## Abstract

The paper presents an extension of authors’ previous model for a 3D hydraulic fracture with Newtonian fluid, which aims to account for the Herschel–Bulkley fluid rheology and to study the associated effects. This fluid rheology model is the most suitable for description of modern complex fracturing fluids, in particular, for description of foamed fluids that have been successfully utilized recently as fracturing fluids in tight and ultra-tight unconventional formations with high clay contents. Another advantage of using Herschel–Bulkley rheological law in the hydraulic fracture model consists in its generality as its particular cases allow describing the behavior of the majority of non-Newtonian fluids employed in hydraulic fracturing. Except the Herschel–Bulkley fluid flow model the considered model of hydraulic fracturing includes the model of the rock stress state. It is based on the elastic equilibrium equations that are solved by the dual boundary element method. Also the hydraulic fracturing model contains the new mixed mode propagation criterion, which states that the fracture should propagate in the direction in which mode \({{\mathrm{\mathrm {II}}}}\) and mode \({{\mathrm{\mathrm {III}}}}\) stress intensity factors both vanish. Since it is not possible to make both modes zero simultaneously the criterion proposes a functional that depends on both modes and is minimized along the fracture front in order to obtain the direction of propagation. Solution for Herschel–Bulkley fluid flow in a channel is presented in detail, and the numerical algorithm is described. The developed model has been verified against some reference solutions and sensitivity of fracture geometry to rheological fluid parameters has been studied to some extent.

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## Acknowledgements

The authors acknowledge the financial support of this research by the Russian Science Foundation (Grant No. 17-71-20139).

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## Appendix

### Appendix

### 1.1 A.1 Herschel–Bulkley fluid model

Let us consider the most general form of the equations of incompressible HB-fluid flow in the 3D case. It includes the continuity equation

and the momentum equation

In (43), (44) \({{\mathrm{\mathbf v}}}= \{ v_i \}\) is the fluid velocity vector and \({\mathbb {P}} = \{ p_{ij} \}\) is the total stress tensor divided into two parts

where *p* is the scalar called the hydrodynamic pressure, \({\mathbb {T}} = \{ \tau _{ij} \}\) is the viscous stress tensor, and \({\mathbb {E}} = diag(1,1,1)\) is the unit (or identity) tensor. The viscous stress tensor \({\mathbb {T}}\) is connected with the strain rate tensor \({\mathbb {D}}=\{ D_{ij} \}\) by the constitutive relations

In (46), (47) the components of the tensor \({\mathbb {D}}\) are

and \(\eta \) is the viscosity function given by the Herschel–Bulkley rheology model

Here, *K* is the flow consistency factor; *n* is the flow behavior index that governs the degree of shear thinning or thickening; \(\tau _0\) is the yield stress. *T* and *D* denote the second invariants of the respective tensors given by the following formulas

The index form of Eqs. (43) and (44) is the following:

### 1.2 A.2 Equations for 2D HB fluid flow in 3D fracture

The fracture width *W* is much smaller than its longitudinal size. Therefore, the flow inside the fracture can be considered locally as the flow in a thin channel between two parallel plates. Without loss of generality let’s assume that the axis \(x_1, x_2\) of the local coordinate system lie in the tangent plane to the fracture surface and the axis \(x_3\) is orthogonal to the fracture surface. Then the transversal fluid velocity component can be assumed small (\(v_3 \approx 0\)) as compared to its longitudinal components. Also the derivatives of the fluid velocity components in the longitudinal directions \({\partial }/ {\partial x_1}\), \({\partial }/{\partial x_2}\) are small in comparison with their derivatives in the transverse direction \({\partial }/{\partial x_3}\). The fluid pressure and its consistency factor are considered constant in the transverse direction. The time derivatives in Eq. (52) are disregarded. The non-stationarity of the fracture propagation model is conditioned by the continuity Eq. (51) through the relationship between the fracture width *W* and the transversal fluid velocity \(v_3\)

Under the assumptions made above, Eqs. (51) and (52) can be simplified. Omitting small terms in Eq. (52), one can obtain for \(i=1,2\)

Equation (52) for \(i=3\) degenerates under the stated assumptions. The integration of Eq. (54) over \(x_3\) gives

where \(A_i\) are some constant values.

Further, the cases of Newtonian and Herschel–Bulkley fluids are discussed separately. For Newtonian fluid, we have \(\eta (D) = \mu = const\), \(\tau _0=0\) and then from Eq. (55) it follows that

Taking into account the boundary conditions at \(x_3=0\) and \(x_3=W\)

one can obtain

From the expressions for fluid fluxes

and Eq. (58) we can get the equations connecting the derivatives of the fluid pressure with the fluxes

Equations (60) coincide with the ones used in Shokin et al. (2015) and Cherny et al. (2016).

