Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal Stress Criterion

Transcription

1 Rock Mechanics and Rock Engineering ORIGINAL PAPER Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal Stress Criterion Quansheng Liu 1 Lei Sun 1 Xuhai Tang 1,2 Bo Guo 3 Received: 27 July 2017 / Accepted: 5 November 2018 Springer-Verlag GmbH Austria, part of Springer Nature 2018 Abstract Accurate simulation of the propagation of hydraulic fractures under in situ stress conditions in three dimensions (3D) is critical for the enhanced design and optimization of hydraulic fracturing in various engineering applications, such as shale gas/oil production and geothermal utilization. To model fracture propagation for geotechnical applications numerically, a maximum principal stress criterion (MPS-criterion) with a weighted average approximation is conventionally applied. However, it is found that the weighted average approximation is inappropriate for hydraulic fracturing under in situ stress conditions, where the presence of both hydraulic pressure and in situ stress can lead to sharp changes of the stress field in the vicinity of the fracture tips. When both hydraulic pressure and in situ stress are considered, the simulated results with the weighted average approximation are inaccurate and are sensitive to the radius of the computational area. In this paper, we present numerical tests to identify this limitation of the weighted average approximation and propose a novel point-based approximation for the MPS-criterion. The performance of the MPS-criterion with the point-based approximation for hydraulic fracturing under in situ stress conditions is confirmed by a numerical test. It can be seen that, compared to the traditional weighted average approximation, the MPS-criterion with the point-based approximation is more stable and accurate for modelling hydraulic fracturing under in situ stress conditions. Keywords Hydraulic fracturing Point-based approximation Maximum principal stress criterion (MPS-criterion) Generalized finite element method (GFEM) Hydro-mechanical coupling List of Symbols σ x, σ y and σ z p w σ tip,1 T 0 σ tip σ i * Xuhai Tang xuhaitang@whu.edu.cn In situ stress in x, y, and z direction Boundary fluid pressure Maximum principal stress at the fracture tip Strength of the material Stress tensor at the fracture tip Stress tensor at the integration point i School of Civil Engineering, Wuhan University, Wuhan , China State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan , China Department of Energy Resources Engineering, Stanford University, Stanford, CA 94305, US η i σ i,1 ng d i ε σ and ε Ω P 1, P 2, P 3, P 4 ω 1 (x), ω 2 (x), ω 3 (x), ω 4 (x) u h (x) Weight function associated with integration point i for the calculation of stress tensor at fracture tip Maximum principal stress at the integration point i Total number of integration points in a computational area Distance from the integration point i to the fracture tip A positive but small number Cauchy stress tensor and strain tensor Tetrahedral element with four vertex nodes Vertex nodes of a tetrahedral element Weight functions at a computational point x of GFEM Global approximation in domain Ω Vol.:( )

2 Q. Liu et al. u i (x) Local approximation associated with node i χ vis Visibility zone Ω φ i (x) Sub-weight functions φ i (x) Shape functions of GFEM associated with node i vol (P 1, P 2, P 3, P 4 ) Volume of a tetrahedral element S Fracture surface S + and S The actual upper side and lower side of fracture surface u + (x) and u (x) Point displacements on the upper and lower sides of the fracture surface δ(x) Apertures of point x on fracture surface δ 0 Initial aperture of fracture p i+1 n and pi n Nodes of fracture tips at steps i + 1 and i Δ n Fracture propagation vector Δijk Fracture element with nodes i, j and k p i, p j and p k Fluid pressures at nodes i, j and k O Centroid point of a fracture element Ω i j Fluid path between node i and node j J i j Fluid pressure gradient between node i and node j z i and z j Vertical coordinates of node i and node j l i j Distance between node i and node j µ Fluid dynamic viscosity coefficient δ i j Equivalent aperture of the flow path between node i and node j b i j Equivalent width of the fluid path between node i and node j q i j Flow rate from node i to node j s Node saturation s n and s n 1 Node saturation at the current time step and previous time step f s A function of saturation p n and p n 1 Fluid pressure at the current time step and previous time step Bulk modulus of fluid K w q t V n and V n 1 K a F F solid, F fluid N E v a r λ p 1, p 2, p 3, p 4 1 Introduction Total flow rate Time increment Volumes of the node at the current time step and previous time step Global stiffness matrix of GFEM Nodal displacement vector of GFEM Load vector of GFEM Vectors of the forces for solid mechanics and forces for fluid mechanics Shape functions for calculating fluid pressure Young s modulus of the rock mass Poisson s ratio of the rock mass Radius of the fracture Distance to the fracture central point Ratio of the in situ stresses and hydraulic pressure Monitoring points Fluid-driven fracture propagation in geomaterials (e.g., soil or rock) exists in many engineering applications, such as heat production from geothermal reservoirs (Salimzadeh et al. 2016) and stability analyses of caprocks during CO 2 injection (Pan et al. 2014). In the past decades, hydraulic fracturing together with horizontal drilling have led to a rapid increase in oil and gas productions from unconventional shale source rocks (Barati and Liang 2014; Wu and Olson 2013). Predicting and controlling the propagation of hydraulic fractures is essential to optimize the energy production (Chen and Carter 2016), as well as to reduce the risk of hazards induced by human activities, such as earthquakes (Bao and Eaton 2016), groundwater contamination (Vengosh et al. 2014), and greenhouse gas emissions (Thomas et al. 2017). Hydraulic fracturing is a hydro-mechanical (H-M) coupling problem that involves three main physical processes (Adachi et al. 2007): (1) mechanical deformation of rocks induced by fluid pressure acting on the fracture surfaces, (2) fracture growth, and (3) fluid flow in the fractures. From the viewpoint of modeling hydraulic fracturing, there are three main challenges: (1) computing the 3D stress strain field and the deformation of the rock mass with discontinuities, (2) determining the fracture initiation condition and

