# Effects of Velocity and Permeability on Tracer Dispersion in Porous Media
## 4 Conclusions  
In this paper, the effects of velocity and permeability on the dispersion coefficient in cores were investigated. Two sets of core flooding experiments were conducted on natural and sand-filled cores, respectively. The relations between the dispersion coefficient, velocity, and permeability were obtained through fitting the experimental results with an analytical model. The main conclusions drawn from the results presented herein are as follows:  
(1) The core experimental results show that the dispersion coefficient is negatively correlated with the permeability and positively correlated with the superficial velocity. Low permeability and high heterogeneity result in large latitudinal dispersion and high dispersion coefficient. If the superficial velocity is high, the pressure drop along the core is large. This causes the fluid to enter the small pores, resulting in large dispersion coefficient. The combination of high velocity and low permeability yields a large dispersion coefficient. Figure 9: Schematic diagram of structures of porous media: (**a**) small pore sizes, (**b**) large pore sizes. The effects of latitudinal dispersion in porous media cannot be ignored in low-permeability cores or formations. (2) The interpretation results show that the dispersion coefficient increased from 0.063 to 0.037 cm\({}^{2}\)/s with the decrease in core permeability from 230 to 1570 mD at the flow rate of 0.02 cm\({}^{3}\)/s. The dispersion coefficient increased from 0.004 to 0.023 cm\({}^{2}\)/s with the increase in flow rate from 0.02 to 0.08 cm\({}^{3}\)/s. The corresponding regression equations indicate linear relations between the dispersion coefficient, core permeability, and fluid velocity, resulting from the micron scale of the grain diameters in the cores. Conceptualization, Y.Y., T.L. ; Methodology, T.L. ; Experiments, Y.L. ([PERSON]), Y.L. ([PERSON]); Mathematical modelling and solution, Z.Y., M.Z., P.D. ; Analysis, Z.Y., P.D. ; Writing--original draft preparation, Y.L. ([PERSON]), Y.L. ([PERSON]), R.W. ; Writing--review and editing, Y.Y., Z.Y., X.Z., J.L. All authors have read and agreed to the published version of the manuscript. This work was supported by National Key Research and Development Plan (No.2018 YFB06 05500) and the Science Foundation of China University of Petroleum (2462019 YJRC013, 2462020 YXZZ 003 and 2462020 XKJS02). Not applicable. The authors declare no conflict of interest. The governing equation of tracer flow at the pore level may be expressed as [3]:  
\[D\frac{\partial^{2}C}{\partial x^{2}}-u\frac{\partial C}{\partial x}=\frac{[ \phi(1-S_{o})+a(1-\phi)\rho_{r}]}{\phi_{f}(1-S_{o}-S_{uc})}\frac{\partial C}{ \partial t} \tag{1}\]  
where \(D\) is the dispersion coefficient (cm\({}^{2}\)/s), \(C\) is tracer concentration in aqueous solution (mg/L), \(u\) is interstitial velocity (cm/s), \(x\) is flow distance (cm), \(\phi_{f}\) is flow porosity, \(\phi\) is porosity, \(S_{o}\) is oil saturation, \(S_{wc}\) is bound water saturation, \(a\) is the Langmuir isothermal adsorption coefficient, and \(\rho_{r}\) is the rock density (g/cm\({}^{3}\)). To obtain the analytical solution for the tracer flow equation at the pore scale, the modified time, \(t^{\prime}\), is introduced as follows:  
\[t^{\prime}=\frac{\phi_{f}(1-S_{o}-S_{wc})}{[\phi(1-S_{o})+a(1-\phi)\rho_{r}]}t \tag{10}\]  
Consequently, the mathematical model for tracer flow at the pore level may be simplified as:  
\[D\frac{\partial^{2}C}{\partial x^{2}}-u\frac{\partial C}{\partial x}=\frac{ \partial C}{\partial t^{\prime}} \tag{11}\]  
\[C(x,0)= \begin{cases}C_{0}&x\leq 0\\ 0&x>0\end{cases}  \tag{12}\]  
\[C(0,t)=C_{0}\quad t>0 \tag{13}\]  
\[C(\infty,t)=0\quad t>0 \tag{14}\]  
where \(C_{0}\) is the tracer concentration in injected solution (mg/L).