## Abstract

Fracturing operations in shale gas reservoirs of the Sichuan–Chongqing area are frequented by casing deformation, failures in delivery of mechanical staging tools and other down-hole complexities. In addition, limitation in volumes of tail-in proppant in the matrix area significantly restricts the conductivity in the near zones of the wellbore. Eventually, flowback performance and productivity of shale gas horizontal wells are negatively affected.

#### Authors

#### Li Deqi^{a}, He Feng^{b}, Ou Weiyu^{b}, Zhu Juhui^{b}, Li Ran^{a} & Pan Yong^{b}

^{a}PetroChina Zhejiang Oilﬁeld Company, Hangzhou, Zhejiang 310013, China. ^{b}CNPC Chuanqing Drilling Engineering Co., Ltd., Chengdu, Sichuan 610051, China

Received 8 November 2017; accepted 25 January 2018

With consideration to the limitations in the implementation of the mechanical staging technique with bridge plug for shale gas development in the Sichuan–Chongqing area, the technique of multi-stage sand filling stimulation in horizontal wells was proposed to solve the above-mentioned problems. By filling sands in fractures, it is possible to divert fluids to maintain long-term high conductivity of fractures, which is the key to satisfactory EOR performances. By introducing the Hertz contact and fractal theory in the analysis of sand plug strength, and in combination of lab engineering simulation test results, the mechanical model for sand plugs in fractures with proppant was constructed. In terms of strength criteria and friction, the stability criteria of sand plug were put forward.

Thus, the permeability fractural model for sand plugs in fractures was perfected. Test results show that the stability of sand plug in the earlier stage of production is mainly affected by fluid washing during flowback, so it is necessary to control the flowback rate strictly. In the later stage of production, the stability is mainly affected by fracture closure stress and flow pressure, so it is necessary to enhance the yield strength of proppant to maintain high conductivity of fractures. In conclusion, the multi-stage sand filling stimulation provides a new technique for multi-stage clustering fracturing operations in shale gas horizontal well development.

## Introduction

Horizontal well large-scale staged volume fracturing, together with bridge plug staged multi-cluster perforation is generally used for enhancing shale gas production around the world [1], [2].

In the Sichuan–Chongqing area, China, shale gas reservoirs are characterized by complicated stress regime, variable burial depth (TVD is generally more than 2300 m or partly more than 3000 m), presence of natural fractures and faults, high operation pressure and relatively great stimulation difficulty. Thus, the application of mechanical staging technology like bridge plug staging is limited in the Sichuan–Chongqing area.

In actual fracturing operations in shale gas reservoirs of the Sichuan–Chongqing area, downhole complexities like casing deformation frequently occurred. More than 50% of casing deformation points were close to Point A, where conventional mechanical staging tool could not pass through, thus making a great deal of horizontal sections fail to be effectively stimulated.

Based on statistics, in the Changning and Weiyuan blocks, 176 fracturing sections were affected by casing deformation, with a total length of about 13700 m. Therefore, it is urgently necessary to develop a new staged fracturing technology free of mechanical packing, with moderate requirements for well trajectory and wellbore size, and capable of infinite staging and maintaining a full-wellbore drifting in the whole fracturing process.

The horizontal well multi-stage sand filling fracturing technology for shale gas development [3] is adopted to stimulate the reservoirs by stages by isolating the intervals with artificial sand plugs which are formed using high-concentration proppants at fracture mounts and near the wellbore. This technology does not use mechanical bridge plug setting, and thus can effectively avoid operational risks possibly induced by bridge plug and save operation cost.

Moreover, the adverse impacts of casing deformation to reservoir stimulation can be minimized. In this paper, the application of horizontal well multi-stage sand filling fracturing in shale gas reservoir stimulation is discussed theoretically and experimentally in terms of the conductivity of sand plugs in fractures and technical parameter control and optimization, and its applicability and effectiveness are verified using field data.