3 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal propagation direction, and (3) coupling the rock deformation, fracture propagation, and fluid flow. Many advanced numerical methods have been developed to simulate the stress strain field and fracture propagation of rock masses. The finite element method (FEM) (Zienkiewicz and Taylor 2008) is a well-established method and has been utilized to simulate 3D fracture propagation for several decades. However, the finite element meshes need to be identical with the fracture surfaces, since the fractures follow the topology provided by the element edges. Therefore, advanced remeshing techniques have been developed for fracturing simulation (Paluszny and Zimmerman 2011; Paluszny et al. 2016). Currently, the FEM with advanced remeshing techniques still has difficulties handling fieldscale reservoirs with complex pre-existing discontinuities. Based on the concept of partition of unity (PU) (Melenk and Babuška 1995; Babuška and Melenk 1997), several approaches have been developed for fracturing simulation without remeshing, including the mesh-free methods (Rabczuk and Belytschko 2004), the generalized finite element method (GFEM) (Duarte et al. 2000; Pereira et al. 2009), the extended finite element method (XFEM) (Moës et al. 1999), the hybrid FE-Meshfree method (Tang et al. 2009; Yang et al. 2015; Liu et al. 2018) and the numerical manifold method (Yang et al. 2016, 2018). These methods are attractive for simulating complex discontinuities in a solid material because they remove the complexity of the computational geometry and allow discretization independent of the underlying geometry. Numerous failure criteria have been introduced for modelling fracturing in numerical methods, including Griffith criterion (Griffith 1921), J-criterion (Rice 1968), strain energy density criterion (Sih 1973, 1974), G-criterion (Hussain and Pu 1974), Coulomb failure criterion (Coulomb 1773), Mohr failure criterion (Mohr 1900), and maximum principal stress criterion (Erdogan and Sih 1963; Schöllmann et al. 2002). The maximum principal stress criterion (MPS-criterion) is widely used to determine the fracture propagation behavior (e.g., whether a new fracture should be generated and along which direction the fracture should propagate), because it is intuitive and convenient. In the application of the MPScriterion in numerical simulations, the weighted average approximation has been generally used to numerically calculate the maximum principal stress at the fracture tip (Yang et al. 2016; Ning et al. 2011; Terada et al. 2007; Bažant and Planas 1998; Bouchard et al. 2000). However, this weighted average approximation suffers from several limitations when hydraulic pressure and in situ stress are considered simultaneously. As shown in Fig. 1, when there is only hydraulic pressure acting on the fracture surfaces, the stress field around the fracture tip gradually changes from tensile to zero; when both hydraulic pressure and in situ stress are considered, the stress field around the fracture tip changes from tensile to compressive in the y direction. When the stress field around the fracture tip changes from tensile to compressive, the maximum principal stress at the fracture tip calculated by the weighted average approximation method is inaccurate and sensitive to the radius of the computational area, which introduces significant numerical error in fracture propagation simulation (see Sect. 6). To overcome these limitations, we developed a novel point-based approximation for the MPS-criterion, which is more stable and accurate. In this paper, we first present the formulation of the MPScriterion with the point-based approximation in Sect. 2. We Fig. 1 The variation of stress states around the fracture tip. a When there is only hydraulic pressure acting on the fracture surfaces, the stress field around the fracture tip gradually changes from tensile to zero. b When both hydraulic pressure and in situ stress are considered, the stress field around the fracture tip changes from tensile to compressive in the y direction pw o σ y tensile y pw compressive o σ y tensile σ y (a) (b)

4 Q. Liu et al. then use the GFEM to model the stress strain field and fracture propagation (Sect. 3), following which, we develop a fluid flow algorithm based on the parallel-plate model to model the fluid flow in fractures (see Sect. 4). The GFEMbased solid-phase solver and the fluid-phase solver are explicitly coupled, named GFEM-Fluid, to handle the H-M coupling process (see Sect. 5). In Sect. 6, we identify the numerical error introduced from the traditional weighted average approximation, and show that the presented pointbased approximation can overcome such limitations. In Sect. 7, a numerical test is designed to confirm the performance of the GFEM-Fluid and MPS-criterion with a point-based approximation, and the numerical results are compared with experimental results. The paper concludes in Sect. 8 with a summary of the major findings. 2 Maximum Principal Stress (MPS) Criterion with a Point Based Approximation for Fracture Advancement The MPS-criterion is widely used in rock engineering to determine the fracture propagation behavior. The MPS-criterion states that: (1) fracture propagates when the maximum principal stress at the fracture tip reaches the strength of the material. (2) fracture propagates perpendicularly to the direction of the maximum principal stress. Mathematically, condition (1) for the fracture growth can be expressed as σ tip,1 = T 0, where σ tip,1 is the maximum principal stress at the fracture tip, and T 0 is the strength of the material. Once the maximum principal stress σ tip,1 exceeds the strength of the material, the fracture tip propagates in the direction perpendicular to the maximum principal stress, as shown in Fig Traditional Weighted Average Approximation To calculate the maximum principal stress at the fracture tip, the weighted average approximation is widely used in many numerical methods, such as the numerical manifold method (Yang et al. 2016; Ning et al. 2011), finite cover method (Terada et al. 2007), adaptive FEM (Bazant and Planas 1998), and FEM with the advanced remeshing technique (Bouchard et al. 2000). In the weighted average approximation, the maximum principal stress at the fracture tip is approximated using the stresses at the surrounding integration points. A circular computational area around the fracture tip is used to identify the surrounding integration points (see Fig. 3). The stress tensor at the fracture tip σ tip is then calculated with the weighted average of all stress tensor at surrounding integration points in the calculation area: (1) ng σ tip = η i σ i, where ng is the total number of integration points in the computational area and σ i is the stress tensor at the i-th integration point. The parameter η i is a weight function associated with the distance from integration point to the fracture tip, which can be expressed as η i = i=1 L i ng j=1 L, i fracture σ y σ σ τ θ σ Fig. 2 The stress components (σ x, σ y and τ xy ) and principal stresses (σ tip,1 and σ tip,2 ) at the fracture tip. The fracture propagation direction (red line) is perpendicular to the direction of maximum principal stress σ tip,1. (Color figure online) calculation area fracture fracture tip integration point di i σ tip = x σ ng i=1 σ σ tip,1 σ tip,1 η σ i tip,2 tip,2 i (i=1,...,ng) Fig. 3 The weighted average approximation for MPS-criterion. The black point is the fracture tip, and the blue points are integration points. The red circle is the calculation area. The stress tensor at fracture tip σ tip is obtained by the weighted average of all stress tensors σ i (i = 1,, ng) at integration points in the calculation area with respect to a weight function η i, which is associated with the distance (d i ) between integration point and the fracture tip. Then, the maximum principal stress at fracture tip σ tip,1 is calculated with the stress tensor σ tip. Once the maximum principal stress σ tip,1 exceeds the strength of the material, the fracture tip propagates in the direction perpendicular to the maximum principal stress (2) (3)

