Elsevier

Applied Mathematical Modelling

Volume 91, March 2021, Pages 590-613
Applied Mathematical Modelling

Direct numerical simulation of proppant transport in hydraulic fractures with the immersed boundary method and multi-sphere modeling

https://doi.org/10.1016/j.apm.2020.10.005Get rights and content

Highlights

  • Multi-sphere DEM is integrated in the immersed boundary framework to handle interactions of complex shaped particles.

  • Settling laws and apparent viscosity models for particle clouds flowing in narrow channels are extracted from simulation.

  • Preferential combination patterns during cylindrical proppant transport are observed from the DNS results.

Abstract

In this paper, a resolved CFD-DEM method based on the immersed boundary method is proposed to simulate the proppant transport process, which is a multi-phase problem with strong fluid-particle coupling mechanisms in the oil and gas industry. A multi-sphere model is integrated into this method to describe complex particle shapes, in which Lagrangian points uniformly distributed on the particle surface are efficiently utilized for solving particle-particle interactions. This approach is validated by several benchmarks, including single-sphere and two-sphere settling tests. A modified driving pressure gradient is also adopted to satisfy bulk velocity constraints for simulating particle settling problems in periodic channels. Transport and settling behaviors of hundreds of sphere and cylinder proppant particles in periodic narrow channels with different widths are investigated, and settling laws and apparent viscosity models for proppant clouds with different shapes are then extracted from the simulation results. Benefiting from the features of multi-sphere modeling, this approach is demonstrated to be both robust and efficient for simulating fluid-particle coupling flow with complex particle shapes.

Introduction

For unconventional oil and gas development, there exists a critical particle conveying process, i.e., proppant transport in hydraulic fractures, which greatly influences the eventual productivity of stimulated reservoirs [1]. From the physical point of view, proppant transport is essentially a multi-phase problem with strong fluid-particle coupling. In addition, motions of proppant particles are constrained in narrow fractures of the rocks. Mechanisms of particle-particle interactions and particle-wall interactions also play a significant role in this process. A better understanding of particle settling and conveying behaviors under these mechanisms assists to optimize injection plans in engineering and improve effective productivity.

In recent decades, numerical simulation has become an efficient approach to study the above process. In large-scale simulations, empirical drag force models are usually applied to calculate the terminal settling velocity of proppant clouds [2], [3], [4], such as Wen and Yu's model [5] and the De Felice model [6]. However, most of the popular drag laws are developed for dispersed particle systems in chemical engineering under certain constraints, which are not appropriate for describing the proppant transport process. First, in most cases, the drag laws are built up for Newtonian fluids, while fracturing fluid can be a power-law and viscoelastic fluid [7]. Second, these laws are established for an open boundary system, while the proppant transport process is constrained in a channel flow. Moreover, unconventional proppant particles are currently designed for better transport capability, such as rod-shaped proppant [8] and X-shaped proppant [9], while most drag laws are used to describe the spherical particle system. In order to overcome the above constraints, a direct numerical simulation technique is necessary to capture full-scale details for this specific problem, which constitutes a powerful tool for building a new drag force model.

In order to resolve full-scale details for this two-phase problem, two critical issues must be solved: fluid-particle coupling and particle-particle interaction. The fluid-particle coupling issue mainly concerns how to exert appropriate boundary conditions on the particle surface and calculate the hydrodynamic force on the particle surface due to fluid-particle interaction. For example, non-slip boundary condition on the particle surface is the most common case. While the particle-particle interaction issue is mainly about how to model the particle-particle contact force for arbitrary-shaped particles.

For the fluid-particle coupling issue, three popular methods exist: the arbitrary Lagrangian-Eulerian (ALE) method [10], the fictitious domain (FD) method [11], [12], and the immersed boundary (IB) method [13], [14], [15], [16],34]. In the ALE method, an unstructured body-fitted grid is generated for the whole computational domain, and it is trivial to exert the non-slip boundary condition on the particle surface. However, because particles shall keep moving during computation, the body-fitted grid needs to be frequently reconstructed, which is time-consuming for 3D cases, and computational accuracy strongly relies on the quality of the unstructured grid. In the FD and IB methods, fluid motion is solved on a Cartesian grid, which does not need to conform to the particle surface, and are thus more efficient than the ALE method. In the FD method, a Lagrangian multiplier is embedded in the fluid governing equation in order to implicitly exert the constraint that, in particle-possessed regions, the fluid velocity should equal to the velocity of a rigid body, and therefore the non-slip boundary condition is naturally satisfied. In the IB method, a source term can be explicitly or implicitly considered in the fluid momentum equation to recover the boundary effect of particles, which is easy to implement and efficient compared to the FD framework.

Uhlmann [14] developed a robust, feasible direct-forcing IB framework for simulating large-scale particulate flows. In that work, the main idea of realizing the IB method is to exert the non-slip boundary condition on the Lagrangian points distributed on a particle surface, which can be regarded as a thin porous shell. After Uhlmann's pioneering work, Breugem [15] improved accuracy to second-order of this framework by applying several modifications, including retraction distance, multi-forcing scheme, and direct account of internal fluid effects. Yu and Shao [17] also utilized the ideas of direct forcing in their direct forcing/fictitious domain (DF/FD) framework based on the original distributed Lagrangian multiplier/fictitious domain (DLM/FD) framework. However, in their work, the non-slip boundary condition is exerted on the Lagrangian points, which are distributed over the whole domain inside of the particle instead of the particle surface. These two variations have the same effects for implementing the non-slip boundary condition, while different treatments are required for calculating the hydrodynamic force exerted on the particles.

