## 3 Drilling Fluid Circulation in the Wellbore
The Robertson-Stiff model includes a gel strength parameter and is used to a limited extent in the petroleum industry. The Herschel-Bulkley model is a power law with ultimate shear stress. Unfortunately, in most applications this model leads to analytically unsolvable expressions; therefore, it is not used for calculations in the oil industry. Some companies use the pseudo-Herschel-Bulkley model, which does not directly relate the viscometric analysis to the hydraulic calculations. The selection of a drilling fluid rheological model is discussed by [PERSON] et al. [69]. Various models are considered: the Newtonian, Bingham Plastic, Casson, Ostwald [PERSON], Herschel-Bulkley, Vom Berg, and Hahn-Eyring, with linear and nonlinear regression methods. The authors conclude that, for the used measurements of drilling fluid properties, the Vom Berg and Hahn-Eyring rheological models are best fitted to the description of drilling fluid rheological parameters. The Herschel-Bulkley viscosity model practical usage for drilling fluids is discussed in the study by [PERSON] and [PERSON] [70]. The main purpose of calculating the drilling fluid flow in the circulation system of the well is to determine the distribution of pressure along its depth. It is used to monitor the fulfillment of the necessary conditions: creating a pressure that prevents the flow of formation fluids and gases into the well and preventing hydraulic fracturing; control of pressure in pumps; control of cuttings removal; and control of the pressure difference in the pipes and in the annular space (pipe strength condition). There are two approaches to obtain the pressure distribution along the well depth. The first approach is to determine the total pressure loss based on precise analytical or empirical formulas for the pressure loss in each element of the circulation system. Multiphase fluid (fluid with inclusions) in this approach is modeled as a single-phase fluid with variable viscosity. The advantages of this approach are the simplicity of implementation and high speed of calculations using exact formulas. An obvious disadvantage is limited application due to the possibility of obtaining exact solutions only for a narrow class of fluid types, flow regimes, and configurations of regions with a significant number of assumptions, as well as the lack of universality of empirical dependencies. The second approach is to numerically solve the equations for the flow of a multiphase fluid in the well circulation system. Depending on the models used, this approach makes it possible to accurately consider various aspects of the multiphase flow of drilling fluid. The more detailed models more accurately describe the drilling fluid flow in the circulation system. At the same time, the use of such models leads to problems with large dimensions that require large computational resources. Multiphase flow consists of two or more different phases (which do not mix with each other). The most studied two-phase flows can be classified according to the type of their phases: gas-solid, liquid-solid, and gas-liquid. Often one of the phases consists of many physical particles (solid particles, drops, bubbles)--such flow is called dispersed. Particles form a dispersed phase, and the medium which contains these particles forms a continuous or carrier phase. The nature of the interaction of the dispersed and continuous phases mainly depends on the size of the particles and their distribution density. If this density is low and the particles are small, then the liquid affects the movement of the particles, but the particles do not affect the flow of the liquid. In this case, there is one-way interaction between the phases. If both the liquid affects the particles and the particles affect the flow of the liquid, then there is a two-way interaction. It is assumed that the interaction of particles with each other can be neglected. The main approaches to the modeling of multiphase flow are Euler-Lagrange and Euler-Euler.