5 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal and L i = 1 d i + ε, where d i is the distance from the i-th integration point to the fracture tip and ε is a positive but small number. Then, the maximum principal stress at fracture tip σ tip,1 is calculated with the stress tensor σ tip. The value and direction of σ tip,1 are the same as the maximum eigenvalue and eigenvector of the stress tensor σ tip. Once the maximum principal stress σ tip,1 exceeds the strength of the material, the fracture tip propagates in the direction perpendicular to the maximum principal stress, as shown in Fig Point Based Approximation When considering both the effect of hydraulic pressure and in situ stress, there are both a tensile zone and a compressive zone near the fracture tip. The stress states of the integration points in the computational area change between tensile and compressive. Because of the compressive stress around the fracture tip, the maximum principal stress at the fractures tip calculated by the weighted average approximation is sensitive to the radius of the computational area. The value and direction of the maximum principal stress are inaccurate when considering the hydraulic pressure and in situ stress simultaneously. The inaccurate maximum principal stress at the fracture tip introduces significant numerical errors in modeling the fracture propagation (see Sect. 6). To overcome these limitations, we develop a novel point-based approximation for the MPS-criterion. In the point-based approximation, the maximum principal stresses of all the surrounding integration points σ i,1 (i = 1,, ng) in the vicinity of fracture tips are first calculated. Integration point j with the maximum value of maximum principal stress among these integration points is extracted. σ j,1 = max ( σ i,1) (i = 1, 2,, ng), (5) where σ j,1 is the maximum principal stress at the integration point j, σ i,1 is the maximum principal stress at the i-th integration point, ng is the total number of integration points in the computational area. The stress tensor at the fracture tip is assumed to be equal to that at integration point j. σ tip = σ j. Then, the maximum principal stress at fracture tip σ tip,1 is calculated with the stress tensor σ tip. The value and direction of σ tip,1 are the same as the maximum eigenvalue and eigenvector of the stress tensor σ tip. Once the maximum principal stress σ tip,1 exceeds the strength of the material, the fracture (4) (6) calculation area fracture fracture tip integration point tip propagates in the direction perpendicular to the maximum principal stress, as shown in Fig Solid Phase i j σ j,1 i,1 = max( σ ) (i=1,...,ng) σ j,1 σ j,2 σ tip,1 σ tip = σ j σ tip,2 Fig. 4 The point-based approximation for MPS-criterion. First, the maximum principal stresses of all integration points σ i,1 (i = 1,, ng) in calculation area are calculated. An integration point j, which has the maximum value of maximum principal stress among these integration points, is picked. The stress tensor at fracture tip σ tip is assumed to be the same as that at integration point j, σ j. Then the maximum principal stress at the fracture tip σ tip,1 is calculated with the stress tensor σ tip. Once the maximum principal stress σ tip,1 exceeds the strength of the material, the fracture tip propagates in the direction perpendicular to the maximum principal stress Fig. 5 The definition of the FE-Element, Bridge-Element, Frac-Element, PU-nodes and FE-nodes The GFEM is constructed using the principles of partition of unity (PU) to simulate rock deformation and fracture propagation without remeshing. In this section, we introduce these principles in a simplified manner. Details on the GFEM can be found in many papers available in the literature (Strouboulis et al. 2000; Duarte et al. 2001, 2007; Pereira et al. 2009). To introduce the GFEM, some of the schematic diagrams are plotted in two dimensions (2D). As shown in Fig. 5, in the context of GFEM, there are three different categories of elements: Frac-Elements, Bridge-Elements and FE-Elements, and there are two kinds of nodes: PUnodes and FE-nodes. The Frac-Elements are the elements

