Study on Phase Behavior of CO₂/Hydrocarbons in Shale Reservoirs Considering Sieving Effect and Capillary Pressure

Abstract

The phase behavior of fluid is essential for predicting ultimate oil recovery and determining optimal production parameters. The pore size in shale porous media is nanopore, which causes different phase behaviors of fluid in unconventional reservoirs. Nanopores in shale media can be regard as semipermeable membrane to filter heavy components (sieving effect) in shale oil, which leads to the different distributions of fluid components and different phase behaviors. In addition, the phase behavior of fluid in nanopores can be significantly altered by large capillary pressure. In this paper, the phase behavior of fluid in shale reservoirs is investigated by a new two-phase flash algorithm considering sieving effect and capillary pressure. Firstly, membrane efficiency and capillary pressure are introduced to establish a thermodynamic equilibrium model that is solved by Rachford–Rice flash calculation and Newton–Raphson method. The capillary pressures in different pore sizes are calculated by the Young–Laplace equation. Then, the influences of sieving effect and capillary pressure on phase behavior are analyzed. The results indicate that capillary pressure can suppress the bubble point pressure of fluid in nanopores. The distributions of fluid components are different in various parts of shale media. In the unfiltered part, density and viscosity of fluid are higher. Finally, it is found that the membrane efficiency can be improved by CO₂ injection. The minimum miscibility pressure for shale oil–CO₂ system is also studied. The developed model provides a better understanding of the phase behavior of fluid in shale oil reservoirs.

Appendices

Appendix A: Initial Equilibrium State

According to known conditions including the pressure (P_II) and mole fraction of components (z_IIⁱ) in the filtered part, pressure in the unfiltered part (P_I) and system temperature (T), the calculation process is given at initial equilibrium state to obtain mole fraction of components in the unfiltered part (z_Iⁱ) and membrane efficiency (ω_f). Based on equality of fugacity of light components and material balance in two parts, N equations can be written as follows:

$$\sum\limits_{i = 1}^{N} {z_{I}^{i} = 1}$$

(25)

$$\left[ {f_{I}^{1} ,f_{I}^{2} ...f_{I}^{N - 1} } \right] = \left[ {f_{II}^{1} ,f_{II}^{2} ...f_{II}^{N - 1} } \right]$$

(26)

where z is mole fraction of component, N is number of components, f is fugacity of component, and I and II denote unfiltered part and filtered part, respectively.

Equation (26) can be rewritten as:

$$F\left( {z_{I}^{1} ,...,z_{I}^{N - 1} } \right){ = }\left[ {f_{I}^{1} - f_{II}^{1} ,f_{I}^{2} - f_{II}^{2} ,...f_{I}^{N - 1} - f_{II}^{N - 1} } \right]{ = }0$$

(27)

In order to solve Eq. (27), Newton–Raphson method is used and the formation is obtained as follows:

$$z_{I}^{t + 1} = z_{I}^{t} - Ja^{ - 1} F\left( {z_{I}^{t} } \right)$$

(28)

where $Ja\left( {i,j} \right) = \frac{{\partial F_{i} }}{{\partial z_{I}^{j} }}$, ${\text{for}}_{{}}^{{}} i = 1,2,...N - 1;_{{}}^{{}} {\text{for}}_{{}}^{{}} j = 1,2,...N - 1$

$$Ja = \left[ {\begin{array}{*{20}c} {\varphi_{I}^{1} P_{I} } & 0 & {...} & 0 \\ 0 & {\varphi_{I}^{2} P_{I} } & {} & 0 \\ {...} & {} & {...} & {...} \\ 0 & 0 & {...} & {\varphi_{I}^{N - 1} P_{I} } \\ \end{array} } \right]$$

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Appendix B: Equilibrium State During Pressure Depletion

According to known conditions including the pressure in the filtered part (P_II), membrane efficiency (ω_f), the sum of mole of each component in both parts (nⁱ), volume of the unfiltered part (V_I) and system temperature (T), the calculation process is given during pressure depletion to obtain the mole of components in both parts (n_Iⁱ, n_IIⁱ), the pressure in the filtered part (P_I). Based on equality of fugacity of light components and material balance in two parts, membrane efficiency and volume conservation, 2 N + 1 equations can be written as follows:

$$\left[ {n_{I}^{1} ,...,n_{I}^{N} } \right] + \left[ {n_{II}^{1} ,...,n_{II}^{N} } \right] = \left[ {n_{{}}^{1} ,...,n_{{}}^{N} } \right]$$

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$$\left[ {f_{I,L}^{1} ,f_{I,L}^{2} ...f_{I,L}^{N - 1} } \right] = \left[ {f_{II,L}^{1} ,f_{II,L}^{2} ...f_{II,L}^{N - 1} } \right]$$