## High conductivity

Compared with the conventional cable-conveyed cluster perforation and pump-down plug technology, the horizontal well multi-stage sand filling fracturing technology is especially advantageous in that the bridge plug is not conveyed via cable and the wellbore is kept in a full opening state in the whole course of fracturing operation. It uses the high-concentration proppant at the late stage of fracturing operation to form artificial sand plugs at the fracture opening and near the wellbore zone, so as to achieve the staging as the conventional mechanical bridge plug can do.

The performance of such staging by sand filling fully depends on whether the sand plugs can satisfy the compressive strength under reservoir closure stress and whether the sand plugs can keep a stability and a high conductivity in the period of drainage. In this paper, two proppant force models were selected to analyze the compressive strength of sand plugs under reservoir closure stress, the mechanical stability of sand plugs in fractures during flowback and the permeability of contact model.

### Compressive strength of sand plugs under reservoir closure stress

Proppant particles are carried by slick water into the formation. As the fluid flow rate reduces, proppant particles gradually deposit in the fractures and form a disorder accumulation [3]. Without any extraneous force, the proppant particles would form an arbitrary accumulation, which has loose texture, insufficient contact and small density [4].

In actual hydraulic fractures, however, as the fracturing fluid leaks off, the fracture width would gradually narrow, and the proppant particles would gradually become dense accumulation from the initial arbitrary accumulation under the combined action of closure stress compression and fluid carrying. According to the experimental research by Ye Danian [5], equant spheres could form dense accumulations with different contact relations in an enclosed compartment. For adequate analysis of the reliability of sand plugs, a body-centered cubic model with a small density and a face-centered cubic model with the maximum density were respectively built (Fig. 1).

**Fig. 1.** Body-centered cubic and face-centered cubic models of equant spheres.

Usually, the proppant particles are different in size and sphericity due to many uncontrollable factors during operations. When the mechanical model of proppant particles is built, the particles are regarded as homogeneous, equant and elastic ideal spheres, so that the proppant particle contact can be converted to point load contact among elastic spheres. In 1881, Hertz first used the mathematical elastic mechanics method to derive the formula for point load contact among smooth-faced elastic spheres [6].

For the dense accumulation models of proppants, the proppants exhibit point contact, and the contact surfaces are round under force, conforming to the deformation continuity condition; the whole contact process is at an elastic stage and is subordinated to the Hooke’s law, therefore, the stress–strain on the contact surface satisfies a linear relationship; the resultant force composed of surface contact pressures equals to the applied load. The point contact of proppant particles is shown in Fig. 2.

**Fig. 2.** Point contact relation among proppant particles.

Based on the mechanical hypothesis for point load contact of Hertz elastic spheres, the radius of elastic sphere contact surface circle (Ra), the contact deformation of two spheres under compressive stress (δ) and the distribution of maximum contact compressive stress (q_{max}) and compressive stress within the contact circle can be derived [7]. Eq. (1) is the nonlinear precise integral expression for elastic contact of two spheres [8]; when the horizontal shift of contact area is neglected, the maximum radius of sphere contact surface circle (Ra), the δ and the maximum contact stress of two spheres at contact circle radius (r) under compressive stress (q_{max}) can be derived.

where, q represents the stress of the two spheres at radius r of contact circle under compressive stress, Pa; similarly, r: the arbitrary radius within contact circle, m; δ: the contact deformation of the two spheres under compressive stress, m; R1 and R2: the radius of the 2 contacting spheres respectively, m; E1 and E2: the Young’s modulus of the 2 contacting spheres respectively, Pa; ν_{1} and ν_{2}: the Poisson’s ratio of the 2 contacting spheres respectively, dimensionless; and ϕ: the integral angle of the two spheres at radius r of their contact deformation circle, (°).