6 Q. Liu et al. crossed by fractures. The Bridge-Elements are adjacent to these Frac-Elements. The other elements are FE-Elements. All nodes within the Frac-Elements are PU-nodes, and the rest are FE-nodes. Additionally, the Bridge-Elements consist of both PU-nodes and FE-nodes. In the GFEM, we consider small displacement elastostatics, which is governed by the equation of equilibrium: σ + b = 0 where in V, σ = C ε, ε = s u. (8) In the above equations, V is the domain of the body, σ is the Cauchy stress tensor, ε is the strain tensor, b is the body force, C is the material moduli tensor, u is the displacement, is the gradient operator, and s is the symmetric gradient operator. The essential and natural boundary conditions are: u = u on G u, n σ = t on G t, where Γ = Γ u + Γ t is the boundary of V with unit normal vector n, and u and t are the prescribed displacements and tractions, respectively. The crack faces considered are traction free. Based on the principles of the partition of unity method (PUM) (Melenk and Babuška 1995; Babuška and Melenk 1997), the construction of the displacement approximation u is aided with a set of non-negative weight functions ω i (x). In the present GFEM, a tetrahedral element Ω with four vertex nodes P = {P 1, P 2, P 3, P 4 } is considered, as shown in Fig. 6. At an arbitrary point, x = {x, y, z} in the domain Ω, the global displacement approximation u h (x) in domain Ω is defined as u h (x) = 4 ω i (x)u i (x), i=1 where u i (x) is the local displacement approximation associated with node i. The local displacement approximations (7) (9) (10) associated with the FE-nodes are set to 1.0 and the local displacement approximations for the PU-nodes are constructed using the least square method, which has been discussed by Yang et al. (2014, 2017). The method of constructing the weight function ω i (x) of the Frac-Elements, Bridge- Elements, and FE-Elements are explained in Sects. 3.1 and 3.2 in detail. It should be noted that the summation of weight functions is equal to 1 in the domain Ω: ω 1 (x) + ω 2 (x) + ω 3 (x) + ω 4 (x) = Construct Weight Function for the Frac Elements (11) For Frac-Elements, the discontinuity in the displacement field across the fracture should be reflected properly. In this study, discontinuous displacement across the fracture is constructed using the visibility criteria (Belytschko et al. 1994, 1996), where the fracture surface is considered to be opaque. The computational points P(x) and P i are visible if the segment [P(x) P i ] does not intersect the fracture surface, as shown in Fig. 7. We define χ Ω as an index set of nodes that are visible associated with the element domain Ω, and the visibility zone χ vis is defined as (Duarte et al. 2000) Ω χ vis Ω = { x i χ Ω [x x i ] fracture surface = }, (12) where x i is the coordinate associated with node P i. The weight functions of the present method are constructed using the Shepard formula (Lancaster and Salkauskas 1986; Duarte et al. 2001). A set of functions φ i (x) is first introduced, which are constructed using FE shape functions on tetrahedral elements: φ 1 (x) = vol(p(x)p 2 P 3 P 4 ) vol(p 1 P 2 P 3 P 4 ), φ 2 (x) =vol(p(x)p 3 P 4 P 1 ) vol(p 1 P 2 P 3 P 4 ) φ 3 (x) = vol(p(x)p 4 P 1 P 2 ) vol(p 1 P 2 P 3 P 4 ), φ 4(x) = vol(p(x)p 1 P 2 P 3 ) vol(p 1 P 2 P 3 P 4 ), (13) Pi P1 P(x2) + fracture surface S p S P(x1) Pl _ Pj P2 3 Fig. 6 A tetrahedral element Ω crossed by a fracture S P 4 Fig. 7 The fracture surface is considered opaque. The computational points P(x 2 ) and P i are visible because the segment [P(x 2 ) P i ] does not intersect the fracture surface. Computational points P(x 1 ) and P i are invisible, because segment [P(x 1 ) P i ] intersects the fracture surface, [P(x 2 ) P i ] S = p Pk

7 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal where vol(p 1 P 2 P 3 P 4 ) is the volume of the tetrahedral element and vol(p(x)p i P j P k ) is the volume of the tetrahedron with vertexes P(x), P i, P j and P k, as shown in Fig. 8. Using the visibility criteria, we define a set of sub-weight functions φ i (x) =φ i (x) when computational points P(x) and vertex P i are visible, and φ i (x) =0 when computational points P(x) and vertex P i are invisible. { φ i (x) = φi (x) x χ vis Ω 0 x χ vis. Ω The weight function associated with node i is then defined as φ i ω i = (x) φ 1 (x)+φ 2 (x)+φ 3 (x)+φ 4 (x). 3.2 Construct Weight Functions for the Bridge Elements and FE Elements (14) (15) The formulation of the weight functions for the Bridge- Elements and FE-Elements used in this study is the same as the formulation in the standard FEM (Zienkiewicz and Taylor 2008): P1 ω 1 (x) = vol(p(x)p 2 P 3 P 4 ) vol(p 1 P 2 P 3 P 4 ), ω 2 (x) =vol(p(x)p 3 P 4 P 1 ) vol(p 1 P 2 P 3 P 4 ), ω 3 (x) = vol(p(x)p 4 P 1 P 2 ) vol(p 1 P 2 P 3 P 4 ), ω 4(x) = vol(p(x)p 1 P 2 P 3 ) vol(p 1 P 2 P 3 P 4 ). 3.3 Geometric Representation of the Fracture (16) In the GFEM, the fracture tip is assumed to be sharp. The fracture is represented by a 3D fracture surface, which is discretized into a set of triangular fracture elements. This methodology was successfully applied in the GFEM (Duarte et al. 2001, 2007; Kim et al. 2010; Pereira et al. 2009, 2010a, b). As shown in Fig. 9, the rock mass and the fracture surface are discretized into a set of tetrahedral elements and triangles, respectively. The numerical integration over the volume elements crossed by fracture surfaces is performed with the aid of a set of sub-elements (Liu et al. 2018). The distortion of these sub-elements is not sensitive to the results since they are not used to define the shape functions, validated by Pereira et al. (2010b). It is noted that the fracture surface defined here is a reference fracture surface, which is used to represent the fracture path numerically. The actual upper side and lower side of the fracture surface (S + and S ) can be calculated according to the reference fracture surface (Sanborn and Prévost 2011), as shown in Fig. 10. S + (x) =S(x)+u + (x), (17) vol(p(x)p4p1p2) P(x) P2 vol(p(x)p1p3p4) vol(p(x)p1p2p3) vol(p(x)p3p4p1) P 4 S _ S + n + S P 3 _ Fig. 8 A tetrahedral element is discretized into four sub-elements by node P(x). vol(p 1 P 2 P 3 P 4 ) is the volume of the tetrahedral element. vol(p(x)p i P j P k ) is the volume of the tetrahedron with vertexes P(x), P i, P j and P k Fig. 10 S + and S are the upper side and lower side of fracture, respectively, which can be calculated based on the reference fracture surface S Fig. 9 The discretization of rock mass and fracture surface + = (a) a rock mass is discretized into tetrahedral GFEM-elements (b) a fracture-surface is discretized into triangular fracture-elements (c) a rock mass with a fracture-surface

