Monte Carlo Simulations of Shear‐Thinning Flow in Geological Fractures

Abstract

Flow modeling of complex fluids in geological formations is of interest in numerous industrial applications. Among them are enhanced oil recovery (Hirasaki et al., 2011;Leung et al., 2014), geothermal circulations in fractured reservoirs (Bächler et al., 2003;Magzoub et al., 2021) and fluid losses during drilling operations (Feng & Gray, 2017), where foams, muds, emulsions, colloidal or non-colloidal suspensions are commonly involved. The use of high-viscosity gels in hydraulic fracturing improves the proppant carrying capacity and favors the generation of wider fractures in comparison to the use of slick water (Pahari et al., 2021). Drilling muds provide cooling and lubrication to the drill bit and are employed as mechanical stabilizers in the construction of the wellbore to pressurize the borehole against collapse. The constitutive law of these fluids does not respect Newton's law of viscosity, because their micro-structure induces a shear-thinning (ST) rheology at the continuum scale (Ansari et al., 2021;Barati & Liang, 2014). The non-Newtonian behavior of these fluids lies in their physical make-up and the ability of mesoscopic components to cross-link chemically (e.g., polymer solutions, see (Wang et al., 2016)) or interact electrostatically (e.g., colloidal suspensions, see Méheust et al., 2011;Parmar et al., 2008).Subsurface geological formations (e.g., crustal rocks) are discontinuous media, consisting in matrix blocks of low permeability separated by fractures, which provide major conduits for flow. The connectivity among fractures and their hydraulic behavior are the features that control the entire formation permeability (Berkowitz, 1994). The simplest model to study the hydraulic behavior of a fracture is the parallel plate model or cubic law. This model has been largely used for its simplicity, although it oversimplifies wall topography. Different approaches have been proposed also to represent rough fractures: deterministic saw tooth (Wilson & Witherspoon, 1974), sinusoidal profiles (Elsworth & Goodman, 1986), or profiles with an assigned aperture probability distribution (Felisa et al., 2018;Lenci & Di Federico, 2020;Neuzil & Tracy, 1981). In minerals and rocks, however, field and laboratory measurements on fracture walls highlight the stochastic self-affine nature of the surface

Reduced‐Order Models Unravel the Joint Impact of Aperture Heterogeneity and Shear‐Thinning Rheology on Fracture‐Scale Flow Metrics

“…This allows us to extend the results obtained by Lenci, Méheust, et al. (2022), which were constrained to a limited set of scenarios.…”

“…$$L/{L}_{\mathrm{c}}\in \left\{1;4;16\right\}$$, assuming a constant value of the characteristic length used in the fracture generator L c = 0.1 m to account for different fracture sizes ($$L\in \left\{0.1\hspace*{.5em}\mathrm{m};0.4\hspace*{.5em}\mathrm{m};1.6\hspace*{.5em}\mathrm{m}\right\}$$) (Lenci, Méheust, et al., 2022),…”

“…(2022) to solve a generalized non‐linear Reynolds equation that extends the model to ST fluids whose rheology includes a transition from a Newtonian plateau at low shear rates to a power law ST behavior at large shear rates. This numerical strategy has been adopted to produce a Monte Carlo (MC) study of the ST flow through rough fractures (Lenci, Méheust, et al., 2022), overcoming the computation burden that is typical of stochastic analysis of non‐linear problems in spatially varying domains and thus allowing to evidence the impact of the fluid's rheology as discussed above. However, exploring the space of controlling parameters for ST flow in geological fractures over a wide range of values of these parameters, using fracture‐scale simulations, would inevitably require a vast number of MCs, and thus a tremendous amount of computational time, due to the dimensionality of the parameter space (consisting of at least three dimensions: fracture closure, fracture correlation length, imposed pressure gradient), the nonlinearity of the model, and the variability of the aperture field's topology over different realizations.…”

“…More recently, the fracture's hydraulic behavior has been shown to be augmented by the interplay between pore‐scale heterogeneity and ST rheology. This potentially leads to fracture transmissivities orders of magnitude larger than the Newtonian counterpart (Lenci, Méheust, et al., 2022). The transmissivity is computed from the ratio of the volumetric flow rate to the applied macroscopic pressure gradient.…”

Displacement Patterns of a Newtonian Fluid by a Shear‐Thinning Fluid in a Rough Fracture

“…In rough fractures, the coupling of shear‐thinning rheology and heterogeneity of fractures induces a more pronounced channelization phenomenon as compared to Newtonian flow in the same geometry (M. Zhang et al., 2019). However, for rough fractures with various aperture heterogeneity, the transmissivity attenuation due to flow channelization is compensated by the fluid shear‐thinning rheology (Lenci, Putti, et al., 2022), which can render the transmissivity up to several orders of magnitude larger than its Newtonian counterpart (note that the transmissivity is not intrinsic, it depends on the imposed macroscopic pressure gradient (Lenci, Méheust, et al., 2022)). Rodríguez de Castro and Radilla (2017) conducted shear‐thinning fluid flow experiments through rough fractures and proposed a method to predict the macroscopic pressure drop.…”