Stable Created Fracture Growth between Two Parallel Boreholes

Abstract

The authors describe numerical modeling of a growing fracture created between two parallel boreholes in an isotropic medium in the nonuniform stress field. The fracture paths are modeled at various deviations of the initial fracture from the axes of boreholes, spacing of the boreholes and the compressive stress field between them. Physical simulation of hydraulic fracturing is implemented using large-size samples.

INTRODUCTION

Hydraulic fracturing method is widely used in underground construction and in mining solid minerals. In-mine hydraulic fracturing is used to measure stresses in the rock mass [1, 2], to increase the gas recovery of coal seams [3–5], to control roof caving [6, 7], and to form cutoff curtains near mine workings [8]. For the effective solution of most of the above problems, it is required to create extended fractures of a certain direction.

Different approaches to implement directional hydraulic fracturing of a seam are known. These include cutting initiation slots of a given geometry, directed load on borehole walls using special devices, creating a local stress field near the well that is favorable for fracture propagation. One of the actual engineering problems is the formation of an extended longitudinal fracture in a seam, for example, due to the synchronous hydraulic fracturing of several closely spaced parallel boreholes. There are factors that prevent their integration into a single system by a hydraulic fracture: stresses in the rock mass with regard to influence of mine working, physical properties of rock, spacing of the boreholes, the presence and geometric parameters of initiation slots, the feed rate and properties of the working fluid, etc

In recent years, a scientific interest in studying this problem has arisen. Based on the results of numerical and physical simulation, the features of fracture propagation in large-size samples were determined depending on the spacing of boreholes, their position relative to effective stresses, minimum and maximum compressive loads [9, 10]. It is proposed to further increase the pore pressure in the medium by pumping fluid through auxiliary holes, as well as to combine this approach with cutting initiation slots in hydraulic fracturing boreholes in order to control the fracture path [11, 12].

It is known that directional riffling on the walls of parallel boreholes affects the fracture paths. A hybrid finite-discrete element method was used to simulate hydraulic fracturing of a borehole containing an initiation slot [13, 14]. The influence of orientation and length of the slot, pumping rate of the working fluid and effective stresses on fracture propagation in one and several boreholes is estimated. The obtained results are in good agreement with the data of laboratory experiments.

The results of numerical and physical studies of hydraulic fracture propagation in three parallel boreholes with directional riffling are presented, and viability of the proposed approach for hydraulic fracture control is noted [15, 16]. It is shown that for certain parameters of the problem: the angle between the direction of maximum horizontal stress $\sigma _H$ and some boreholes is more than 30° and $(\sigma _H - \sigma _h ) / \sigma _h > 0.5$, where $\sigma _H$ is the vertical stress, the fracture turns quickly in the direction of action $\sigma _H$. The problem of formation of a single hydraulic fracture from five parallel boreholes in a two-dimensional formulation was considered in [17].

This paper aims at numerical studying of a growing fracture created between two parallel boreholes in an isotropic medium in the nonuniform stress field and physical simulating of synchronous hydraulic fracturing in large-size artificial samples. The initial fracture in the numerical model was located at an angle to the axis of boreholes, which allowed studying the stability of fracture propagation along their walls depending on various problem parameters.

1. COMPUTATIONAL DOMAIN PARAMETERS AND NUMERICAL SIMULATION OF HYDRAULIC FRACTURE

Hydraulic fracturing was mathematically modeled in a three-dimensional formulation using the extended finite element method XFEM implemented in the ABAQUS software package [18–20]. This method helps to obtain solutions that include a displacement jump using discontinuous functions, and near the fracture tip—special asymptotic functions. This approach allows simulating the fracture propagation in the direction that is determined from the analysis of the stress state in the vicinity of its tip. An implementation of XFEM based on phantom nodes and cohesive damage rule was used [18, 21].

