## Abstract

Multi-stage hydraulic fracturing of horizontal wells is the main stimulation method in recovering gas from tight shale gas reservoirs, and stage spacing determination is one of the key issues in fracturing design. The initiation and propagation of hydraulic fractures will cause stress redistribution and may activate natural fractures in the reservoir. Due to the limitation of the analytical method in calculation of induced stresses, we propose a numerical method, which incorporates the interaction of hydraulic fractures and the wellbore, and analyzes the stress distribution in the reservoir under different stage spacing.

### Authors

Yi-Jin Zeng^{1} • Xu Zhang^{1} • Bao-Ping Zhang^{1}

^{1}Sinopec Research Institute of Petroleum Engineering, Beijing 100101, China

Edited by Yan-Hua Sun

Received: 11 May 2015 / Published online: 18 September 2015

© The Author(s) 2015.

Simulation results indicate the following: (1) The induced stress was overestimated from the analytical method because it did not take into account the interaction between hydraulic fractures and the horizontal wellbore. (2) The hydraulic fracture had a considerable effect on the redistribution of stresses in the direction of the horizontal wellbore in the reservoir. The stress in the direction perpendicular to the horizontal wellbore after hydraulic fracturing had a minor change compared with the original in situ stress. (3) Stress interferences among fractures were greatly connected with the stage spacing and the distance from the wellbore. When the fracture length was 200 m, and the stage spacing was 50 m, the stress redistribution due to stage fracturing may divert the original stress pattern, which might activate natural fractures so as to generate a complex fracture network.

## 1 Introduction

Multi-stage fracturing in horizontal wells is an important and effective completion method for compact, low permeability reservoirs (Cai et al. 2009; Yao et al. 2013; Koløy et al. 2014; Li et al. 2014; Sorek et al. 2014; Zhang and Li 2014). The essence of hydraulic fracturing is to inject high-pressure fluids into a reservoir to create induced fractures around the wellbore. As a reliable and economic formation stimulation technology, it has been successfully used in shale gas reservoirs. Due to low matrix porosity and low permeability of shale gas reservoirs, to make long well lengths for effective drainage areas in traditional designs, it is not practical for tight shales (Zhou et al. 2010; Zeng et al. 2010; Wu et al. 2011, 2012; Wang et al. 2014a). In order to create complex fractures in shale reservoirs through multi-stage hydraulic fracturing in horizontal wells, analysis of stage spacing and stress interference among fractures needs to be done.

Stress distribution around a horizontal wellbore is very complex, affected by filtration of fracturing fluids, pore pressure, etc. (Fischer et al. 1994; Economides 2006; Civan 2010; Fang and Khaksar 2011; Zhang et al. 2012; Ziarani and Aguilera 2012; Hou et al. 2013; Pan et al. 2014; Chen et al. 2015). Meanwhile, previous stages of hydraulic fracturing would affect later stages of fracturing (Xu 2009; Wei et al. 2011; Chuprakov et al. 2011; Wang et al. 2014b).

Some researchers investigated optimization of hydraulic fracturing based on a one-factor analysis of fracture parameters and evaluated effects of fracture parameters on productivity of horizontal wells (Zhu et al. 2013). A new grid refinement was used to optimize fracturing parameters. Effects of the horizontal-section length, number of fractures, fracture length, and conductivity on the productivity of horizontal wells were also studied (Chen et al. 2013). An optimization method for perforation spacing was established, it included a mathematical model of the induced stress field on the basis of a homogeneous and isotropic 2D plane strain model and the shift between the maximum and minimum horizontal principal stress (Yin et al. 2012). Qu et al. put forward a design method for optimizing horizontal well fracturing parameters (Qu et al. 2012). Shang et al. established a wellbore stress distribution model and a fracture pressure calculation model (Shang et al. 2009).

In the above-mentioned research, an analytical method based on classic fracture mechanics was used to analyze characteristics of the fracture-induced stress field in hydraulic fracturing, but this model did not consider interaction between fractures and the horizontal wellbore, so did not explicitly give characteristics of different stages of spacing and stress interference among fractures.

On the basis of analyzing limitations of the classical analytical method, a comparative analysis of the numerical method is carried out. With the numerical model, we analyze in situ stress change and interference among fractures.

## 2 Fracture geometry and net pressure for multi-stage fractured horizontal wells

### 2.1 Fracture geometry

During hydraulic fracturing, a fracturing fluid is injected continuously into the formation, and fractures will propagate dynamically. However, it is difficult to predetermine the fracture geometry using the pressure distribution function. While using the criterion of K_{I} = K_{Ic} at each moment, the hydraulic mechanical coupling problem can be solved.

From the theory of fracture mechanics and the Castigliano theorem, the width of a fracture under plain strain conditions can be calculated from the following equation:

where ∆p(y) = p(y) − σ_{h} is the net pressure in the fracture; p(y) is the fluid pressure; σ_{h} is the minimum horizontal in situ stress perpendicular to the fracture plane; E is the elastic modulus of the rock; a is the fracture half length; ξ is the temporary fracture half length during integration, as shown in Fig. 1.