Using the Equations (58) and (60) one can obtain shear rate value

For HB-fluid, the viscosity function \(\eta (D)\) depends on the velocity components that are functions of \(x_3\). For integrating Equations (55), let us move to a coordinate system, in which the direction of \(x_1\)-axis coincides with the direction of the velocity vector \({\mathbf {u}}\). In this coordinate system, we have

Integrating Equation (55) and taking into account the boundary conditions (57) and the expressions for the viscosity function (62), one can obtain

where

The latter formula for \(v_d(x_3)\) takes into account the typical velocity profile of the HB fluid flow between two parallel plates as shown in Fig. 21. This formula is correct if the pressure gradient is enough to overcome the yield stress

Otherwise the fluid is motionless. Similarly, one can obtain the expression for the second velocity component \(v_2\). Now the expressions for the fluid fluxes can be written in the following form:

Extracting the first degree of the pressure derivatives \({\partial p}/{\partial x_i}\) in the right-hand side of Equation (65) gives

where \(\eta _{a}\) is an apparent viscosity. It can be expressed in terms of the pressure derivatives \({\partial p}/{\partial x_i}\) as well as in terms of the fluxes.

The expression of the apparent viscosity through the pressure derivatives can be obtained from the Equation (63)

Rewriting Equation (67) in another coordinate system results in the substitution of the pressure derivative \({\partial p}/{\partial x_i}\) by the pressure gradient \(\nabla p\), which is invariant relative to coordinate transformations

where

To get the formula for the apparent viscosity in terms of the fluid fluxes, let us sum the squares of the right-hand sides of Equation (65) for \(q_1\) and \(q_2\) and express the pressure gradient \(|\nabla p|\) in terms of the fluid flow vector module \(|\mathbf{q}| = (q_1^2 + q_2^2)^{1/2}\), which is invariant relative to coordinate transformations similar to \(|\nabla p|\). Then we obtain

Formula (69) is a 2D generalization to the case of Herschel–Bulkley rheology of the expressions obtained earlier by other researchers for a Power-law fluid. For the case of the Power-law fluid (\(\tau _0 \equiv 0\)) the Equation (66) with the apparent viscosity expressed by formula (69) has been used in Sousa et al. (1993), Garagash (2006), and the expression (68) has been employed in Ouyang et al. (1997) and Rungamornrat et al. (2005). Here as well as in Cherny and Lapin (2016) formula (68) is chosen because then there is no need to calculate the fluid flux while the fluid flow equations are solving.

Note that like in case of Newtonian fluid one can obtain the value of shear rate using the Equations (63), (66) (69), but the calculations are quite more complicated

Integrating equation (51) over \(x_3\) and taking into account relationships (53) and (59), one can obtain the equation for the fracture width *W*

The substitution of the expression for the fluxes (66) into Equation (71) gives the equation for the fluid pressure

One should note that one of the major difficulties while modeling the fluid flows with non-zero yield stress (such as Bingham of Herschel–Bulkley fluids) is the necessity to trace the boundary between the region of non-zero strain rate (46), where the Eqs. (49), (52), (62), (65)–(66), (67), (68), (69), (72) are applied and the region (47) where the fluid flow should be treated as the rigid body motion (\(D_{ij}=0\)). For the modeling of the regions with the zero and non-zero strain rate tensor (yielded and unyielded regions) within the framework of the same equations various modifications of the expression for the viscosity (49), are used, and they prevent its degeneration when \(D\rightarrow 0\) (Mitsoulis 2007).

In the problems of fluid flow in the hydraulic fracture the search for the boundary of the unyielded region is easier than in the problems of fluid flow in the regions with the given boundaries. The unyielded region appears in each point of the fracture (\(x_1, x_2\)) in the middle of its cross-section (between the fracture sides \(0<x_3<W\)). The boundaries of the region are calculated explicitly, while solving the problem of fluid flow between two parallel plates (65). This solutions is used to obtain equation (72) As it can be concluded from (64) the pressure gradient is bounded from zero. The area where strain rate tensor is zero can fill the whole cross section, then fluid at this crack point (\( x_1, x_2 \)) stops, and equation (72) is not valid. But the calculations made in Sect. 5.3, show that such a situation can not be realized in the simulation of the fractures considered. This means that at all points of the fracture the fluid flows, and the pressure gradient always exceeds \( \frac{2 \tau _0}{W} \). This allows us to use equation (72) with the expression for viscosity (68) without additional modifications.

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Cherny, S., Lapin, V., Kuranakov, D. *et al.* 3D model of transversal fracture propagation from a cavity caused by Herschel–Bulkley fluid injection.
*Int J Fract* **212**, 15–40 (2018). https://doi.org/10.1007/s10704-018-0289-4

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DOI: https://doi.org/10.1007/s10704-018-0289-4