In calculations, we used an area of $10\times 10\times 10$ containing two parallel boreholes 0.1 m in diameter, interconnected by an initial fracture of length $l$, which has deviation $\alpha$ from the borehole axis. For $\alpha = 2.5^{\circ}$ and 5° we assumed $l=0.5$ m, for $\alpha = 10^{\circ}$—$l=0.2$ m to prevent errors in calculation of the fracture path due to its significant bending along the borehole wall. The direction of boreholes coincided with $z$-axis, and the spacing was $d$. Minimum element size in calculations—0.015 m. Compressive stresses $S_{xx}$, $S_{yy}$, $S_{zz}$ are set at the external boundary of computational domain. Elastic modulus of the medium $E=10$ GPa, Poisson’s ratio $\nu = 0.3$, dynamic fluid viscosity $\eta = 0.001$ Pa$\cdot$s, critical fracture energy $G_{c}=120$ N/m, compressive strength $\sigma _c = 1$ MPa. Geometric parameters of the medium and boundary conditions of the problem are shown in Fig. 1.

Fig. 1

Parameters of computational domain of the medium in central sections (a) $x0z$ and (b) $y0z$: $d$—spacing of boreholes; $l$—initial fracture length; $\alpha$—angle between initial fracture and borehole axis; $S_{xx}$, $S_{yy}$, $S_{zz}$—normal loads on the faces of a cubic model

2. RESULTS OF NUMERICAL EXPERIMENTS

We consider two parallel boreholes at a distance $d=0.5$ m from each other. Figure 2a shows the calculated 3D crack at its initial deflection $\alpha = 10^{\circ}$ in hydrostatic compression field $S_{xx}=S_{yy}=S_{zz}=1.0$ MPa, Fig. 2c—intersection of this fracture with the plane of symmetry $x=5$ m located between two boreholes and with the borehole surface, Fig. 2d—lines of intersection of the fracture with planes $z=4.0$, 4.6 and 5.0 m. Figures 2b, 2e and 2f are plotted similarly. The following parameters were used here: $\alpha = 5^{\circ}$; compression field $S_{xx}=S_{zz}=3.0$ MPa, $S_{yy}=4.5$ MPa; $z=4.36$, 4.45, 4.51, 4.60 and 5.00 m. It was found that in the case of a hydrostatic field, the fracture mainly propagates along the boreholes. If the vertical stress exceeds horizontal ones, the fracture is reoriented transversely to the borehole plane, even at a relatively small deviation angle of 5°.

Fig. 2

Shapes of hydraulic fractures between parallel boreholes at a distance of 0.5 m from each other: (a) hydrostatic compression $S_{xx}=S_{yy}=S_{zz}=1.0$ MPa, $\alpha = 10^{\circ}$; (b) $S_{xx}=S_{zz}=3.0$ MPa, $S_{yy}=4.5$ MPa, $\alpha =5^{\circ}$; (c) and (e) intersection of fractures with symmetry plane $x=5$ m (solid line) and with borehole (dashed line); (d) and (f) intersection of fractures with different planes $z$

Fig. 3

Intersection of a fracture with (a) symmetry plane $x=5$ m and (b) plane $z=6$ m under hydrostatic compression $S_{xx}=S_{yy}=S_{zz}=1.0$ MPa at $\alpha = 10^{\circ}$ (1), 5° (2) and 2.5° (3)

Figure 3 shows the intersections of fractures with a symmetry plane $x=5$ m (Fig. 3a) and with a plane $z=6$ m (Fig. 3b) at different angles $\alpha$ in a hydrostatic compression field of 1 MPa. With growth of $\alpha$, the fracture deflection between boreholes increases (see Fig. 3b). At small angles of $\alpha$, fractures are more quickly oriented parallel to the borehole plane and then propagate in this direction.