For the particle-particle interaction issue, the discrete element method (DEM) [18] is usually adopted to calculate the contact force between particles after modeling the particle shape. Regarding modeling a particle of arbitrary shape, there are mainly two approaches: polyhedral approximation and multi-sphere approximation [19]. Galindo-Torres [35] also proposed an approach named sphero-polyhedra method, which can be considered as a combination of the two approximations. Generally, compared to the polyhedral approximation, the multi-sphere model is more efficient in contact detection between complex particles. It is worth noting that in the framework of the immersed boundary method, it is natural to apply the multi-sphere model to describe complex particles. In order to exert non-slip boundary conditions, many Lagrangian points are already uniformly distributed on the particle surface, and these points can be regarded as seeds in the multi-sphere model and utilized to reconstruct the particle shape. It is mentioned in the literature [20] that using the multi-sphere model leads to artificial roughness on the particle surface, which makes a very smooth particle surface uneven. However, this problem is serious only in low-resolution cases. In this work, the number of Lagrangian points is always sufficient, so that the effect of artificial roughness can be ignored.

Although multi-sphere modeling is widely applied in pure DEM simulation describing interactions between complex-shaped particles, it is rarely implemented in the resolved fluid-particle coupling simulation, in spite of its high efficiency for implementation under the direct-forcing immersed boundary framework and strong extendibility for various-shaped particles. In this study, we integrate multi-sphere modeling into the immersed boundary framework for direct numerical simulation of the proppant transport process with different particle shapes. Furthermore, in order to study the long-term behaviors of proppant in periodic channels, special treatments are proposed in order to satisfy certain constraints for bulk velocity. It is worthy of noting that in this work non-slip boundary condition is applied on the particle surface, which is suitable for conventional proppant particles without special surface-coated treatments.

The remainder of this paper is organized as follows. Section 2 introduces the basic governing equations for fluid and particle motions, as well as the basic ideas of the immersed boundary method and multi-sphere model. Three benchmark tests, including fixed sphere and single/two-sphere settling, are tested in Section 3 for validation. In Section 4, settling and transport behaviors of hundreds of sphere/cylinder proppant particles in periodic channels are studied. Finally, conclusions and directions for future work are presented Section 5.

Section snippets

Governing equations for fluid and particle

In the framework of the resolved CFD-DEM method [14], fluid motion is governed by Navier-Stokes equations as follows:·uf=0,uft+·(ufuf)=1ρf(P+Pe)+νfΔuf+f,where uf is the vector of fluid velocity; ρf is the fluid density (incompressible in this form); P is the fluid pressure; ∇Pe is the constant driving pressure gradient; νf is the kinetic fluid viscosity; f is the forcing term due to the boundary effect in the immersed method; ∇ • and ∇ are the divergence and gradient operators,

Validation tests

In this section, four validation tests are designed to validate our code. First, fluid motion around a fixed sphere is studied, and the dimensionless Darcy number is calculated and compared with the result in the literature. Second, a single-sphere settling problem is simulated and compared with the experimental results at different Reynolds numbers reported in the literature. Third, a two-sphere settling problem, known as the DKT (Drafting-Kissing-Tumbling) phenomenon, is simulated and

Case studies

In this section, two groups of cases are designed to study the transport behavior of sphere/cylinder proppant clouds in narrow fractures. First, initial and boundary conditions for this problem are defined, and a modified driving pressure gradient is adopted to fulfill the bulk velocity constraints. Second, the relationship between the settling velocity and the fracture width is obtained for a sphere/cylinder cloud. Third, the relationship between the apparent viscosity and the fracture width

Conclusions

In this work, a resolved CFD-DEM framework based on the immersed boundary method and multi-sphere model is proposed to study the settling and transport behavior of spherical and cylindrical proppant particles in periodic channels. Through comparing our simulation results with four benchmark tests in previous literature, the proposed framework is validated. In order to study the long-term behaviors of particle clouds, a modified pressure gradient scheme is proposed for the periodic boundary

References (36)

Cited by (18)

  • A volumetric-smoothed particle hydrodynamics based Eulerian-Lagrangian framework for simulating proppant transport

    2023, Journal of Petroleum Science and Engineering
    Citation Excerpt :

    Contrary to the resolved CFD-DEM, the unresolved CFD-DEM is suitable for the situations when the particle size is considerably smaller than that grid cells (Kloss et al., 2012). Both the resolved (Poła et al., 2017; Zeng et al., 2021) and unresolved (Tang et al., 2022; Vega et al., 2021; Wu and Sharma, 2019; Yamashiro and Tomac, 2021; Yi et al., 2018) CFD-DEM have been successfully used to study the proppant transport. However, the CFD-DEM is still difficult to be applied to large scale or practical simulations due to its extremely high requirements of computation powers (Siddhamshetty et al., 2020).

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