## 3 Drilling Fluid Circulation in the Wellbore  
The drilling fluid performs various functions: carrying rock cuttings to the surface, maintaining the stability of the wellbore, balancing formation pressure, cooling the bit, and supplying hydraulic energy to the bit to increase the rate of penetration (see Figure 3, the thin blue arrows represent the drilling fluid flow, big blue and orange arrows show the heat transfer--cooling the fluid in the top and heating in the bottom, respectively, gray arrows illustrate the formation pressure, dark blue arrow goes for fluid (mass) transfer through the wall). Drilling fluid can be based on water, oils (petroleum), air, steam, and foam. Water and oil have insufficient viscosity and density. Viscosity is needed to carry the cuttings, and density is needed to resist the pore pressure in the rock. Therefore, special solid particles are added to the fluid. Each type of drilling fluid contains three components: liquid, active, and inert solid particles. Active particles react chemically with the liquid, affecting its rheological properties. Figure 3: The drilling fluid circulation scheme. A paper by [PERSON] [8] is devoted to a review of the literature on modeling the flow of drilling fluid in wells and the flow of crude oil in the oil pipelines. The work contains an extensive bibliography. A list of works devoted to the experimental determination of the liquid properties is presented. The experimental studies available in the literature are described, confirming the complex rheology of drilling fluids, including its dependence on temperature, pressure, time, and the magnitude of shear deformation. The author describes the main rheological fluid models: Bingham, Casson, Herschel-Bulkley, and others. As a general conclusion, the author notes that all the cited studies contain strict simplifications, and, therefore, their results can be used for a narrow range of real problems. For example, in most works it is assumed that the drilling fluid is incompressible and single-phase and that its properties do not depend on pressure and temperature. Moreover, the validation and verification of theoretical models is often neglected. Modeling the Newtonian and non-Newtonian fluid flow in circular tubes and annular space is discussed by [PERSON] et al. [55]. [PERSON] et al. [56] involve in the consideration the rotation of the drill string. Similar topics are discussed also by [PERSON] et al. [57] and [PERSON] [58]. The monograph by [PERSON] [59] is devoted to the features and the modeling of various flows of non-Newtonian fluids. The flow of Bingham fluids is discussed in the study by [PERSON] and [PERSON] [60], where the problems of flow in channels and cavities with wall shapes varying in time and space are considered. [PERSON] et al. [61] propose a new rheological model for a heavy-oil flow in oil-field pipelines. The monograph by [PERSON], [PERSON], and [PERSON] [62] is devoted to the drilling and cement fluids, including their properties and production methods. Studies by [PERSON] et al. [63], [PERSON] [64], [PERSON] [65], and [PERSON] and [PERSON] [66] are devoted to hydrodynamic calculations in drilling. The hydraulic codes are presented that allow determining the flow rate and rheological properties of a fluid that satisfy various conditions. Examples of hydraulic calculations can also be found in [67; 68]. The most used fluid models are Newtonian (viscous stresses are linearly correlated to the local strain rate); Bingham viscoplastic (behaves as a rigid body at low stresses but flows as a viscous fluid at high stress); and power-law models (power-law relationship between stress and shear rate, which better reproduces the behavior of a fluid at low shear rates). There are also three main fluid models, which are some combination of the Bingham and power-law models: Casson, Robertson-Stiff, and Herschel-Bulkley models. The two-parameter Casson model is widely used in industry but rarely applied to the drilling fluids.