Figure 4 shows the intersections of fractures with the symmetry plane $x=5$ m (Fig. 4a) and with the plane $z=5.5$ m (Fig. 4b) at $\alpha = 5^{\circ}$ and two different nonuniform fields in the medium: $S_{xx}=S_{zz}=3.0$ MPa, $S_{yy}=4.5$ MPa and $S_{xx}=S_{zz}=1.0$ MPa, $S_{yy}=1.5$ MPa. The ratio between the maximum and minimum compressive stresses was preserved, but the stresses decreased in relation to tensile strength of the material $\sigma _c$. It was found that at high stresses relative to $\sigma _c$, the fracture is quickly reoriented in the direction perpendicular to the borehole plane.

Figure 5 shows the intersections of fractures with a symmetry plane $x=5$ m (Fig. 5a) and with the plane $z=5.7$ m (Fig. 5b) at $\alpha = 10^{\circ}$ and hydrostatic compression of the medium of 1 MPa for distances between borehole centers $d=0.50$ and 0.25 m. It was found that a decrease in distance $d$ leads to smaller deviations of the fracture from the plane that connects the centers of boreholes.

Dependencies in Fig. 5c, 5d are plotted similarly, but for this the following parameters were used: compression field $S_{xx}=S_{zz}=3.0$ MPa, $S_{yy}=4.5$ MPa, $z=5.5$ m. A significantly smaller fracture deflection is observed at $d=0.25$ m; however, unlike hydrostatic compression, the fracture is not retained in the plane that connects the centers of boreholes, and it gradually extends to the upper boundary of the borehole.

Fig. 4

Intersection of a fracture with (a) symmetry plane $x=5$ m and (b) plane $z=5.5$ m at $\alpha = 5^{\circ}$ in case of nonuniform stress field:1—$S_{xx}=S_{zz}=3.0$ MPa; $S_{yy}=4.5$ MPa;2—$S_{xx}=S_{zz}=1.0$ MPa, $S_{yy}=1.5$ MPa

Fig. 5

Intersection of fracture with (a) symmetry plane $x=5$ m and (b) plane $z=5.7$ m at $\alpha = 10^{\circ}$ and hydrostatic compression $S_{xx}=S_{yy}=S_{zz}=1.0$ MPa for borehole-to-borehole distances of 0.50 (1) and 0.25 m (2); intersection of a fracture with (c) symmetry plane $x=5$ m and (d) plane $z=5.5$ m at $\alpha = 10^{\circ}$ and stresses $S_{xx}=S_{zz}=3.0$ MPa, $S_{yy}=4.5$ MPa for $d=0.50$ (1) and 0.25 m (2)

3. RESULTS AND DISCUSSION

Additional numerical studies were carried out to analyze the calculated fracture paths and the influence of problem parameters on them. We consider two boreholes with applied internal pressure $P$ in the stress field $S_{xx}$, $S_{yy}$, $S_{zz}$ (Fig. 6a). Figure 6b$--$d shows stresses $\sigma _{xx}$, $\sigma _{yy}$, $\sigma _{zz}$ along segment $AB$ normalized to $\vert S_{xx}\vert$ at $S_{yy}=1.5S_{xx}$, $S_{zz}=S_{xx}$, $P=5S_{xx}$ and distance between borehole centers $d=0.5$ m. We assume that positive stresses correspond to extension of the medium, and negative ones—to compression. For the fracture to remain in the plane of boreholes, the following condition must be met: along segment $AB$ stress $\sigma _{yy}$ must be higher than $\sigma _{zz}$, as we assume that the fracture propagates perpendicular to the minimum compressive stress (Fig. 6b). In this case, the condition is satisfied in the vicinity of points $A$ and $B$ outside the area marked with dashed lines. As the fracture grows inside the area, its deviation from the plane of boreholes is expected, because $\sigma _{zz} > \sigma _{yy}$. This explains the deflection in numerically calculated fractures that is closer to the symmetry plane $x=5$ m between boreholes.

