The advancement in drilling technologies within the last two decades has allowed deep vertical drilling and long horizontal and multilateral wells. A common operational issue in drilling such long wellbores is efficient hole cleaning and transportation of the generated drill cuttings to the surface. Efficient hole cleaning is a complex problem as it involves a simultaneous analysis of cuttings characteristics, fluid rheology and the geometry of the annulus space. Simplified analytical models, supported by lab experimental observations and numerical simulations have been proposed to determine the optimum flow rate for efficient hole cleaning. These models discard the influence of some parameters due to simplified assumptions. The use of particle based numerical models presented advantages in terms of considering the interaction of the particles and more realistically modeling the coupled flow and particles’ interaction. Yet, more studies are needed to fully understand the complex nature of cuttings transportation.

In this study, we first present the results of the settling velocity of a single particle in a stationary incompressible Newtonian fluid falling vertically in an infinite medium, a pipe and an annulus space. The Basset–Boussinesq–Oseen equation (BBO) analytical solution using the differential transformation method (DTM) and DTM- Padé approximations as well as the semi-analytical solution using MATLAB were used to calculate the settling velocity and track the particle motion as a function of particle size and density as well as fluid viscosity at different time. The results were compared with experimental results and the Eulerian-Lagrangian based numerical simulations conducted using MFiX software. In overall, good agreements were observed between the results of the semi-analytical, experimental and numerical simulations. However, higher orders of approximations were required in analytical models to converge the solution and yield better results, but still may show some fluctuations.

The findings of single particle analysis were extended to multiple particle simulations. The downward motion of a pack of identical 3 mm spherical particles composed of 171 particles with 8gr/cc density inside a pipe and an annulus space were simulated numerically and the velocity of the pack was determined as a function of particle size and density as well as fluid viscosity. Also, the time when the lower and upper boundaries of the pack reached the bottom of the annulus were defined. The results suggest that relationships exist between the motion (i.e. the vertical location) of the single particle and the boundaries of the multiple particles’ pack. We also simulated the settlement of heterogeneous particles, i.e. two packs of 1 mm and 3 mm particles, respectively, and two packs of anhydride and sandstone particles falling simultaneously in a stationary fluid in an annulus space. Here also relationships observed between vertical motion of single particle and multiple heterogeneous packs. This result may help to better define the corresponding depth of the cuttings observed at the surface as, due to their densities and sizes, cuttings corresponding to deeper depth may reach the surface earlier, hence, defining the novelty of this research.

在过去的二十年中，钻井技术的发展使深层垂直钻井以及长水平井和多边井成为可能。在钻这么长的井眼时，一个常见的操作问题是有效的孔清洁以及将生成的钻屑输送到地面。高效的孔清理是一个复杂的问题，因为它需要同时分析钻屑特性，流体流变性和环形空间的几何形状。已提出了简化的分析模型，并通过实验室实验观察和数值模拟来确定有效清洗孔的最佳流速。这些模型由于简化的假设而放弃了某些参数的影响。基于粒子的数值模型的使用在考虑粒子的相互作用以及更实际地对耦合的流动和粒子的相互作用进行建模方面具有优势。然而，需要更多的研究来充分理解切屑运输的复杂性。

在这项研究中，我们首先介绍单个粒子在固定不可压缩牛顿流体中垂直沉降在无限介质，管道和环空中的沉降速度的结果。使用微分变换法（DTM）和DTM-Padé近似的Basset-Boussinesq-Oseen方程（BBO）解析解以及使用MATLAB的半解析解来计算沉降速度并跟踪粒子运动在不同时间的粒径和密度以及流体粘度。将结果与实验结果以及使用MFiX软件进行的基于欧拉-拉格朗日的数值模拟进行了比较。总体而言，在半分析，实验和数值模拟的结果之间观察到良好的一致性。然而，

单粒子分析的发现扩展到了多粒子模拟。数值模拟了由171个密度为8gr / cc的颗粒组成的相同的3mm球形颗粒在管道和环空内的向下运动，并根据颗粒大小和密度以及流体粘度。另外，定义了包装的上下边界到达环的底部的时间。结果表明，单个粒子的运动（即垂直位置）与多个粒子堆积的边界之间存在关系。我们还模拟了异质颗粒的沉降，即分别两包1 mm和3 mm的颗粒，两包酸酐和砂岩颗粒同时落在环形空间的固定流体中。在这里，还观察到单个粒子和多个异质堆积的垂直运动之间的关系。该结果可能有助于更好地定义在表面处观察到的切屑的相应深度，因为由于其密度和大小，与更深深度相对应的切屑可能更早到达表面，因此，定义了这项研究的新颖性。