(31)

$$V_{I} = V\left( {P_{I} ,T_{I} ,n_{I} } \right) = V_{I,initial}$$

(32)

$$f_{I,L}^{N} { = }\frac{{f_{II,L}^{N} }}{{1 - \omega_{f} }}$$

(33)

The above equations can be simplified as:

$$F\left( {n_{I} ,P_{1} } \right){ = }\left[ {F_{1} ,...,F_{N + 1} } \right]{ = }\left[ {f_{I}^{1} - f_{II}^{1} ,f_{I}^{2} - f_{II}^{2} ,...,f_{I}^{N - 1} - f_{II}^{N - 1} ,f_{I,L}^{N} - \frac{{f_{II,L}^{N} }}{{1 - \omega_{f} }},V\left( {P_{I} ,T_{I} ,n_{I} } \right) - V_{I,initial} } \right]{ = }0$$

(34)

In order to solve Eq. (34), Newton–Raphson method is used and the formation is obtained as follows:

$$\left[ {n_{I} ,P_{1} } \right]^{t + 1} = \left[ {n_{I} ,P_{1} } \right]^{t} - J^{ - 1} F\left( {\left[ {n_{I} ,P_{1} } \right]^{t} } \right)$$

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where

$$J = \left[ {\begin{array}{*{20}c} {\frac{{\partial F_{1} }}{{\partial n_{I}^{1} }}} & {...} & {\frac{{\partial F_{1} }}{{\partial n_{I}^{N} }}} & {\frac{{\partial F_{1} }}{{\partial P_{1} }}} \\ {...} & {...} & {...} & {...} \\ {\frac{{\partial F_{N} }}{{\partial n_{I}^{1} }}} & {...} & {\frac{{\partial F_{N} }}{{\partial n_{I}^{N} }}} & {\frac{{\partial F_{N} }}{{\partial P_{1} }}} \\ {\frac{{\partial F_{N + 1} }}{{\partial n_{I}^{1} }}} & {...} & {\frac{{\partial F_{N + 1} }}{{\partial n_{I}^{N} }}} & {\frac{{\partial F_{N + 1} }}{{\partial P_{1} }}} \\ \end{array} } \right]$$

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$$\frac{{\partial F_{a} \left( {n_{I} ,P_{1} } \right)}}{{\partial n_{I}^{b} }}{ = }\frac{{\sum\limits_{N} {n_{I}^{i} } - n_{I}^{a} }}{{\left( {\sum\limits_{N} {n_{I}^{i} } } \right)^{2} }}\varphi_{I}^{a} P_{I} - \frac{{\sum\limits_{N} {n_{I}^{i} } - n_{I}^{a} - \sum\limits_{N} {n^{i} } + n^{a} }}{{\left( {\sum\limits_{N} {n^{i} } - \sum\limits_{N} {n_{I}^{i} } } \right)^{2} }}X_{a} \varphi_{II}^{a} P_{II} ,^{{}} a \in \left[ {1,N} \right],b \in \left[ {1,N} \right],a = b$$

(36)

$$\frac{{\partial F_{a} \left( {n_{I} ,P_{1} } \right)}}{{\partial n_{I}^{b} }}{ = }\frac{{ - n_{I}^{a} }}{{\left( {\sum\limits_{N} {n_{I}^{i} } } \right)^{2} }}\varphi_{I}^{a} P_{I} - \frac{{n^{a} - n_{I}^{a} }}{{\left( {\sum\limits_{N} {n^{i} } - \sum\limits_{N} {n_{I}^{i} } } \right)^{2} }}X_{a} \varphi_{II}^{a} P_{II} ,^{{}} a \in \left[ {1,N} \right],b \in \left[ {1,N} \right],a \ne b$$

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$$\frac{{\partial F_{j} \left( {n_{I} ,P_{1} } \right)}}{{\partial P_{1} }}{ = }\frac{{n_{I}^{j} }}{{\sum\limits_{N} {n_{I}^{i} } }}\varphi_{I}^{j} ,_{{}} j \in \left[ {1,N} \right]$$

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$$\frac{{\partial F_{N + 1} \left( {n_{I} ,P_{1} } \right)}}{{\partial n_{I}^{j} }} = \frac{{z_{L} RT}}{{P_{1} }},_{{}} j \in \left[ {1,N} \right]$$

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$$\frac{{\partial F_{N + 1} \left( {n_{I} ,P_{1} } \right)}}{{\partial P_{1} }} = - \frac{{z_{L} RT}}{{P_{1}^{2} }}\sum\limits_{N} {n_{I}^{i} }$$

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