A Stochastic Analysis of the Non-Newtonian Hydraulic Behaviour of Rough Fractures
Résumé
Fluids involved in activities occurring in fractured underground reservoirs, either related to naturalresource recovery (e.g., hydrofracturing, drilling, geothermal exploitation) or environmental reme-diation schemes, often exhibit complex rheology. The micro-structure of foams, muds, emulsions, orcolloidal suspensions induces shear-thinning in the continuum scale mechanical behaviour, whichcan be described by the Ellis rheology. This three-parameter model has a Newtonian low-shear ratebehaviour of apparent viscosity0, a high-shear rate power-law trend with exponentn, and a transi-tion between the two regulated by a characteristic stress1⁄2. Such fluids often flow in rock fractureshaving rough walls characterized by long-scale correlations in the topography, i.e., a self-affine scaleinvariance at all scales. The facing walls of a given fracture are also mated at scales larger than a characteristic correlation length scale. Such geometries can be reproduced numerically utilizing anFFT-based algorithm. The fracture closure is then measured as the ratio of the aperture field’s rough-ness amplitude to the mean fracture aperture. The investigation of the non-Newtonian hydraulicbehaviour of such natural or artificial fractures implies a considerable mathematical and numeri-cal effort to properly account for non-linearities and medium geometry. A full stochastic analysisof large fractures with a variety of statistical descriptive parameters via Monte Carlo simulationsis almost prohibitive considering a fully 3-D simulation of the flow. The flow of a shear-thinningfluid through a variable aperture fracture can instead be described under the assumptions of thelubrication theory, a depth-averaged formalism that reduces the model formulation to a single two-dimensional non-linear PDE. A numerical code has been implemented adopting the finite volumemethod, with the fracture discretized on a staggered grid, defining the pressures at the centre of eachfinite volume and the aperture at each side. The system of non-linear equation is solved adopting theNewton-Krylov method, considering a continuation strategy to face strong non-linear cases (verylownvalues), and solving the linearized symmetric system of equations via variable-fill-in incom-plete Cholesky preconditioned conjugate gradient algorithm. A Monte Carlo framework is adoptedto study the influence of rheology, fracture dimension and pressure gradient on fracture hydraulicbehaviour, generatingN M C= 1000realizations of the aperture field. The approach allows char-acterizing the hydraulic behaviour via ensemble statistics, such as the PDFs of the velocity fieldsand the dependence of the fracture transmissivity on fracture closure, and how it is impacted by thefluid’s shear thinning behaviour. Fracture flow is mainly cocurrent, presenting narrow PDFs withnearly exponential decay. Evident channelling and localization effects are associated with stronglyheterogeneous aperture fields and very shear-thinning fluids. In these cases, the probability distri-butions of velocity components PDFs show wide tails deviating from the exponential decay, andthe fracture transmissivity is much higher compared with the Newtonian case of identical meanaperture
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