The possible transport of fibers by fluid flow in fractures is investigated experimentally in transparent models using flexible polyester thread (mean diameter $280 μ\mathrm{m}$) and Newtonian and shear thinning fluids. In the case of smooth parallel walls, fibers of finite length $\ell = 20-150 \mathrm{mm}$ move at a constant velocity of the order of the maximum fluid velocity in the aperture. In contrast, for fibers lying initially at the inlet side of the model and dragged by the flow inside it, the velocity increases with the depth of penetration (this results from the lower velocity - and drag - in the inlet part). In both cases, the friction of the fiber with the smooth walls is weak. For rough self-affine walls and a continuous gradient of the local mean aperture transverse to the flow, transport of the fibers by a water flow is only possible in the region of larger aperture ($\bar{a} \gtrsim 1.1 \mathrm{mm}$) and is of "stop and go" type at low velocities. Time dependent distorsions of the fiber are also often observed. When water is replaced by a shear thinning polymer solution, the fibers move faster and continuously in high aperture regions and their friction with the walls is reduced. Fiber transport becomes also possible in narrower regions where irreversible pinning occurred for water. In a third rough model with no global aperture gradient but with rough walls and a channelization parallel to the mean flow, fiber transport was only possible in shear-thinning flows and pinning and entanglement effects were studied.
We analyse a problem of a hydraulic fracture driven by a non-Newtonian shear-thinning fluid. Fluid viscosity is described by the four-parameter truncated power-law model. By varying the parameters of the rheological model we investigate spatial and temporal evolution of fluid flow inside the crack and the resulting fracture geometry. A detailed quantitative and qualitative analysis of the underlying physical phenomena is delivered. The results demonstrate that rheological properties of fluids significantly affect the process of hydraulic fracture not only by the values of viscosity, but also b…
In this paper we analyse a problem of a hydraulic fracture driven by a non-Newtonian shear-thinning fluid. For the PKN fracture geometry we consider three different rheological models of fluid: i) the Carreau fluid, ii) the truncated power-law fluid, iii) the power-law fluid. For each of these models a number of simulations are performed. The results are post-processed and compared with each other in order to find decisive factors for similarities/dissimilarities. It is shown that under certain conditions even the basic power-law rheology can be a good substitute for the Carreau characteristic…
Flow and transport are studied in transparent model fractures with rough complementary self-affine walls with a relative shear displacement $\vec{u}$. The aperture field is shown to display long range correlations perpendicular to $\vec{u}$: for flow in that direction, the width and geometry of the front of a dyed shear-thinning polymer solution displacing a transparent one have been studied as a function of the fluid rheology and flow rate. The front width increases linearly with distance indicating a convection of the fluids with a low transverse mixing between the flow paths. The width also…
Pressure-imposed rheometry is used to examine the influence of surface roughness on the rheology of immersed and dry frictional spheres in the dense regime. The quasi-static value of the effective friction coefficient is not significantly affected by particle roughness while the critical volume fraction at jamming decreases with increasing roughness. These values are found to be similar in immersed and dry conditions. Rescaling the volume fraction by the maximum volume fraction leads to collapses of rheological data on master curves. The asymptotic behaviors are examined close to the jamming t…
The miscible displacement of a shear-thinning fluid by another of same rheological properties is studied experimentally in a transparent fracture by an optical technique imaging relative concentration distributions. The fracture walls have complementary self-affine geometries and are shifted laterally in the direction perpendicular to the mean flow velocity {\bf U} : the flow field is strongly channelized and macro dispersion controls the front structure for Péclet numbers above a few units. The global front width increases then linearly with time and reflects the velocity distribution between…
The permeability of two-dimensional fractures with self-affine fractal roughness is studied via analytic arguments and numerical simulations. The limit where the roughness amplitude is small compared with average fracture aperture is analyzed by a perturbation method, while in the opposite case of narrow aperture, we use heuristic arguments based on lubrication theory. Numerical simulations, using the lattice Boltzmann method, are used to examine the complete range of aperture sizes, and confirm the analytic arguments.
