## 5 Mass Transfer between the Drilling Fluid and Soil in the Open Section of the Well  
The \"mass transfer\" denotes here both the absorption of the drilling fluid filtrate by the formation, and the influx of formation fluids (water, oil, or gas) into the well. Solving the problem of mass transfer allows setting the sources or sinks in the model for calculating the circulation of the drilling fluid in the annular space. The change in rheological properties of the drilling fluid due to mass transfer are considered. If the fluid filtrate displaces the formation fluid in the near-wellbore zone, then its loss is estimated. The main result of solving the problem is determining the filtration rate at the well-reservoir interface. The calculated changes in the drilling fluid rheology affects all drilling process. Additionally, for the heat transfer problem the filtration rate inside the reservoir in the near-wellbore space is calculated. When studying the issues of mass transfer in oil and gas production, the filtration equations in the porous medium ([PERSON]'s law, continuity equation, equations of state) with equations of diffusion and mass transfer in a porous medium (generalized Fick's law, equation of mass balance, equations of kinetics, and isotherms of matter exchange between phases) are used. As a result, the system of differential equations is solved. The solution determines the velocity field associated with the pressure gradient at a given porosity and density, which may depend on the pressure distribution in the well and in the formation. Models of axisymmetric and plane-radial filtration are usually used for simulation of mass transfer between an open well part and the surrounding soil. The stationary flat-radial filtration models allow the analytical solution under certain assumptions. Moreover, these models are given by the boundary value problems for ordinary differential equations. Usually, these 2D models are used to obtain a quick estimate of the required values. When using an axisymmetric filtration model, the formation fluid influx and drilling fluid loss are considered without internal mass transfer between the phases. The filtration velocity field can be found after calculating the pressure distribution in the area. When the drilling fluid filtrates into formation, the formation fluid is displaced without mixing. The generally used approach is the separate consideration of the filtration problem for each layer as it was performed by [PERSON] [103], in which the top and bottom boundaries of the layers are assumed to be impenetrable. Some models involve anisotropic filtration parameters. The formation properties for fluid loss and influx in such models may significantly differ. One of the main functions of drilling fluid is to hold the wellbore pressure greater than the formation pressure, thus preventing an influx (or so-called kick) of formation fluids into the well. Dynamic numerical simulation and analyzing the hydrodynamic properties of drilling fluid allows detecting a kick and developing effective well control strategies; see, for example, [PERSON] and [PERSON] [104]. The mathematical model for the formation-wellbore mass transfer can have different levels of complexity. According to the number and quality of available physical experiments and measurements, it can be made more complicated and therefore more difficult to implement for the complex multiphysical calculations. An overview of the relevant studies is presented by [PERSON] in [105]. In this study, the pressure decay measurements are used to estimate the mass transfer of gas (methane or carbon dioxide) into the liquid phase. The transient-state equilibrium diffusion model is used. The transient fully coupled model for the two-phase flow of CO\({}_{2}\) and water-based fluid in a wellbore is presented by [PERSON] et al. in [106]. The authors consider the mass and heat transfer and dynamic coupling between the wellbore and reservoir.