**1. Introduction**

As the technology of enhanced oil recovery by gas injection has already been applied worldwide, the research of the transmit mechanism between injected-gas and oil is important to the optimization of gas injection plan. Diffusion is an important phenomenon during the process of gas injection displacement. Because of diffusion, gas molecules will penetrate into the oil phase, while the oil will penetrate into the gas phase. Oil and gas could get balance with time. Diffusion affects the parameters of system pressure, component properties and balance time, which thus affect the efficiency of displacement. Molecular diffusion, which we usually refer to, includes mass transfer diffusion and self-diffusion. Mass transfer diffusion mainly occurs in non-equilibrium condition of the chemical potential gradient ( *<sup>i</sup>* ) .The moleculars move from high chemical potential to low chemical potential of molecular diffusion until the whole system reaching equilibrium. The self-diffusion refers to free movement of molecules (or Brownian motion) in the equilibrium conditions. Mass transfer diffusion and self-diffusion can be quantitatively described by the diffusion coefficient. Up till now, there is no way to test the molecular diffusion coefficient directly. As for the question how to obtain the diffusion coefficient, it is a requirement to establish the diffusion model firstly, and then obtain the diffusion coefficient by analysis of experiments' results.

### **2. Traditional diffusion theory**

### **2.1 Fick's diffusion law**

Fick's law is that unit time per through unit area per the diffusive flux of materials is proportional directly to the concentration gradient, defined as the diffusion rate of that component A during the diffusion.

$$J\_A \propto \frac{dc\_A}{dz} \quad \text{or} \quad J\_A = -D\_{AB}\frac{dc\_A}{dz} \tag{1}$$

Where, *JA*—mole diffusive flux, 2 1 *kmol m s* ;

*z* —distance of diffusion direction;

Research on Molecular Diffusion Coefficient

**3. Molecular diffusion coefficient model** 

**3.1 Establishment of diffusion model** 

Fig. 1. Physical model schematic drawing

as shown in fig.1.

of Gas-Oil System Under High Temperature and High Pressure 3

Fick's second law describes the concentration change of diffusion material during the process of diffusion. From the first law and the second law, we can see that the diffusion coefficient D is independent of the concentration. At a certain temperature and pressure, it is a constant. Under such conditions, the concentration of diffusion equation can be obtained by making use of initial conditions and boundary conditions in the diffusion process, and then the diffusion

In 2007, through the PVT experiments of molecular diffusion, Southwest Petroleum University, Dr. Wang Zhouhua established a non-equilibrium diffusion model and obtained a multi-component gas diffusion coefficient. The establishment of the model is shown in fig.1, with the initial composition of the known non-equilibrium state in gas and liquid phase. During the whole experiment process, temperature was kept being constant. The interface of gas - liquid always maintained a balance, considering the oil phase diffuses into the vapor phase. When the diffusion occurs, the system pressure, volume and composition

As shown in fig.1, *<sup>i</sup> x* and *<sup>i</sup> y* are i-composition molar fraction of liquid and gas phase respectively. *Coi* and *Cgi* are i-composition mass fraction of liquid and gas phase respectively. ni is the total mole fraction of i-composition, mi is the total mass fraction of i-

*L t <sup>o</sup>* / , is the rate of movement of gas-liquid interface. *z* , *<sup>o</sup> z* and *<sup>g</sup> z* are coordinate axis

*<sup>b</sup>* , defined as

composition. *Lo* and *Lg* are the height of liquid and gas phase respectively.

coefficient could be gotten by solving the concentration of diffusion equation.

of each phase will change with time until the system reaches balance.

*<sup>A</sup> dc dz* —concentration gradient of component A at z-direction, <sup>3</sup> *kmol m m* / / ;

*DAB* —the diffusion coefficient of component A in component B, 2 1 *m s* .

Therefore, Fick's law says diffusion rate is proportional to concentration gradient directly and the ratio coefficient is the molecular diffusion coefficient. The Fick's diffusion law is called the first form.

Gas diffusion:

$$N\_A = f\_A = -D \frac{d\varepsilon\_A}{dz} \tag{2}$$

For:

$$
\sigma\_A = \frac{n\_A}{v} = \frac{p\_A}{RT} \tag{3}
$$

We can obtain:

$$N\_A = -\frac{D}{RT} \frac{dp\_A}{dz} \tag{4}$$

$$N\_A \int\_0^z dz = -\frac{D}{RT} \int\_{p\_A}^{p\_i} dp\_A \tag{5}$$

$$N\_A \cdot z = \frac{D}{RT}(p\_A - p\_i) \tag{6}$$

$$N\_A = \frac{D}{RTz}(p\_A - p\_i) \tag{7}$$

Define *<sup>G</sup> <sup>D</sup> <sup>k</sup> RTz* ( *Gk* -mass transfer coefficient) ,then:

$$N\_A = k\_G \left(p\_A - p\_i\right) \tag{8}$$

Similarly, we can obtain the liquid phase diffusion, which is written as follows:

$$N\_A = k\_L \left(\boldsymbol{\varepsilon}\_i - \boldsymbol{\varepsilon}\_A\right) \tag{9}$$

Where *<sup>L</sup> <sup>D</sup> <sup>k</sup> z*

Fick also presented a more general conservation equation:

$$\frac{\partial \mathbf{c}\_1}{\partial t} = D \left( \frac{\partial^2 \mathbf{c}\_1}{\partial z^2} + \frac{\mathbf{l}}{A} \frac{\partial A}{\partial z} \frac{\partial \mathbf{c}\_1}{\partial z} \right) \quad t > 0, 0 < \mathbf{x} < L \tag{10}$$

When area A is constant, eq. 10 become a basic equation of one-dimensional unsteady state diffusion, which is also known as Fick's second law.

Fick's second law describes the concentration change of diffusion material during the process of diffusion. From the first law and the second law, we can see that the diffusion coefficient D is independent of the concentration. At a certain temperature and pressure, it is a constant. Under such conditions, the concentration of diffusion equation can be obtained by making use of initial conditions and boundary conditions in the diffusion process, and then the diffusion coefficient could be gotten by solving the concentration of diffusion equation.