8 Q. Liu et al. S (x) =S(x)+u (x), (18) where u + (x) and u (x) are the displacements at computational point x on the upper side and the lower side of fracture surface, respectively. The aperture of fracture is defined as the difference in displacement from the S + to the S side of the discontinuity surface (Sanborn and Prévost 2011). The apertures at point x on fracture surfaces can be obtained from: δ(x) =δ 0 + u + (x) u (x), where δ 0 is the initial aperture, which can be zero or another value depending on the actual situation. In GFEM, generally it is assumed that the aperture of fracture is relatively small compared to the rock mass. 3.4 Updating the Fracture Surfaces (19) During the fracturing simulation, the number of fracture tips and fracture surfaces increases. The fracture propagation behavior is determined using the MPS-criterion with the point-based approximation (see Sect. 2). The nodes located on the fracture tip are defined as tip nodes, which will propagate to new tip nodes if fracture propagates (see Fig. 11a). Then, the set of fracture elements is updated by adding new fracture surface segments between the old fracture tips and the newly created fracture tips (Fig. 11b). The location of the new tip nodes is determined as influence domain of node i is the union of all fracture elements associated with node i (see Fig. 12). Taking fracture element Δijk as an example, p i, p j and p k are the fluid pressures at nodes i, j and k, respectively, and we assume that p i > p j > p k. The fracture element is divided into three sub-domains from its centroid point, O (see Fig. 13), and Ω i j is defined as the fluid flow path between node i and node j. Assuming a linear distribution of fluid pressure within the triangular surface, the pressure gradient within Ω i j can be calculated as J i j = p i p j + ρg( z i z j ) (21) where z i and z j are the vertical coordinates of nodes i and j, respectively, and l i j is the distance between node i and node j. In this case, the flow rate from node i to node j is calculated by the cubic law (Batchelor 1967; Witherspoon et al. 1980): q i j = δ i j 3 J i j 12μ l i j, b i j, (22) where µ is the fluid dynamic viscosity coefficient, δ i j is the equivalent aperture of the flow path, and b i j is the equivalent width of the fluid path between node i and node j, which p i+1 n = p i n + Δ n, (20) where p i+1 n and pi n are the fracture tip nodes at steps i and i + 1 and Δ n represents the propagation vector. 4 Fluid Phase 4.1 Governing Equations of the Fluid Phase The fracture elements with implicit apertures are the only fluid pathways. The fluid flow in each fracture element is modeled as a parallel-plate flow (Witherspoon et al. 1980). The Fig. 12 The influence domain of node i and the fracture elements with implicit apertures Fig. 11 Representation of fracture updating: a the generation of new tip nodes: Δ n is the propagation vector, p i n and pi+1 n are the fracture tip nodes at steps i and i + 1; b new fracture and fracture elements, tip i and tip i+1 are the fracture tips at steps i and i + 1

