRV). The results of uncoupled modeling are not presented in detail, but in this case it is shown that coupled modeling is necessary to obtain

that a number of fractures were created, with fairly regular intervals.

**4. History matching of field injection pressure — Uncoupled**

The concept used for approximation of geomechanical effects in an uncoupled model and equations developed for production modeling [1, 2] can be used also for injection modeling. For injection cases change in pressure is always positive and consequently the effective mean stress is always larger when poroelastic effects are considered. For uncoupled modeling it can

by a similar function applied to porosity.

ces [13].

**3. The field study**

history match (see [1, 2] for details).

**geomechanical injection models**

$$T\_r = \frac{K\_m A\_n + K\_f A\_f}{K\_m A\_n} = 1 + \ R\_{fa} \frac{w\_f^3}{12 \ K\_m w} = f\left(P\_f - P\_{fuc}\right) \tag{3}$$

*Hf* in Equation (1) is the estimate of fracture half-height based on the 2-D Perkins-Kern geometry assumption of vertical fracture with smooth closure at the top and bottom [10]. Permeability reduction facto*r Rfa* in Equation (2) is a correction factor that accounts for deviations from Poiseuille law such as roughness, tortuosity and non-Darcy flow. It can be several orders of magnitude less than 1. Fracture opening or closing pressure (*Pfoc*) is normally taken as the initial minimum horizontal total stress acting perpendicular to fracture face. Fracture opening or closing pressure may be actually greater (or lower) than the initial minimum horizontal total stress due to poroelastic and thermo elastic effects. The method allows one to create tables of dynamic transmissibility multipliers (as a function of pressure or stress) which are then used in a conventional reservoir simulator to propagate the high permeability (in the fracture plane) in time. As such, the method does not directly solve any fracture mechanics equations; although an estimate of fracture width can be obtained from Equation (3) from the known *Tr* at any grid cell in fracture plane and in time. Typically a single function is used, but variations of confining stress can be modeled by the use of multiple functions with different *Pfoc* values. Finally, the created fracture volume can be accounted for by a similar function applied to porosity.

Obviously, the method is approximate in several respects. The fracture opening is computed from PK geometry for a fixed height while in reality the height varies along the distance from the well. The fracture opening also varies in the vertical direction. In spite of these approxi‐ mations, the method provides a very realistic approximation to the results obtained by models based on fracture mechanics [4], and is capable of history matching complex injection sequen‐ ces [13].
