A New Multiscale Computational Model for Low Salinity Waterflooding in Clay Bearing Sandstones

Abstract

We develop a new multiscale model to compute effective properties such as relative permeability, contact angle and partition coefficients in low salinity enhanced oil recovery processes for two-phase flow in sandstones containing reactive surfaces of kaolinite clay. In this setting, we construct a three-scale approach which entails the local nanoscale description ruled by the electro-chemistry of a confined electrolyte solution containing Na⁺, $Ca^{2+}$, $H^+$, $Cl^-$ and $OH^-$ ions residing between bounded crude-oil droplets at residual saturation and clay substrate. Our analysis focuses on the case of surface complexation geochemical reactions between the ionic species of the invading water and the electrically charged kaolinite and oil–water interfaces. In this scenario, we construct a local electric double layer problem for the electric potential based on a non-symmetric Poisson–Boltzmann equation supplemented by nonlinear boundary conditions with the magnitude of the surface charge strongly dictated by the geochemical reactions. By invoking the local mechanical equilibrium of the electrolyte solution and solving numerically the nonlinear problem using the finite element method, we compute the local ionic profiles and reconstruct numerically the disjoining pressure and adsorption isotherms for each ionic species for a wide range of brine compositions and pH of the water phase. Furthermore, combining the disjoining pressure results with the Frumkin/Derjaguin wetting theory allows to compute the dependence of the contact angle on wettability, pH and salinity. Subsequently, the formal homogenization procedure is adopted to upscale the pore-scale flow and ion transport to the macroscale giving rise to a new Darcy scale coupled flow/transport model. The hyperbolic part of the nonlinear homogenized model is solved analytically in an 1D example of enhanced oil recovery.

Appendices

A. Dependence of the retardation coefficient on clay content

In what follows, we develop a direct relation between the clay volumetric surface area $A^c_V$, specific surface area $A_S^c$ and the clay content CC.

Denoting $M_c$ and $M_s$ the mass of the clay and sandstone, $\rho _c$ and $\rho _s$ the corresponding densities we have by definition

$$\begin{aligned} \rho _c := \displaystyle \frac{M_c}{|Y_c|} \qquad \text{ and } \qquad \rho _s := \displaystyle \frac{M_s}{|Y_s|} \end{aligned}$$

(126)

where $|Y_c|$ and $|Y_s|$ are the volumes occupied by the clay and sandstone matter, respectively. The clay specific surface area $A_S^c$ and the clay content CC are defined by

$$\begin{aligned} A_S^c:=\displaystyle \frac{A^{c}}{M_c} \quad \text{ and } \quad CC := \displaystyle \frac{M_c}{M_c + M_s} \end{aligned}$$

(127)

where $A^{c}$ is the surface area of the clay particles in the domain.

Thus denoting $|Y_f|$, $|Y_c|$ and $|Y_s|$ the volume occupied by the interparticle pores, clay and sandstone, respectively, with $|Y| =|Y_f|+|Y_c|+|Y_s|$. We have

$$\begin{aligned} \varphi _f := \displaystyle \frac{|Y_f|}{|Y|} ; \quad \varphi _c := \displaystyle \frac{|Y_c|}{|Y|} \quad \text{ and } \quad \varphi _s := \displaystyle \frac{|Y_s|}{|Y|} \end{aligned}$$

(128)

where $\varphi _f$, $\varphi _c$ $\varphi _s$ are the volume fractions of the interparticle pores, clay and inorganic matter, respectively. Using the definition (126)–(128) the volumetric surface area of the clay $A^c_V$ can be expressed in the form

$$\begin{aligned} A^c_V := \displaystyle \frac{A^{c}}{|Y|}= \displaystyle \frac{A^{c}}{M_c}\displaystyle \frac{M_c}{|Y|}=A^c_S\displaystyle \frac{M_c}{|Y_s|}\displaystyle \frac{|Y_s|}{|Y|} =A^c_S \ \rho _c \ \varphi _c \end{aligned}$$

(129)

Also

$$\begin{aligned} 1-\varphi _s= & {} 1-\displaystyle \frac{|Y_s|}{|Y|}= 1-\left( \displaystyle \frac{|Y_s|}{|Y_c|+|Y_s|}\right) \left( \displaystyle \frac{|Y_c|+|Y_s|}{|Y|}\right) = 1-\left( \displaystyle \frac{|Y_s|}{|Y_c|+|Y_s|}\right) \left( \displaystyle \frac{|Y|-|Y_f|}{|Y|}\right) = \nonumber \\= & {} 1-\left( \displaystyle \frac{|Y_s|}{|Y_c|+|Y_s|}\right) \left( 1-\varphi _f\right) = 1-\left( 1-\displaystyle \frac{|Y_c|}{|Y_c|+|Y_s|}\right) \left( 1-\varphi _f\right) \nonumber \\= & {} 1-\left( 1-\displaystyle \frac{1}{1+\displaystyle \frac{|Y_s|}{|Y_c|}} \right) \left( 1-\varphi _f\right) \end{aligned}$$

(130)