Figure 6c shows vertical stresses $\sigma _{yy}$ and stress $\sigma _{zz}$ (dashed line) under various borehole pressure. As we can see, if the pressure in boreholes is low ($P=2S_{xx}$), then stress curve $\sigma _{yy}$ will be absolutely below horizontal stresses $\sigma _{zz}$, which can result in fracture reorientation. At high pressure ($P=10S_{xx}$), it is possible to achieve the conditions when vertical stresses $\sigma _{yy}$ will be maximum between boreholes, and the fracture will propagate between them. It should be noted that the stress at the boundary of boreholes at points $C$ and $D$ could be quite high. This will cause fracturing at these points and fracture propagation in the vertical direction (in this case, at point $C$ the stress $\sigma _{xx} / |S_{xx}| = 7.7$, which considerably exceeds stresses $\sigma _{yy}$ between boreholes). Leaving the line connecting two boreholes sharply deteriorates the conditions of fracture propagation in the plane of boreholes (see Fig. 6d).

Fig. 6

(a) Statement of the problem and (b) stresses normalized to $S_{xx}$ under different conditions:1—$\sigma _{yy}$ along segment $AB$;2—$\sigma _{zz}$ along boreholes; 3—$\sigma _{xx}$ along segment $AB$; (c) stress $\sigma _{yy}$ along segment $AB$ under pressure $P=10S_{xx}$ (1), $5S_{xx}$ (2) and $2S_{xx}$ (3) (value of $\sigma_{zz}$ is marked with dashed line); (d) same as in (c), but along segment $CD$

During numerical experiments, it was revealed that in conditions of a hydrostatic and, in some cases, nonuniform stress field in the medium, fracture paths on the borehole surface are quickly oriented and propagate along the borehole axis. numerical example. This can be examplified numerically.

We consider a model with one borehole, on the walls of which there is an initial square-shaped fracture $0.05\times 0.05$ m in size with a deviation of 5° and 10° toward plane $xz$. Fracture growth is ensured by a gradual increase in pressure on the borehole walls, while fluid was not pumped into the fracture ("dry fracturing"). Pressure in the medium was $S_{xx}=S_{zz}=1.0$ MPa, $S_{yy}=2.0$ MPa. Fractures are immediately oriented parallel to the borehole plane and propagate along it (Fig. 7a).

Fig. 7

(a) Intersection of fractures with the borehole wall when it is loaded with pressure in nonuniform stress field $S_{xx}=S_{zz}=1.0$ MPa, $S_{yy}=2.0$ MPa for fracture deviations of 5° and 10° relative to plane $xz$ and (b) local area near the fracture tip on the borehole surface with effective stresses

Fig. 8

Model of artificial block with boreholes in power circuit equipped with two flat jacks: (a) top view; (b) central section of model in $z0y$-plane

Figure 7b shows a local area near the fracture tip on the borehole surface. Stresses $\sigma _{11}$ and $\sigma _{22}$ affect this surface element. In this case, $\sigma _{11}$ is horizontal compressive stress $\sigma _{zz}$ along the borehole axis, $\sigma _{22}$ equals axysymmetric stresses. Due to the applied pressure, condition $\sigma _{22} > \sigma _{11}$ is satisfied on the borehole surface (stress sign is accounted in the inequality) causing the fracture growth along the borehole, which was observed in the calculations.

4. LAB-SCALE HYDRAULIC FRACTURE SIMULATION

Hydraulic fracturing was physically simulated on a setup designed for testing large-size cubic samples. Its layout and basic characteristics are given in [22, 23]. Two samples with a rib length of 420 mm were made from M300 sandcrete (Portland cement and quartz sand in a ratio of 1:2). After pouring into special molds that ensure the parallelism of faces, the samples were kept for 21 days until complete hardening. Then, two boreholes with a diameter $D=12$ mm were drilled in each sample with a center-to-center distance of 30 ($2.5D$) and 66 ($5.5D$) mm, respectively. Boreholes were not connected to the initial fracture, as was done in the numerical modeling. Metal fittings 85 mm long fixed with epoxy resin were installed in these boreholes to supply working fluid (Fig. 8).

