# A Computational Study of Polymer Solutions Flow Regimes during Oil Recovery from a Fractured Model
## Materials and Methods
### Numerical Procedure  
Direct numerical simulations were used to investigate two-phase flows during oil recovery from a fractured structure. A two-phase flow of immiscible fluids, namely oil and aqueous polymer solution, was considered. At present, there is a quite large number of methods for calculating such kinds of flows; however, for direct numerical simulation of two-phase flows in microchannels, the Eulerian methods are mainly used, such as the volume of fluid method (VOF) [23] and the Level set method [24]. This is primarily due to the dominant role of capillary forces and interactions of two-phase flows with the walls in microchannels. In this work, preference was given to the VOF method, which is characterized by high persistence of the numerical solution. This is very important when simulating flows in an extended network of microchannels at low capillary numbers. The basic equations of the mathematical model within the VOF method are as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla\left(\rho\overset{\rightarrow}{V} \right)=0, \tag{1}\]  
\[\frac{\partial}{\partial t}\left(\rho\overset{\rightarrow}{V}\right)+\
abla \cdot\left(\rho\overset{\rightarrow}{V}\overset{\rightarrow}{V}\right)=- \
abla p+\
abla\cdot\left(\tau\right)+\overset{\rightarrow}{F}_{s}, \tag{2}\]  
\[\frac{\partial\alpha}{\partial t}+\
abla\cdot\left(\overset{\rightarrow}{ \alpha}\overset{\rightarrow}{V}\right)=0, \tag{3}\]  
where \(\alpha\) is the volume fraction of polymer aqueous solution in the computational cell, \(\overset{\rightarrow}{F}_{s}\) is the volume force of interfacial tension at the oil-water interface, \(\overset{\rightarrow}{V}\) is the flow velocity, \(p\) is the pressure, \(\rho\) is the density, and \(\tau\) is the viscous stress tensor. Within the direct numerical simulation method, the density is found as follows:  
\[\rho=\rho_{\text{P}}\alpha+(1-\alpha)\rho_{\text{o}}, \tag{4}\]  
where \(\rho_{\text{P}}\) is the density of the aqueous polymer solution. Since the polymer concentration is very low, the density does not differ from that of pure water; \(\rho_{\text{o}}\) is the density of crude oil. As can be seen from the viscosity measurement results, presented in the previous section, even at low concentrations, the polymer solution is non-Newtonian. The considered polymer solutions are characterized by a pseudo-viscoplastic rheology. Figure 6: TEM images of diamond powders (**a**) structure, obtained after drying an aqueous suspension of nanodiamonds (**b**). To simulate the rheological behavior, in this work, the generalized viscosity approach of the medium was used [25]. According to this approach, viscous stresses are defined as follows:  
\[\tau_{ij}=\mu\big{(}\varphi,\dot{\gamma}\big{)}\left(\bigg{(}\
abla\overset{ \rightarrow}{V}+\
abla\overset{\rightarrow}{V}\overset{\rightarrow}{V} \bigg{)}-\frac{2}{3}\
abla\cdot\overset{\rightarrow}{V}E\right), \tag{5}\]  
where \(\mu\big{(}\varphi,\dot{\gamma}\big{)}\) is the effective viscosity coefficient of the polymer solution, which depends significantly on the polymer concentration \(\varphi\) and the shear rate \(\dot{\gamma}\), and \(E\) is the unit tensor. Within the VOF method, the expression for the effective viscosity is:  
\[\mu(\varphi,\dot{\gamma})=\mu_{p}(\varphi,\dot{\gamma})\alpha+(1-\alpha)\mu_{ \mathrm{o}}, \tag{6}\]  
where \(\mu_{p}\big{(}\varphi,\dot{\gamma}\big{)}\) is the dependence of effective viscosity of a polymer solution on the polymer concentration and shear rate; \(\mu_{\mathrm{o}}\) is the viscosity of crude oil. Experimental data have shown that the viscosity of the polymer solution is well described by the power-law model:  
\[\mu_{p}\big{(}\varphi,\dot{\gamma}\big{)}=K(\varphi)\dot{\gamma}^{(n(\varphi) -1)}. \tag{7}\]  
Dependences of rheological parameters \(n(\varphi)\) and \(K(\varphi)\) on the polymer concentration are shown in Table 2.  
In addition to the rheology, the addition of polymer has a significant effect on the wettability and interfacial tension (see Table 4).