**Fig. 1** Geometry of a hydraulic fracture.

When the net pressure is distributed smoothly and continuously along the fracture, it can be defined by a continuous function *p*(*y*) = *p*_{w} *f*(*y*). The stress intensity factor *K*_{I} at the fracture tip can be expressed as

where p_{w} is the wellbore pressure; f(x) is the eigen function of the pressure distribution.

According to the mean value theorem of integral, Eq. (2) becomes

is a constant, and

considering the fracture propagation criterion K_{I} = K_{Ic}.

Substituting Eq. (3) into (1) and integrating it give

where K_{Ic} is the fracture toughness of the target zone. From Eq. (4), it can be seen that the fracture cross section is a slim ellipse determined by the fracture half length a, E, K_{Ic}, and independent of the pressure distribution in it.

### 2.2 Net pressure

We assume that the fracturing fluid is incompressible, and the formation is impermeable, then the fracture volume is equal to the volume of the fluid injected into the formation.

where q is the injection rate; t is the injection time.

The fracture parameters can be obtained as

Using fluid mechanics theory, the distribution of the net pressure can be approximated by a linear equation:

Substituting Eq. (9) into (2) gives the wellbore pressure p_{w}:

The fluid pressure distribution *p*(*y*) and the tip pressure *p*_{a} are

Equations (11) and (12) indicate that the tip pressure p_{a} is not 0, and the wellbore pressure p_{w} should be always greater than the minimum horizontal stress σ_{h} to maintain the fracture propagation considering the pressure drop.

## 3 In situ stress model for multi-stage fractured horizontal wells

Due to shale formations of low permeability, multi-stage hydraulic fracturing of horizontal wells is used to interact with natural fractures and weak bedding planes, to create as many crossing fractures as possible to maximize the stimulated reservoir volume. During stage fracturing, the main hydraulic fractures are created one after another. When a main fracture is formed, there will exist an induced stress field around the fracture, which influences the in situ stress field around it. The superposition of the induced stress field and the in situ stress field will affect the initiation and propagation of subsequent fractures. At present, classical fracture mechanics theory is used to calculate the stress field after fracturing.

### 3.1 Limitation of the classical analytical method

According to the theory of fracture mechanics, a two-dimensional model was established, based on the assumption of homogeneity, isotropy, and plane strain conditions, to calculate the induced stress field, as shown in Fig. 2.

**Fig. 2** Stress field induced by a hydraulic fracture.

We assume that the fracture is vertical, its longitudinal section is elliptical, and the height is H. z-axis is along the fracture height direction. x-axis is along the horizontal wellbore, and y-axis is along the direction of the maximum horizontal stress. It is assumed that the tensile stress is positive and the compressive stress is negative. The induced stresses at an arbitrary point (x, y, z) are as follows (Yin et al. 2012):

with

c=H/2,

where σ_{x}, σ_{y}, and σ_{z} are the three normal stress components induced by a fracture, MPa; τ_{xz} is the shear component, MPa; p is the fluid pressure, MPa; ν is the Poisson ratio.

The relationships between these parameters are as follows:

In situ stresses are composed of σ_{x}, σ_{y}, and σ_{z}. The stress field around a later fracture may be a summation of induced stresses of the former fractures and the in situ stresses. According to the principle of superposition, the stress field around the nth fracture was studied by Zhang and Chen (2010a, b, c):

where σ′_{H(n)}, σ′_{h(n)}, and σ′_{v(n)} are the three principal stresses around the nth fracture, MPa; σ_{x(in)},σ_{y(in)}, and σ_{z(in)} are the induced stress components around the nth fracture caused by the ith fracture, MPa.

When a new fracture is formed, it will produce an induced stress field which can be calculated by Eq. (13). Superimposing the newly induced stress field on the old one gives the final stress field (Eq. 15). This method just considers the effect of hydraulic fractures on the stress field but ignores the influence of the horizontal well. So it is limited and cannot provide the exact value of the stress field.

Because the stress field around the horizontal wellbore changes dramatically, the effect of the horizontal wellbore on the stress field distribution should be considered. Numerical calculation can achieve the above purpose.

### 3.2 Numerical method for in situ stress field in horizontal wells

The initial stress field around the wellbore will be disturbed and redistributed in the shale reservoir during drilling and completion operations. When the first hydraulic fracture is formed, the in situ stress field around the hydraulic fracture and the wellbore will be redistributed. So interaction between fractures and the horizontal wellbore should be considered when calculating the variation of stress in horizontal well fracturing.

Considering interaction between fractures and the horizontal wellbore, analysis of the in situ stress field can be simplified as a plane strain problem. In this analysis, the stress induced by the change of reservoir temperature is ignored, and the fluid flowing in fractures is incompressible.

**Fig. 3** Stress fields considering interaction between fractures and the horizontal wellbore

Figure 3 shows a model of stress fields considering interaction between fractures and the horizontal wellbore. In order to reduce calculation time, we considered symmetrical geometry, in which we would calculate the stress field in the finite element model.

## 4 Comparison of classical analytical and numerical methods

The stress difference around a fracture before and after fracturing can be expressed as

are the minimum and maximum horizontal stresses around the fracture after fracturing, respectively.

The main parameters used in numerical simulation are shown Tables 1 and 2.

**Table 1** Mechanical parameters of the shale reservoir.

**Table 2** Input parameters.