The quasi-brittle nature of rocks challenges the basic assumptions of linear hydraulic fracture mechanics (LHFM): linear elastic fracture mechanics and smooth parallel plates lubrication fluid flow. We relax these hypotheses and investigate the growth of a plane-strain hydraulic fracture in an impermeable medium accounting for a rough cohesive zone and a fluid lag. In addition to a dimensionless toughness and the time-scale of coalescence of the fluid and fracture fronts as in the LHFM case, the solution now also depends on the in-situ-to-cohesive stress ratio and the intensity of the flow dev…
Transport properties of three-dimensional self-affine rough fractures are studied by means of an effective-medium analysis and numerical simulations using the Lattice-Boltzmann method. The numerical results show that the effective-medium approximation predicts the right scaling behavior of the permeability and of the velocity fluctuations, in terms of the aperture of the fracture, the roughness exponent and the characteristic length of the fracture surfaces, in the limit of small separation between surfaces. The permeability of the fractures is also investigated as a function of the normal and…
The effect on the transport properties of fractures of a relative shear displacement $\vec u$ of rough walls with complementary self-affine surfaces has been studied experimentally and numerically. The shear displacement $\vec u$ induces an anisotropy of the aperture field with a correlation length scaling as $u$ and significantly larger in the direction perpendicular to $\vec u$. This reflects the appearance of long range channels perpendicular to $\vec u$ resulting in a higher effective permeability for flow in the direction perpendicular to the shear. Miscible displacements fronts in such f…
Miscible tracer dispersion measurements in transparent model fractures with different types of wall roughness are reported. The nature (Fickian or not) of dispersion is determined by studying variations of the mixing front as a function of the traveled distance but also as a function of the lateral scale over which the tracer concentration is averaged. The dominant convective dispersion mechanisms (velocity profile in the gap, velocity variations in the fracture plane) are established by comparing measurements using Newtonian and shear thinning fluids. For small monodisperse rugosities, front …
Dispersion experiments are compared for two transparent model fractures with identical complementary rough walls but with a relative shear displacement $\vecδ$ parallel ($\vecδ\parallel \vec{U}$) or perpendicular ($\vecδ \perp \vec{U}$) to the flow velocity $\vec{U}$. The structure of the mixing front is characterized by mapping the local normalized local transit time $\bar t(x,y)$ and dispersivity $α(x,y)$. For $\vecδ \perp \vec{U}$, displacement fronts display large fingers: their geometry and the distribution of $\bar t(x,y)U/x$ are well reproduced by assuming parallel channels of hydraulic…
Tracer diffusion and hydrodynamic dispersion in two-dimensional fractures with self-affine roughness is studied by analytic and numerical methods. Numerical simulations were performed via the lattice-Boltzmann approach, using a new boundary condition for tracer particles that improves the accuracy of the method. The reduction in the diffusive transport, due to the fractal geometry of the fracture surfaces, is analyzed for different fracture apertures. In the limit of small aperture fluctuations we derive the correction to the diffusive coefficient in terms of the tortuosity, which accounts for…
Dynamics of fluid-fluid displacements in a model rough fracture beyond the quasistatic limit: A spectral approach
Institute of Environmental Assessment and Water Research (IDAEA), Spanish National Research Council (CSIC)（环境评估与水资源研究所，西班牙国家研究委员会）；Universitat de Barcelona（巴塞罗那大学）；University of Barcelona Institute of Complex Systems (UBICS)（巴塞罗那大学复杂系统研究所）
Accurately resolving the coupled momentum transfer between the liquid and solid phases of complex fluids is a fundamental problem in multiphase transport processes, such as hydraulic fracture operations. Specifically we need to characterize the dependence of the normalized average fluid-particle force $\langle F \rangle$ on the volume fraction of the dispersed solid phase and on the rheology of the complex fluid matrix. Here we use direct numerical simulations (DNS) to study the creeping flow ($Re\ll 1$) of viscoelastic fluids through static random arrays of monodisperse spherical particles us…
Complexity of fluid flow in a rough fracture is induced by the complex configurations of opening areas between the fracture planes. In this study, we model fluid flow in an evolvable real rock joint structure, which under certain normal load is sheared. In an experimental study, information regarding about apertures of the rock joint during consecutive 20 mm displacements and fluid flow (permeability) in different pressure heads have been recorded by a scanner laser. Our aim in this study is to simulate the fluid flow in the mentioned complex geometries using the lattice Boltzmann method (LBM)…
We study the dynamics of proppants carried by fluid driven into an evolving penny-shaped fracture. The behaviour of the slurry flow is investigated in two phases: pressurised injection and elastic closure. During injection the slurry is modelled using a frictional rheology that takes into account the shear-induced migration and jamming of the proppants. Making pragmatic assumptions of negligible toughness and cross-fracture fluid slip, we find self-similar solutions supporting a range of proppant concentration profiles. In particular, we define an effective viscosity, which equates the fractur…
We investigate through numerical simulations of the Navier-Stokes equations the influence of the surface roughness on the fluid flow through fracture joints. Using the Hurst exponent $H$ to characterize the roughness of the self-affine surfaces that constitute the fracture, our analysis reveal the important interplay between geometry and inertia on the flow. Precisely, for low values of Reynolds numbers Re, we use Darcy's law to quantify the hydraulic resistance $G$ of the fracture and show that its dependence on $H$ can be explained in terms of a simple geometrical model for the tortuosity $τ…
Loss of circulation while drilling is a challenging problem that may interrupt operations, reduce efficiency, and may contaminate the subsurface. When a drilled borehole intercepts conductive faults or fractures, lost circulation manifests as a partial or total escape of drilling, workover, or cementing fluids, into the surrounding rock formations. Loss control materials (LCM) are often used in the mitigation process. Understanding the fracture effective hydraulic properties and fluid leakage behavior is crucial to mitigate this problem. Analytical modeling of fluid flow in fractures is a tool…
High flow rate, water-driven hydraulic fractures are more common now than ever in the oil and gas industry. Although the fractures are small, the high injection rate and low viscosity of the water, lead to high Reynolds numbers and potentially turbulence in the fracture. Here we present a semi-analytical solution for a blade-shaped (PKN) geometry hydraulic fracture driven by a turbulent fluid in the limit of zero fluid leak-off to the formation. We model the turbulence in the PKN fracture using the Gaukler-Manning-Strickler parametrization, which relates the the flow rate of the water to the p…
Granular material flowing on complex topographies are ubiquitous in industrial and geophysical situations. In this paper, we study the small-scale model of a granular layer flowing on a rough incline. The shape of a granular front is solved analytically by using a 1D Savage-Hutter's model based on depth-averaged mass and momentum equations with the fractional expression for the frictional rheology $μ(I)$. Unlike previous studies where a "plug flow" is assumed, a free shape factor $α$ describing the vertical velocity profile, is taken into account to determine the solution. Such a way, we put i…