9 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal (Itasca 2005; Liu et al. 2018). In this case, the flow rate is calculated as ( ) fsi + f sj q i j = q j i = δ i j 3 J i j 2 12μ b i j, (23) where f s is a function of the saturation (Lin and Lee 2009) and Fig. 13 The flow path associated with a fracture element Δijk. The fracture element is divided into three sub-domains by the centroid point O. The fluid path Ω i j is the path of fluid flow between node i and node j. q i j is the flow rate between node i and node j i δi V i ijk V k ijk V j ijk Fig. 14 The volumes of nodes associated with a fracture element Δijk. V i Δijk, Vj Δijk and Vk are the volumes of nodes i, j and k associated Δijk with fracture element Δijk. The volume of each node is equal to onethird of the volume of fracture element can be calculated as b i j = Ω i j l i j. The sign convention is defined as follows: flow-out is negative and flow-in is positive. 4.2 Updating the Fluid Pressure and Saturation It should be noted that the hydraulic nodes here can be considered as a virtual cavity with a certain volume. The volume of node i associated with fracture element Δijk, V i Δijk, is shown in Fig. 14. The total volume of node i is the summation of the volumes of node i associated with all the connecting fracture elements. A dimensionless factor, domain saturations = V fluid, is introduced here to identify whether the V node node domain is filled with fluid, where V fluid is the volume of fluid and V node is the volume of the node domain. The fluid mass flow into the node domain is either used to change fluid pressure or to change the saturation (Itasca 2005). When the node domain is fully saturated, the fluid pressure changes with the injected fluid mass and the fluid rate is updated using Eq. (22). Otherwise, when the node domain is not filled with fluid, the fluid flow is used to fill the domain and the fluid pressure will not increase. The effective flow rate should decrease when the saturation decreases k δk j δj f s =s 2 (3 2s) 0 s 1, (24) where s is the node saturation. The total flow rate of node i can be calculated as the summation of the flow rates at node i associated with all the connecting fracture elements. The fluid pressure at a node can be updated as (Itasca 2005) { p n 1 p n + K = w q Δt K ΔV V n w s = 1 V m (25) 0 s < 1, where p n and p n 1 are the fluid pressure at the current time step and previous time step, respectively. K w is the bulk modulus of the fluid, q is the total flow rate, and t is the time increment. In addition, V n and V n 1 are the volumes of the node at the current time step and previous time step, respectively. We have V = V n V n 1 and V m = Vn +V n 1. 2 The saturation of node i can be updated as (Itasca 2005) s n = s n 1 + q Δt V ΔV V m, where s n and s n 1 is the saturation of node i at current and previous time step. It should be indicated that if s n >1, we set s n =1, and Eq. (25) is used to update the fluid pressure. If s n <1, p is set zero, and Eq. (26) is used to update the saturation. Note that there is a convergence criterion herein, that is, the hydraulic time step size should be less than a critical value (Lisjak et al. 2017): [ ] Δt Δt V = min, (27) K w kn where V is the node volume and kn is the permeability factor of the n-th flow path connected to the hydraulic node i, which is calculated as k n = δ2. For all nodes, the minimum 12μ value of Δt is used in the algorithm. 4.3 Boundary Conditions (26) There are two kinds of boundary conditions of fluid flow: prescribed pressure condition and prescribed flow rate condition, as shown in Fig. 15.

10 Q. Liu et al. 1. Prescribed pressure condition If a prescribed pressure value, p 0, is applied on the specified hydraulic nodes, the fluid pressure at these hydraulic nodes is set to p 0 instead of being computed using Eq. (25). p = p Prescribed flow rate condition If a prescribed flow rate value, q 0, is assigned to the specified hydraulic nodes, the total flow rate at these nodes can be set as q 0 instead of being computed using Eq. (23). q = q 0. (29) Fluid injection and extraction are obtained by prescribing positive and negative q 0 values, respectively. 5 Coupling of GFEM Fluid 5.1 Basic Assumptions (28) We make the following assumptions to couple the GFEM with the fluid algorithm: (1) the rock mass is impermeable, and fluid flow only occurs in the fractures; (2) the fluid flow is assumed to be laminar, viscous, and compressible; and (3) the apertures of the fractures are calculated from Eq. (19), which is related to the initial aperture and the rock deformation. 5.3 The GFEM Implementation for Internal Fluid Flow The system of the equilibrium equation can be written in matrix form as follows: Ka = F, (30) where K is the global stiffness matrix, a =[a 1 a 2 a n ] T is the matrix of nodal displacements, and F is the equivalent load vector of the structure and can be expressed as follows: F = F solid + F fluid, (31) where F solid and F fluid are vectors of the external forces for solid mechanics and internal forces for fluid mechanics, respectively. F fluid can be represented as F fluid = s Npds, (32) where N is the shape function of this GFEM and p is the fluid pressure applied to the fracture surfaces. As shown in Fig. 17, a body Ω is crossed by a fracture S. The fluid pressure is applied normally to the upper and lower fracture surfaces (S +, S ), and the distribution of the fluid pressure can be simplified as a linearly distributed load. p i, p j and p k are the fluid pressures at the nodes 5.2 Coupling Procedure The GFEM-Fluid consists of two parts: (1) the solid solver, which simulates the rock mass deformation and fracture propagation, and (2) the fluid solver, which simulates the fluid flow process and updates the fluid pressure. The schematic diagram of the coupling between the two solvers is presented in Fig. 16. Fig. 16 The coupling between the solid solver and fluid solver in GFEM-Fluid Fig. 15 Prescribed pressure condition and prescribed flow rate condition are applied on a specified hydraulic node i: a the pressure at node i is always set to p 0 ; b the flow rate into node i is set to q 0 p=p0 i q=q0 i (a) A prescribed pressure condition p=p 0 is applied on node i. (b) A prescribed flow rate condition q=q 0 is applied on node i.

