Longitudinal dispersion in heterogeneous layered porous media during stable and unstable pore-scale miscible displacements

Highlights

• Numerical simulations are carried out to model longitudinal dispersion at pore-scale.

• Longitudinal dispersion coefficient is discovered to be a function of heterogeneity and layering.

• Different dispersion regimes are identified based on the range of the Peclet number.

• Scale-dependency of dispersion is governed by the layering configuration and viscosity ratio.

Abstract

Dispersion and mixing of fluids during miscible displacement in porous media are known to be strongly affected by heterogeneity and viscous fingering. In this study, direct pore-level numerical simulations are employed to model flow and solute transport during both stable and unstable miscible displacements in two-dimensional packings of circular grains. Longitudinal dispersion coefficients are then computed by fitting the numerically obtained concentration profiles to the analytical solution of the convection-dispersion equation. We observed that diffusive, mixed diffusive-advective, and mechanical dispersion regimes are present during stable displacements whereas the middle diffusive-advective regime is absent in unstable displacements due to the evolution of viscous fingers. Viscous fingering also increases the longitudinal dispersion coefficient several orders of magnitude compared to the stable displacement. The results reveal that the scale-dependency of longitudinal dispersion is a function of layering direction and viscosity contrast. During stable displacement in porous domains with no layering or with the layering perpendicular to the displacement direction, the longitudinal dispersion coefficient increases along the length of flow until it reaches an asymptotic value. When layering is in the direction of displacement, however, the scale-dependency behavior is a function of the layering pattern. In unstable displacements, however, longitudinal dispersion always exhibits an asymptotic behavior. Finally, it is observed that the effect of heterogeneity and layering on the magnitude of the longitudinal dispersion coefficient diminishes as viscosity ratio increases to higher adverse values.

Introduction

The dynamics of mixing and transport of solute in porous media is the major focus of subsurface transport studies in groundwater hydrology and miscible recovery of hydrocarbon reservoirs. Mixing and dispersion in pore space is the combined result of molecular diffusion and convective spreading. Molecular diffusion is the transport of solute particles due to a concentration gradient, whereas convective spreading arises from the fluctuations of velocity field inside the porous media. These variations in the velocity magnitude and direction cause the solute particles to travel different paths with different velocities within a porous medium, thus, generating concentration contrasts. This mechanism increases the size of the concentration front and provides a large surface area for molecular diffusion. It is the enhanced mass transfer by molecular diffusion across this enlarged concentration front that makes convective spreading irreversible and contributes to the true mixing (Dentz and Le Borgne, 2011). If there is no molecular diffusion, convective spreading is a reversible phenomenon and only leads to apparent mixing (Jha et al., 2009). The process of mixing that occurs during solute transport in pore-scale is usually described as hydrodynamic dispersion in macroscale and can be quantified by dispersion coefficient. Dispersion coefficient ($\overline{\mathbf{K}}$) is a tensor quantity that measures the degree of mixing which originates from both diffusion and convective spreading. Experimental measurements of the longitudinal dispersion coefficient (K_L, the diagonal component of $\overline{\mathbf{K}}$ along the general direction of flow) are usually carried out using breakthrough experiments (Correa et al., 1990; Jiao and Hötzl, 2004; Arya et al., 1988; Bretz et al., 1988), NMR spectroscopy (Seymour and Callaghan, 1997; Stapf and Packer, 1998; Lebon et al., 1997), and particle tracking velocimetry (Moroni and Cushman, 2001). An alternative approach is to numerically calculate K_L using digital core analysis. Digital core analysis enables us to describe the detailed properties of a porous medium at the pore-scale (micro/nano scale) and calculate K_L at the macroscale. It can be achieved by the numerical simulation of flow and transport in a pore-scale digital representation of a porous medium. Several authors have used different methods to simulate dispersion in various types of porous media. For instance, some authors (Yao et al., 1997; Rolle and Kitanidis, 2014; Mostaghimi et al., 2012; Bijeljic et al., 2011; Maier et al., 2000; Maier et al., 2003; Salles et al., 1993; Sallès et al., 1993; Coelho et al., 1997) have used stochastically reconstructed models to represent porous media during dispersion simulation while others (Bijeljic et al., 2004; Bruderer and Bernabé, 2001; Bruderer-Weng et al., 2004; Makse et al., 2000; Koplik et al., 1988; de Arcangelis L et al., 1986; Babaei and Joekar-Niasar, 2016) solved the flow and transport equations in the extracted equivalent pore networks of porous media.