The material of physical models was studied and their mechanical properties were determined on the equipment provided by the Center for Collective Use of Geomechanical, Geophysical and Geodynamic Measurements, SB RAS. The results are presented in Table 1.

A fracture in the sample was created by simultaneously pumping hydraulic oil MGE-46V into both boreholes. The open borehole interval was 215 mm. Table 2 shows the loading conditions of physical models, maximum pressures recorded in the first and second cycles of hydraulic fracturing, as well as injection volumes in each cycle.

Table 1. Mechanical properties of geomaterial for two physical models

Table 2. Experimental and pressure parameters of hydraulic fracturing

After the experiment, the sample was partially sawn with a circular saw to a depth of 70 mm, then it was split with metal wedges. The rupture plane was located at a distance of 250 mm from the upper boundary of the block. Figure 9 shows photographs of broken surfaces of physical models with traces of formed fractures, as well as the results of manual tracing of detected fractures.

At close spacing of the boreholes, their connection by a hydraulic fracture is observed in conditions when stress $S_{yy}$ exceeds $S_{xx}$. The fracture is predominantly formed in one of the boreholes, which is due to the low supply rate of working fluid and a pressure drop in the hydraulic system at the initial moment of fracture formation In the second experiment, despite the maximum stress $S_{xx}$, boreholes were not connected by the fracture. A fracture between boreholes at the upper boundary of the block is noticeable, but it is most likely associated with the presence of fittings, the lower edge of which can cause fracture formation in the near-surface sample area. If we analyze the divergence of fractures on the rupture plane of the model, a similar behavior was noted in other numerical and physical experiments [12, 16]. When comparing the fracture paths in two tests, we note that in the first case, the most extended fracture wing is directed toward the upper boundary of the block, in the second case, it tends to reach the side face. This is explained by the value of horizontal compressive stresses $S_{xx}$ compared with the others, which is minimum in the first test and maximum in the second.

Fig. 9

Broken surfaces of physical models in the $y$0$x$-plane for borehole spacing of (a) $d=2.5D$ and (b) $5.5D$; on top—original photographs, on bottom—showing results of manual tracing of fractures (solid line) and artificial fractures that formed during the block breaking (dashed line)

CONCLUSIONS

The characteristic features of propagation of the initial fracture created between two parallel boreholes in the nonuniform stress field are determined. The influence of such parameters as the angle of deviation of the initial fracture from the axis of boreholes, spacing of the boreholes and the compressive stress field between them on the fracture path was estimated.

In the hydrostatic stress field, the fracture propagates along the boreholes. As the angle of deviation $\alpha$ grows, the fracture deflection between boreholes increases. At small angles of $\alpha$, the fracture is quickly oriented parallel to the borehole plane and then propagates in this direction. In case when the vertical compression exceeds the horizontal ones, even at a relatively small deviation angle of 5°, the fracture is reoriented perpendicular to the borehole plane. The fracture is reoriented the faster, the stronger compressive stresses differ from the tensile strength $\sigma _c$ of the material.

Reducing the spacing of boreholes in the hydrostatic stress field leads to smaller deviations of the fracture from the plane that connects their centers even at $\alpha = 10^{\circ}$. We can observe how the fracture gradually reaches the upper boundary of the borehole in the nonuniform stress field at a minimum distance of 0.25 m.

The results of physical simulation of hydraulic fracturing of parallel boreholes showed that they become connected by a fracture at a relatively close spacing ($d=2.5D$), and in the case of $d=5.5D$ such connection does not take place despite the maximum stress $S_{xx}$ in the experiment.

ACKNOWLEDGMENTS

The authors are grateful to the Center for Collective Use of Geomechanical, Geophysical and Geodynamic Measurements, SB RAS for providing equipment for geomechanical core studies

FUNDING

The work was supported by the Russian Foundation for Basic Research and by the Government of the Novosibirsk Region, project no. 20-45-540005.