11 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal i, j and k of fracture elements, respectively, which are obtained using the fluid solver. σ z R 50 6 Comparison Between the Weighted Average Approximation and Point Based Approximation for the MPS Criterion z o y x a 10 H 50 In this section, a numerical test is designed to compare the traditional weighted average approximation and the present point-based approximation in the context of GFEM-Fluid. The limitations of the traditional weighted average approximation under the presence of both hydraulic pressure and in situ stress are identified, while these limitations can be eliminated using the present point-based approximation. As shown in Fig. 18, a circular fracture is embedded centrally in a cylindrical linear elastic rock mass with a constant internal fluid pressure of p w = 20 MPa acting on the fracture. The cylindrical rock mass has a radius of 50 m and height of 50 m. The radius of this circular fracture is a = 10 m. The material properties of the rock are: Young s modulus of E = 4 GPa, Poisson s ratio of v = 0.3 and strength of the material T 0 = 1 MPa. Four sets of tests are designed with different in situ stress, σ z, applied in the z direction, where the ratios of σ z and hydraulic pressure (λ = σ z /p w ) are 0, 0.25, 0.5 and The analytical solution for fracture aperture distribution under the effect of fluid pressure and in situ stress can be expressed as follows (Sneddon 1950): δ = 4(p w σ z )(1 ν2 ) a (33) πe 2 r 2, Fig. 18 The geometry of a fluid-pressurized fracture in a cylindrical rock mass where δ is the fracture aperture, a denotes the radius of fracture, and r represents the distance to the fracture central point, E and v are Young s modulus and Poisson s ratio of the rock mass, respectively. The variation of fracture aperture at the central point with different λ is shown in Fig. 19. For λ = 0, the distribution of the aperture along the radius of the circular fracture is illustrated in Fig. 20. As depicted in Fig. 21, the numerical and analytical solutions of the fracture aperture are in good agreement. The stress and strain at the integration points are solved using the GFEM simulator. The xoz-slice, yoz-slice and xoyslice of the maximum principal stress counter at integration points are shown in Fig. 22, when λ is 0, 0.25, 0.5 and 0.75, respectively. The stress state of the integration points in the vicinity of the fracture tip suffers from sharp changes in the presence of both hydraulic pressure and in situ stress. It is observed that there is a tensile stress concentration zone σ z i j hydraulic fracture k S + i k S + j p k p i pj p i p k i p j S _ j k _ (a) (b) Fig. 17 Fluid pressure acting on the upper and lower fracture surfaces (S +, S ) Fig. 19 Fracture apertures at center point of fracture with different ratio of in situ stresses and hydraulic pressure, λ = 0, 0.25, 0.5, 0.75

12 Q. Liu et al. and four times the mesh size. The results of these two methods are compared in Table 1 and Fig. 24. The limitations of the traditional weighted average approximation can be summarized as Fig. 20 The contours of fracture aperture (m) with GFEM-Fluid when σz /pw = 0 Fig. 21 Comparison between analytical and numerical solution of fluid-pressurized fracture aperture (in red) at the fracture tips, where the values of maximum principal stress around fracture tips are positive and decrease with the increase of λ. There is also a compressive zone (in blue) near the fracture, where the values of maximum principal stress are negative, because of the both effect of hydraulic pressure and in situ stress. The maximum principal stresses at the fracture tips are calculated using both the traditional weighted average approximation and the new point-based approximation based on the stress states at integration points. Four monitoring points with coordinates p1(10, 0, 0), p2( 10, 0, 0), p3(0, 10, 0) and p4(0, 10, 0) on the fracture tip are selected to illustrate the difference between the two methods, as shown in Fig. 23. The sensitivity analysis regarding the radius of the computational area for both methods is also studied, with the radius of the calculation area two As shown in Table 1, when the radius of the calculation area are two and four times the mesh size, the values and directions of maximum principal stresses at the monitoring points calculated by the weighted average approximation are different. However, the values and directions of maximum principal stresses at the fractures tips calculated by the point-based approximation are the same when the radius of calculation area is different. This shows that the results of the weighted average approximation are sensitive to the computational area, while the results calculated by the point-based approximation are more stable and not sensitive to the computational area. 2. Because of the compressive zone around the fracture tip, the values of the maximum principal stress at fracture tips calculated by the weighted average approximation are much smaller than those calculated by the pointbased approximation. Furthermore, the values of the maximum principal stress obtained from the weighted average approximation fluctuate and are sometimes compressive (negative) (see Fig. 20), which is unreasonable and inaccurate. Because, as shown in Fig. 19, the fractures are always open under both hydraulic pressure and in situ stress, the stress state at the fracture tip should always be tensile (positive). On the other hand, the maximum principal stresses calculated by the point-based approximation are always positive and decrease with the increasing in situ stress, which is more reasonable. 3. The analytical propagation direction should be parallel to the fracture surface, thus the directions of maximum principal stresses at the fracture tips should be parallel to the z direction. As shown in Table 1, the direction of the maximum principal stresses with the point-based approximation is more accurate than that obtained by the weighted average approximation. In conclusion, the traditional weighted average approximation introduces significant errors (e.g., the propagation direction) in modeling hydraulic fracturing when hydraulic pressure and in situ stress are both considered. The stress states at the fractures tips calculated by the weighted average approximation are sensitive to the computational area, and the value and direction of the maximum principal stress is inaccurate. The point-based approximation introduced in this paper can overcome these limitations and is more accurate. It should be noted that the shape of fracture tip may have an effect on the stress state around the fracture tip. In this paper, we assume that the fracture tip is sharp, which has been pointed out in Sect. 3.3.

13 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal Fig. 22 The xoy-slice, xoz-slice and yoz-slice of maximum principal stress contour at integration points with different ratios of in situ stresses and hydraulic pressure (λ = 0, 0.25, 0.5 and 0.75). The black circle is the calculation area. There is a tensile stress concentration zone (in red) near the fracture tips, where the values of maximum principal stress are positive and decrease with the increase of λ. There is also a compressive zone (in blue) near the fracture, where the values of maximum principal stress are negative. The maximum principal stress at the fracture tip is then calculated with the stress states at the surrounding integration points that are located in the calculation area. (Color figure online) 7 Numerical Test An application test is presented to confirm the performance of the GFEM-Fluid and the MPS-criterion with point-based approximation in simulating 3D hydraulic fracturing, considering the both hydraulic pressure and in situ stress. In the experiment of Jiao et al. (2015), a cubic plaster specimen with pre-set fractures and central vertical holes was prepared, which was then fractured by injecting high-pressure water in the central vertical holes, without applying any other loads. A similar physical experiment was carried out by Chen et al. (2010), in which an anisotropic in situ stress was applied. The geometry of the specimen is shown in Fig. 25. For each plaster specimen, two symmetrical pre-set fractures are prepared with inclination angles of 30. The length and depth of the pre-set fractures are 20 and 90 mm, respectively.

