Reply to Discussion on “Analysis of Bingham fluid radial flow in smooth fractures”

Lingcho Zou, Ulf Håknsson, Vldimir Cvetkovic

a Division of Resources,Energy and Infrastructure,Department of Sustainable Development,Environmental Science and Engineering,Royal Institute of Technology,Stockholm,10044, Sweden

b Division of Soil and Rock Mechanics, Department of Civil and Architectural Engineering, Royal Institute of Technology, Stockholm,10044, Sweden

c Skanska AB, Stockholm,11274, Sweden

Keywords:Rock grouting Radial flow Bingham fluid Approximation solution

ABSTRACT Recently, Hoang et al. (2021) discussed our paper Zou et al. (2020). In our paper, we made a statement that Dai and Bird(1981)’s solution for two-dimensional (2D) radial Bingham fluid flow between parallel plates violates mass balance.Hoang et al.pointed out that Dai and Bird(1981)’s solution does not violate the mass balance because Dai and Bird (1981)’s solution and our analysis are based on different assumptions, i.e. with consideration of the vertical velocity component in the continuity equation or not,which leads to two different approximation models. In this sense, the mass balance of Dai and Bird(1981)’s solution should not be checked using our solution as a reference. In this reply, we add remarks on the two approximation models and their implication for rock grouting analysis. The discussion by Hoang et al. and this reply are helpful to thoroughly eliminate the existing confusion regarding the two solutions in the rock grouting research community.

1. Introduction

We appreciate Hoang et al. (2021) for their interest and efforts made for the discussion on our paper ‘Analysis of Bingham fluid radial flow in smooth fractures’.The solution of Bingham fluid flow in smooth fractures is the basis for theoretical analysis of cement grouting in rock fractures.In our paper(Zou et al.,2020),we made a statement that Dai and Bird(1981)’s solution for two-dimensional(2D) radial Bingham fluid flow between parallel plates violates mass balance. Hoang et al. (2021) pointed out that Dai and Bird(1981)’s solution does not violate mass balance because Dai and Bird (1981)’s solution and our analysis are based on different assumptions, i.e. with consideration of the vertical velocity component in the continuity equation or not,which leads to two different approximation models.

In the literature, the confusion regarding Dai and Bird (1981)’s solution and its application to rock grouting has been raised because no explicit mathematical models, i.e. the governing equations, were presented either in Dai and Bird (1981) nor in the following rock grouting literature,e.g.Gustafson et al.(2013).In the discussion by Hoang et al. (2021), the two approximation models with associated solutions were clearly summarized in their Eqs.(5)-(7), which is helpful for clarification. If Dai and Bird (1981)’s solution is viewed as a different model from ours, its mass balance should not be checked using our solution as a reference; in this sense, our statement that Dai and Bird (1981)’s solution violates mass balance is indeed unnecessary as pointed out by Hoang et al.(2021).

The results of the two models were compared for different Bingham numbers (Bn) in Hoang et al. (2021), showing that the difference of results between the two types of models is not significant in general,which is consistent with our comparison results of the two types of rock grouting solutions shown in Figs.6 and 7 in Zou et al. (2020).

In this reply,we would like to add the following remarks on the two approximation models and their implication for rock grouting analysis.The discussion by Hoang et al.(2021)and this reply should be helpful to eliminate the existing confusion regarding the two solutions in the rock grouting research community.

2. Approximation for the complexity

We would like to emphasize that cement grouting in rock fractures is a complex process involving complex fluid flow in complex geometrical structures. Simplified models for radial flow of Bingham fluids in smooth fractures were used to develop theoretical models for rock grouting analysis.

The radial velocity is dependent on the radius in the radial flow configuration,and it reduces with the increasing radius.This means that the Reynolds number is not constant:it can be very high near the inlet (i.e. injection borehole) and very small near the outlet.Therefore, in principle, the kinematical effects should not be ignored to capture the realistic physical process. However, it is impossible to analytically solve the full continuity and momentum equations at present. The asymptotic expansion method has been used to obtain approximation solutions for radial flow of Bingham fluids in the configuration of squeeze flow (e.g. Muravleva, 2017).The model(Eqs.(5)-(7))presented in the discussion by Hoang et al.(2021) is the zero-order approximation of the continuity and momentum equations.The model presented in Zou et al.(2020)is also the zero-order approximation by further omitting the vertical velocity component in the continuity equation. Both models are approximations of realistic kinematical effects, as discussed by Muravleva (2017).

The shape of the plug flow region varies in the two different approximation models. For instance, the plug flow region in the model presented in Zou et al. (2020) is independent of the radius;in the model(Eqs.(5)-(7))presented in Hoang et al.(2021),the plug flow region increases with the radius. In higher orders of approximation models, the plug flow region can be more complex (e.g.Muravleva, 2017). The reason for the obtained different shapes of plug flow region in the two approximation models is indicated in the discussion by Hoang et al. (2021), which is due to the application of different boundary conditions for the shear stress. Specifically, we use the boundary condition on the surface of the plug flow region,while the other model sets shear stress equal to zero in the middle of the fracture aperture in the plug flow region. We think that the shear stress in the plug region is undefined according to the definition of the Bingham model,which only defines that the shear stress is below the yield stress in the plug flow region (it implies that the shear stress can be any value below the yield stress in the plug flow region)(see Eq. (3)in Zou et al. (2020)or Eqs. (1)and(2)in Hoang et al.(2021)).Moreover,the Bingham model is an idealized rheological model and its validity for any realistic fluids remains an open question(Barnes,1999).Therefore,the real shape of the plug flow region for realistic fluids/grouts or even the existence of a rigid plug flow region remains unknown at this point.

3. Implication for rock grouting analysis

With the understanding that Dai and Bird(1981)’s solution and our analysis are two separate approximation models, we should clarify that the two types of analytical solutions for rock grouting,presented in Gustafson et al. (2013) and Zou et al. (2020), respectively, are both approximation solutions, i.e. neither of them is an exact solution.However,as noted in the discussion by Hoang et al.(2021),the solution presented in Zou et al.(2020)is much simpler compared to that based on Dai and Bird (1981)’s solution. More importantly, the solution presented in Zou et al. (2020) is more relevant to the boundary condition commonly applied in rock grouting with controlled injection pressure,where the solution for the pressure is explicit with respect to the boundary pressures(see Eqs. (A3) and (A4) in Zou et al. (2020)). In contrast, Dai and Bird(1981)’s solution is more relevant to the boundary condition of known constant flowrate, where the solution for the pressure needs to be solved from the flowrate equation,i.e.Eq.(4)in Hoang et al. (2021). Considering that the difference between solutions of the two models is relatively small(see Fig.6 in Hoang et al.,2021)and that the predicted grout propagation lengths using the two models are very close (see Figs. 6 and 7 in Zou et al., 2020), it is recommended using the simpler solution presented in Zou et al.(2020)for rock grouting analysis in practice.

In summary, the two types of approximation models are both based on restrictive assumptions. They are at best applicable only for scoping calculations and some degree of uncertainty for analysis of rock grouting in engineering projects has to be expected. Validation of the two models using more rigorous direct numerical simulations (e.g. using an augmented Lagrangian approach or Uzawa/projection method) or physical experiments (e.g. Shamu et al., 2020) might be possible at small scales, but it remains unavailable at engineering application scales. Validation of the analytical models and development of more reliable numerical tools for design and operation of rock grouting applications are important topics for future studies.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.