The load applied on proppant particles mainly comes from reservoir closure stress, fluid pressure and particle gravity. The apparent density of proppants only ranges 2.6–2.8 g/cm^{3} and the particles are very small, so the particle gravity itself can be neglected in calculating the applied load. The closure stress of hydraulic fracture is all along perpendicular to the fracture surface (Fig. 3). Let the fracture dip be θ, the closure stress is σ. Taking the embedding of proppant particles on fracture surface into account [9], [10], [11], [12], the maximum net load applied on the proppant sphere with radius R by fracture surface closure stress and fluid pressure p_{c} is p:

where, σ, σ_{h} and σ_{v} represent the fracture closure stress, horizontal stress and vertical stress, respectively, Pa; similarly, θ: the fracture dip, (°); p: the net load applied on proppant particles, N; p_{c}: the fluid pressure, Pa; and R: the proppant particle radius, m.

**Fig. 3.** Closure stress vs. fracture dips.

The contact among proppant spheres satisfies elastic sphere contact, therefore, its failure criterion follows the maximum shear strength theory, with expression as follows:

According to Chen Guohui [13], when elastic spheres contact, the maximum shear is τ_{max}, then,

Therefore, the above mechanical model can be used to judge whether the proppant particles can bear the reservoir applied load under yield strength (σ′), and whether their strengths are not damaged under load conditions.

### Stability of sand plugs under fluid action

Proppant particles form the above mentioned contact models in hydraulic fractures. Such models not only bear the fracture closure stress, but also bear the scouring action of fracturing fluid and flowback fluid in the course of stimulation. The primary acting force of sand plugs in fractures to resist the scouring action of fluids comes from the frictional resistance on rough surface. Based on the Coulomb friction theory [14], the frictional resistance among proppant particles is positively correlated to the normal stress applied on them; the frictional resistance for each model is shown in Fig. 4.

**Fig. 4.** Frictional resistances of body-centered cubic and face-centered cubic models of equant spheres.

As to body-centered cube, because the contact points are symmetric each other, α = 45°; therefore, in the coordinate system shown in Fig. 4, the maximum static frictional resistance F_{1} is:

where, F_{1} represents the maximum static frictional resistance of the body-centered cubic model, N; μ_{f} represents the static friction coefficient of proppant particles, dimensionless; and α represents the included angle between net load on proppant particle and friction surface, (°).

As to face-centered cube, apart from 8 contact spheres in the body-centered cube, 4 contact spheres parallel to fluid are also included. In the coordinate system shown in Fig. 4, F_{2} is expressed as:

where, F_{2} represents the maximum static frictional resistance of the face-centered cubic model, N.

When fluid flows through proppant particles, fluid viscosity, gas–liquid ratio and flow rate would all affect the scouring effect of fluid on proppant particles. Based on the fluid mechanics theory, under fluid scouring effect, the sand plugs in fractures are mainly affected by viscous resistance and pressure drag, and the total scouring force on single proppant particle is F_{D}, then,

where, R represents the proppant particle radius, m; similarly, F_{D}: the total scouring force on single proppant particle, N; C_{D}: the resistance coefficient, dimensionless; ρ: the fluid density, kg/m^{3}; and v: the flow rate of fluid among proppant spheres, m/s.

Therefore, the relative magnitude of scouring force FD and the maximum static frictional resistance can be used to judge whether the sand plugs in fractures can keep stable in the course of injection and flowback.

### Calculation of sand plug permeability based on contact model and fractal theory

Based on the contact among proppant particles in body-centered cube and face-centered cube, the maximum porosity of them under the conditions free of closure stress and fluid pressure is derived.

In Fig. 5, for purpose of intuitively expressing the contact relation of the spheres, the spheres are particularly reduced, and small circular columns are used to represent the tangent relation of the spheres. As to body-centered cubic and face-centered cubic models, there are cubic side length expressions m_{1} and m_{2} respectively, i.e.:

**Fig. 5.** Contact relations of body-centered cubic and face-centered cubic models of equant spheres.