14 Q. Liu et al. Fig. 22 (continued) These experiments are now modelled numerically. Two numerical models are discussed: in Model I, the in situ stress is ignored; in Model II, a compressive in situ stress, σ x = 0.4 MPa, is applied in the x-direction. A constant pressure, p w = 0.8 MPa, is applied to the wellbore surface in both models. The mechanical and hydraulic parameters are listed in Table 2. The macro-mechanical parameters are taken from Jiao et al. (2015). In Model I, the propagation of the hydraulic fractures and the fluid pressure distribution are shown in Fig. 26, in which the hydraulic fractures finally split the specimen into two parts. The fractures created in Model I are approximately planar. The fracture paths in the horizontal cross-sections are shown in Fig. 27a. The pre-set fractures propagate approximately along the direction of their initial inclination angles. These numerical results agree well with the experimental results (Jiao et al. 2015) (see Fig. 28). In Model II, the compressive in situ stress, σ x = 0.4 MPa, is applied in the x-direction to numerically investigate the influence of the in situ stress. The propagations of hydraulic fractures and the fluid pressure distribution are shown in Fig. 29. The fracture propagation paths in the horizontal cross-section under an anisotropic in situ stress are shown in Fig. 27b. It is observed that the hydraulic fractures clearly

15 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal p 2 (-10,0,0) p 3 (0,10,0) O fracture p 4 (0,-10,0) p 1 (10,0,0) fracture tip Fig. 23 The location of the four monitoring points on the fracture tip. The circular fracture is discretized into a set of triangular fracture elements. The nodes located on the fracture tip are defined as tip nodes, from which the fracture propagates. Four monitoring points are selected among these fracture tip nodes to monitor the maximum principal stress propagate and turn towards the x-direction, which is parallel to the direction of maximum principal stress. The phenomenon that hydraulic fractures gradually reorient into the direction of the maximum principal stress is consistent with the observation by Chen et al. (2010) (Fig. 30), Almaguer et al. (2002) and Wang (2015). 8 Conclusions We identify the limitations of the MPS-criterion with the weighted average approximation in modeling hydraulic fracturing. We also present a point-based approximation for the MPS-criterion, which is more stable and accurate. Several numerical tests are utilized to validate the application of the MPS-criterion with the point-based approximation in the context of GFEM-Fluid for fracturing simulation, especially for hydraulic fracturing. The influence of in situ stresses is also considered. In conclusion, the MPS-criterion with point-based approximation has the following advantages: 1. The stress state at the fractures tip calculated by weighted average approximation is sensitive to the computational area, while that calculated by point-based approximation is more stable and not sensitive to the computational area. 2. The values of maximum principal stress calculated with the weighted average approximation fluctuate and are sometimes compressive (negative) under both fluid pressure and in situ stress, which is unreasonable and inaccurate. The present point-based approximation can overcome these limitations. Table 1 The maximum principal stresses at monitoring points calculated by weighted average approximation and point-based approximation when calculation area radius are two and four times the mesh size Calculation area The calculation area radius is 2 times of mesh size The calculation area radius is 4 times of mesh size Method Weighted average approximation Point-based approximation Weighted average approximation Point-based approximation λ Monitoring points Value Direction Value Direction Value Direction Value Direction 0 p p p p p p p p e e p p p p p p p p The direction here is the angle between maximum principal stress and z direction

16 Q. Liu et al. 8.0x10 5 p1 1.6x x10 5 p 1 stress maximum principal 4.0x x x x x x x10 6 p 2 p 3 p 4 maximum principal stress 1.2x x x10 6 p 2 p 3 p 4-1.2x λ λ (a) weighted average approximation (b) point-based approximation Fig. 24 The value of maximum principal stress at monitoring points solved by the traditional weighted average approximation and the present point-based approximation, when the radius of calculation area is two times the mesh size Fig. 25 A plaster specimen with pre-set fractures and a central hole σy 30 σx 20 σx y x σy (a) The geometry of theplaster specimen (b) Pre-set fractures Table 2 Parameters for hydraulic fracturing in a plaster specimen with a vertical wellbore and pre-set fractures Parameters Value Injection pressure, p w (MPa) 0.8 In situ stress in x-direction, σ x 0.4 Rock parameters Young s modulus, E (GPa) 4.0 Poisson s ratio, ν 0.3 Strength of the material, T 0 (MPa) 0.3 Fluid parameters Bulk modulus of fluid, K w (GPa) 2.2 Viscosity, µ (Pa s) Initial value of aperture, δ 0 (m) The direction of the maximum principal stress calculated with the point-based approximation is more accurate than that with weighted average approximation when the hydraulic pressure and in situ stress are considered. 4. The MPS-criterion with the point-based approximation can be utilized in the context of the generalized finite element method (GFEM), which is well-established for fracturing simulation without remeshing.

17 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal Fig. 26 The evolution of fracture surface and fluid pressure distribution of Model I 13

18 Q. Liu et al. Fig. 27 The fracture paths in the cross-sections observed in two models. In Model I, without in situ stress, the fractures propagate diametrically and propagate approximately along the direction of their initial inclination angles. In Model II, because of in situ stress applied in the x-direction, the fractures propagate and turn into the x-direction σx=0mpa σy=0mpa σy=0mpa σx=0mpa σx=0.4mpa σy=0mpa σy=0mpa σx=0.4mpa (a) Model I (b) Model II Fig. 28 Experiment results for Model I (Jiao et al. 2015)

19 Modelling Hydraulic Fracturing with a Point-Based Approximation for the Maximum Principal Fig. 29 The evolution of fracture surface and fluid pressure distribution of Model II 13

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