Foam Generation Hysteresis in Porous Media: Experiments and New Insights

Abstract

Foam application in subsurface processes including environmental remediation, geological carbon-sequestration, and gas-injection enhanced oil recovery (EOR) has the potential to enhance contamination remediation, secure $\hbox {CO}_{2}$ storage, and improve oil recovery, respectively. Nanoparticles are a promising alternative to surfactants in creating foam in harsh environments. We conducted $\hbox {CO}_{2}$-in-brine foam generation experiments in Boise sandstones with surface-treated silica nanoparticle in high-salinity conditions. All the experiments were conducted at the fixed $\hbox {CO}_{2}$ volume fraction and fixed flow rate which changed in steps. The steady-state foam apparent viscosity was measured as a function of injection velocity. The foam flowing through the cores showed higher apparent viscosity as the flow rate increased from low to medium and high velocities. At very high velocities, once foam bubbles were finely textured, the foam apparent viscosity was governed by foam rheology rather than foam creation. A noticeable hysteresis occurred when the flow velocity was initially increased and then decreased, implying multiple (coarse and strong) foam states at the same superficial velocity. A normalized generation function was combined with CMG-STARS foam model to cover full spectrum of foam behavior in the experiments. The new model successfully captures foam generation and hysteresis trends in presented experiments in this study and data from the literature. The results indicate once foam is generated in porous media, it is possible to maintain strong foam at low injection rates. This makes foam more feasible in field applications where foam generation is limited by high injection rates that may only exist near the injection well.

Appendices

Appendix 1: STARS Foam-Model Summary

In the STARS foam model, when foam is present, the gas relative permeability $\left( {k_\mathrm{{rg}} } \right)$ is reduced by a resistance factor of FM,

$$\begin{aligned} k_\mathrm{{rg}}^\mathrm{f} =k_\mathrm{{rg}} \cdot \hbox {FM}=k_\mathrm{{rg}} \Big /\left( {1+fmmob\mathop \prod \limits _{i=1}^{n} F_{i} } \right) , \end{aligned}$$

(4)

where fmmob is maximum mobility reduction factor when all conditions are favorable. The $F_i$ functions reflect the effect of different physical parameters on foam behavior in porous media. We only use dry-out, shear-thinning, and generation functions in this paper.

$$\begin{aligned} k_\mathrm{{rg}}^\mathrm{f} =k_\mathrm{{rg}} /\left( {1+fmmob \cdot F_{\mathrm{dry-out}} \cdot F_\mathrm{{gen}} \cdot F_{\mathrm{shear}}} \right) , \end{aligned}$$

(5)

The dry-out function represents foam coalescence and is defined as,

$$\begin{aligned} F_{\mathrm{dry-out}} =0.5+\frac{1}{\pi }arctan\left( {epdry\left( {S_\mathrm{w} -fmdry} \right) } \right) , \end{aligned}$$

(6)

where fmdry is the water saturation at which foam experiences significant coalescence. epdry controls the sharpness of foam transition from high-quality to low-quality regime. For the large values of epdry, foam collapses at a very narrow range of water saturation and fmdry approaches to the limiting water saturation $\left( {S_\mathrm{{w}}^*}\right)$ (Farajzadeh et al. 2015). In the recent version of STARS (2015), fmdry and epdry were substituted with sfdry and sfbet, respectively. Then, sfdry can be represented as a function of surfactant concentration, oil saturation, salt concentration, and capillary number. Disabling the other functionalities, keeps sfdry constant playing the same role as fmdry does.

The shear-thinning function is defined as,

$$\begin{aligned} F_{\mathrm{{shear}}} =\left\{ {{\begin{array}{ll} 1&{}\quad N_\mathrm{{ca}} \le fmcap\\ \left( {fmcap/N_\mathrm{{ca}} } \right) ^{epcap}&{}\quad N_\mathrm{{ca}} >fmcap \\ \end{array} },} \right. \end{aligned}$$

(7)

fmcap is the reference rheology capillary number and typically set to the smallest value that foam may encounter in the simulations (Cheng et al. 2000; Boeije and Rossen 2013; Farajzadeh et al. 2015). epcap represents the shear-thinning exponent. For the Newtonian behavior, $epcap=0$. A positive epcap corresponds to the shear-thinning behavior. $N_\mathrm{{ca}}$ is capillary number and is defined as

$$\begin{aligned} N_\mathrm{{ca}} =k\left| {\nabla \varPhi } \right| /\sigma . \end{aligned}$$

(8)

where $\nabla \varPhi$ is the gradient of phase potential, k is absolute permeability, and $\sigma _\mathrm{{wg}}$ is interfacial tension between water and gas.

The foam generation is modeled as a power-law function of capillary number as below,

$$\begin{aligned} F_\mathrm{{gen}} =\left\{ {{\begin{array}{ll} 0&{}\quad N_{\mathrm{ca}} \le fmgcp \\ \left( \left( {N_\mathrm{{ca}} -fmgcp} \right) /fmgcp \right) ^{epgcp}&{}\quad N_\mathrm{{ca}} >fmgcp \\ \end{array} }} \right. , \end{aligned}$$

(9)

where fmgcp is critical foam generation capillary number and epgcp is generation exponent. For $epgcp=0$, foam generation is independent of capillary number. Lotfollahi et al. (2016) assumed a constant bubble generation rate to compute dimensionless foam texture implicit in STARS model.

Appendix 2: Water and Gas Relative Permeability Functions

Two-phase foam-free water-gas relative permeabilities are calculated from the following expressions

$$\begin{aligned} k_\mathrm{{rw}} =k_\mathrm{{rw}}^\mathrm{o} S_\mathrm{{n}}^{{n_\mathrm{{w}}}} , \end{aligned}$$

(10)

$$\begin{aligned} k_\mathrm{{rg}} =k_\mathrm{{rg}}^{\ rm o} \left( {1-S_\mathrm{{n}} } \right) ^{n_\mathrm{{g}}}, \end{aligned}$$

(11)

where $k_\mathrm{{rw}}$ and $k_\mathrm{{rg}}$ are water and gas relative permeabilities, $k_\mathrm{{rw}}^\mathrm{o}$ and $k_\mathrm{{rg}}^{\ rm o}$ are water and gas endpoint relative permeabilities, and $n_\mathrm{{w}}$ and $n_\mathrm{{g}}$ are water and gas Corey’s exponents. $S_\mathrm{{n}}$ is normalized water saturation and is defined as,

$$\begin{aligned} S_\mathrm{{n}} =\left( {S_\mathrm{{w}} -S_\mathrm{{wr}} } \right) /\left( {1-S_\mathrm{{wr}} -S_\mathrm{{gr}} } \right) , \end{aligned}$$

(12)

where $S_\mathrm{{wr}}$ and $S_\mathrm{{gr}}$ are residual water and residual gas saturations, respectively.