Dispersion coefficient magnitude during solute transport in porous media strongly depends on the mechanisms that augment convective spreading and enhance diffusion. This includes viscous fingering and the variations of the velocity field due to the medium's structural heterogeneity (Nicolaides et al., 2015). When there is an adverse viscosity ratio, hydrodynamic instabilities will develop at the concentration front. The resulting viscous fingers increase the size of concentration front and generate sharp concentration gradients due to the stretching, steepening, and tip-splitting mechanisms (Tan and Homsy, 1988). Both experimental (Jiao and Hötzl, 2004; Brigham et al., 1961) and Darcy-scale numerical (Connolly and Johns, 2016; Johns and Garmeh, 2010) studies have shown that the additional mixing caused by the enhanced molecular diffusion at this perturbed concentration front results in a larger dispersion coefficient. Some authors (Jha et al., 2013; Jha et al., 2011a; Jha et al., 2011b), however, argued that at high adverse viscosity ratios, flow focusing across the flow domain due to the channeling of less viscous fluid slows down the creation of additional interface by tip-splitting and hinders mixing. In heterogeneous porous media, the presence of microscale (i.e. nonuniform distribution of shape and size of pores and throats) and macroscale (i.e. layering) heterogeneity creates some preferred paths for flow inside the media and gives rise to a nonuniform velocity distribution. It makes the leading edge of the concentration front disordered and introduces additional mixing. Previous laboratory measurements and numerical studies have demonstrated that an increase in the variance of pore size distribution (Bretz et al., 1988), complexity of the pore structure (Brigham et al., 1961), Dykstra–Parsons coefficient of permeability fields (Arya et al., 1988; Connolly and Johns, 2016; Johns and Garmeh, 2010; Adepoju et al., 2013), permeability variance (John et al., 2010), and longitudinal correlation length of permeability (Arya et al., 1988; Johns and Garmeh, 2010; John et al., 2010) leads to higher dispersion coefficients. At pore-scale, higher dispersion coefficients have been observed when medium heterogeneity becomes more complex from sandstones to carbonates (Bijeljic et al., 2011), or the variance (Bruderer and Bernabé, 2001; Bruderer-Weng et al., 2004) and correlation length (Bruderer-Weng et al., 2004; Babaei and Joekar-Niasar, 2016) of pore-network models increase.

An important characteristic of dispersion is the scale-dependency along the flow length (i.e. variation of K_L along the porous medium length). Due to the presence of the solid phase, the velocity field variations in the medium are significant and cause the concentration front to grow as the solute particles travel. Hence, a larger surface area is available for the molecular diffusion of solute which increases the dispersion until the solute particles have traveled enough distance (or time) to completely sample the velocity field. Experimental measurements (Arya et al., 1988; Moroni and Cushman, 2001) as well as Darcy-scale (Connolly and Johns, 2016; Johns and Garmeh, 2010; Adepoju et al., 2013; John et al., 2010; Mahadevan et al., 2003) and pore-scale (Bijeljic et al., 2011; Maier et al., 2000; Maier et al., 2003; Bijeljic et al., 2004; Bruderer and Bernabé, 2001; Bruderer-Weng et al., 2004; Garmeh et al., 2009; Zhu and Fox, 2002) numerical simulations at the unit viscosity ratio (i.e. stable flow) have shown that K_L increases with the displacement time or the distance traveled by solute particles until asymptotic dispersion is established. At the pore-level, this asymptotic length scale is shown to be about five to six pore diameters for random bead packs (Moroni and Cushman, 2001), about seven grain diameters for unconfined sphere packings (Maier et al., 2003), and hundreds of pores for consolidated media (Bijeljic et al., 2004). At the Darcy scale, however, due to the presence of large scale heterogeneities such as permeability layering, the asymptotic length for dispersion is much higher (in order of a few to hundreds of meters) (John et al., 2010; Mahadevan et al., 2003). For layered permeability fields (Lake and Hirasaki, 1981), it was observed that if the ratio of the longitudinal travel time of solute particles owing to advection to their transverse travel time due to dispersion is smaller than 0.2, longitudinal dispersion coefficient never reaches its asymptotic value. In the presence of viscous fingering, numerical simulations of dispersion during Darcy flow (Jha et al., 2013; Jha et al., 2011a; Jha et al., 2011b) have revealed that there is an interplay between channeling and viscous fingering as the transport of solute proceeds. At early times, a higher rate of mixing was observed due to the additional surface area created by fingering. At later times during the flow at high adverse viscosity ratios, however, channeling dominates the flow and slows down the creation of the additional interface area thus making the mixing less efficient (Jha et al., 2013; Jha et al., 2011a; Jha et al., 2011b).

Despite the extensive research on mixing and dispersion in porous media, there are a few studies on the combined effect of medium heterogeneity and viscous fingering on the values of K_L and its scale-dependency. Most of the previous studies focus on the scale-dependency of dispersion during stable flow (unit viscosity ratio) and the others that account for viscous fingering (Nicolaides et al., 2015; Connolly and Johns, 2016; Johns and Garmeh, 2010; Jha et al., 2013; Jha et al., 2011a; Jha et al., 2011b) are focused on the modeling of dispersion at Darcy-scale which neglects the variability of heterogeneity in scales smaller than the size of the continuum gridblocks. In this study, we model dispersion in pore-scale by solving the point equations of flow and transport in 2-dimensional pore-scale models of granular porous media with different levels of heterogeneity and layering. Numerical simulations at different Peclet numbers are carried out to investigate the effect of heterogeneity and layering on the scale-dependency of K_L during both unit and adverse viscosity ratio flows. In what follows, we first present the governing equations, the physical models of the porous media, the numerical approaches, and the calculation procedure of K_L in section two. Results of the dispersion simulation at both stable and unstable flows are discussed in section three. Finally, we summarize the main conclusions of the present study in section four.