From the definitions (126)–(127), the expressions $|Y_s|/|Y_c|$ above can be rewritten in the form

$$\begin{aligned} \displaystyle \frac{|Y_s|}{|Y_c|}=\displaystyle \frac{\rho _c M_s}{\rho _s M_c} \quad \text{ and } \quad \displaystyle \frac{M_s}{M_c}= \displaystyle \frac{\displaystyle \frac{M_s}{M_c+M_s}}{\displaystyle \frac{M_c}{M_c+M_s}}= \displaystyle \frac{1-\displaystyle \frac{M_c}{M_c+M_s}}{\displaystyle \frac{M_c}{M_c+M_s}} =\displaystyle \frac{1-CC}{CC} \end{aligned}$$

(131)

Combining (129)–(131) we have

$$\begin{aligned} \varphi _c = 1-\varphi _s - \varphi _f = \displaystyle \frac{1-\varphi _f}{1+\displaystyle \frac{\rho _c(1-CC)}{\rho _s CC}} \quad \Rightarrow \quad A^c_V = \displaystyle \frac{A^c_S \ \rho _c(1-\varphi _f)}{1+\displaystyle \frac{\rho _c(1-CC)}{\rho _s CC}} \end{aligned}$$

(132)

In Fig.24 we display the volumetric surface area ($A^c_V$) as a function of clay content (CC) for three different specific surface areas ($A^c_S$). For the (Fig. 24a and b), we consider $\varphi =0.5$ and 0.75, respectively. The results show that the volumetric surface area increase linearly with the clay content and specific surface area. Moreover, the increase of the porosity decreases the $A^c_V$.

Fig. 24

Volumetric surface area (VSA) as a function of clay content (CC) for three different specific surface areas (SSA) and two porosities $\varphi =0.5$, 0.75, respectively

B. Contact Angle

Here, we revise the Frumkin/Derjaguin stability theory which establishes the correlation between the contact angle of an oil droplet on a substrate and local balance between capillary and disjoining pressures. Thus, $P_c$ denoting the capillary pressure, defined by the pressure difference between non-wetting and wetting phases, the modified Young–Laplace equation is given by Basu and Sharma (1996), Hirasaki (1991)

$$\begin{aligned} P_c=\Pi _d + 2 \sigma ^{w,nw} J \end{aligned}$$

(133)

where $\sigma ^{w,nw}$ is the interfacial tension between wetting and the non-wetting phases and J is the surface curvature of the oil droplet.

The film tension $\sigma ^{f}$ of the aqueous electrolyte solution is given by

$$\begin{aligned} \sigma ^f = \sigma ^{ow} + \sigma ^{sw} + \Pi _d H \end{aligned}$$

(134)

with $\sigma ^{ow}$ and $\sigma ^{sw}$ the surface tensions at the aqueous/oil and aqueous/clay interfaces, respectively.

For a fixed temperature T and chemical potential $\mu$, the thermodynamic definition of the total disjoining pressure is given by Basu and Sharma (1996), Hirasaki (1991a), Hirasaki (1991b)

$$\begin{aligned} \Pi _d = - \displaystyle \frac{\partial }{\partial H}\left( \sigma ^{ow} + \sigma ^{sw}\right) \end{aligned}$$

(135)

Hence, integrating the above expression from the flat region where $J=0$ and $H=H(P_c)$ to $H\rightarrow \infty$ gives

$$\begin{aligned}&\int _{\infty }^{H(P_c)}d\left( \sigma ^{ow} + \sigma ^{sw}\right) = - \int _{\infty }^{H(P_c)}\Pi _d \mathrm{{d}}z \quad \Rightarrow \quad \left( \sigma ^{ow} + \sigma ^{sw}\right) - \left( \sigma ^{ow}_{\infty } + \sigma ^{sw}_{\infty }\right) \nonumber \\&\quad = - \int _{\infty }^{H(P_c)}\Pi _d \mathrm{{d}}z \end{aligned}$$

(136)

where $\sigma ^{ow}_{\infty }$ and $\sigma ^{sw}_{\infty }$ denote the tension in the bulk fluid. Now combining (134) and (136) furnishes

$$\begin{aligned} \sigma ^f - \left( \sigma ^{ow}_{\infty } + \sigma ^{sw}_{\infty }\right) = \Pi _d(d)d(P_c) - \int _{\infty }^{H(P_c)}\Pi _d \mathrm{{d}}z \end{aligned}$$

(137)

The contact angle can be computed from Young equation in the form

$$\begin{aligned} \cos \theta = \displaystyle \frac{\sigma ^f - \sigma ^{sw}_{\infty }}{\sigma ^{ow}_{\infty }} \end{aligned}$$

(138)

Thus, using (138) in (137) we derive the representation for the contact angle in terms of the disjoining pressure

$$\begin{aligned} \cos \theta = 1+ \displaystyle \frac{\Pi _d(d)d(P_c)}{\sigma ^{ow}_{\infty }} - \displaystyle \frac{1}{\sigma ^{ow}_{\infty }} \int _{\infty }^{H(P_c)}\Pi _d \mathrm{{d}}z \end{aligned}$$

(139)

where in the flat region $J=0$ using (133) we have

$$\begin{aligned} \Pi _d(d)=P_c \end{aligned}$$

(140)

Finally, combining (139) and (140) we arrive at the desired representation for the contact angle

$$\begin{aligned} \cos \theta = 1+ \displaystyle \frac{P_c d(P_c)}{\sigma ^{ow}_{\infty }} - \displaystyle \frac{1}{\sigma ^{ow}_{\infty }} \int _{\infty }^{H(P_c)}\Pi _d \mathrm{{d}}z \end{aligned}$$

(141)