Publisher's Note: Drag law for an intruder in granular sediments [Phys. Rev. E 95, 032901 (2017)].

This corrects the article DOI: 10.1103/PhysRevE.95.032901.

Drag law for an intruder in granular sediments.

We investigate the drag experienced by a spherical intruder moving through a medium consisting of granular hydrogels immersed in water as a function of its depth, size, and speed. The medium is observed to display a yield stress with a finite force required to move the intruder in the quasistatic regime at low speeds before rapidly increasing at high speeds. In order to understand the relevant time scales that determine drag, we estimate the inertial number I given by the ratio of the time scales required to rearrange grains due to the overburden pressure and imposed shear and the viscous number J given by the ratio of the time scales required to sediment grains in the interstitial fluid and imposed shear. We find that the effective friction µ_{e} encountered by the intruder can be parametrized by I=sqrt[ρ_{g}/P_{p}]v_{i}, where ρ_{g} is the density of the granular hydrogels, v_{i} is the intruder speed, and P_{p} is the overburden pressure due to the weight of the medium, over a wide range of I where the Stokes number St=I^{2}/J≫1. We then show that µ_{e} can be described by the function µ_{e}(I)=µ_{0}+αI^{ß}, where µ_{0}, α, and ß are constants that depend on the medium. This formula can be used to predict the drag experienced by an intruder of a different size at a different depth in the same medium as a function of its speed.

Evolution of Porosity and Channelization of an Erosive Medium Driven by Fluid Flow.

We demonstrate that a homogeneous porous medium composed of sedimentary particles develops channels due to curvature driven growth of fluid flow coupled with an increase in porosity. While the flux is increased linearly, the evolution of porosity is observed to be intermittent with erosion occurring at the boundaries between low and high porosity regions. Calculating the spatial distribution of the flow within the medium and the fluid stress given by the product of the fluid flux and the volume fraction of the particles, we find that the system organizes itself to be locally near the threshold needed to erode the weakest particles. A statistical model simulating the coupling of the erosion, transport, and deposition of the particles to the local fluid flow and porosity is found to capture the overall development of the observed channels.

Experimental study of stable imbibition displacements in a model open fracture. I. Local avalanche dynamics.

We report the results of an experimental investigation of the spatiotemporal dynamics of stable imbibition fronts in a disordered medium, in the regime of capillary disorder, for a wide range of experimental conditions. We have used silicone oils of various viscosities µ and nearly identical oil-air surface tension and forced them to slowly invade a model open fracture at different constant flow rates v. In this first part of the study we have focused on the local dynamics at a scale below the size of the quenched disorder. Changing µ and v independently, we have found that the dynamics is not simply controlled by the capillary number Ca∼µv. Specifically, we have found that the wide statistical distributions of local front velocities, and their large spatial correlations along the front, are indeed controlled by the capillary number Ca. However, local velocities exhibit also very large temporal correlations, and these correlations depend more strongly on the mean imposed velocity v than on the viscosity µ of the invading fluid. Correlations between local velocities lead to a burstlike dynamics. Avalanches, defined as clusters of large local velocities, follow power-law distributions-both in size and duration-with exponential cutoffs that diverge as Ca→0, the pinning-depinning transition of stable imbibition displacements. Large data sets have led to reliable statistics, from which we have derived accurate values of critical exponents of the relevant power-law distributions. We have investigated also the dependence of their cutoffs on µ and v and related them to the autocorrelations of local velocities in space and time.

Experimental study of stable imbibition displacements in a model open fracture. II. Scale-dependent avalanche dynamics.

We report the results of an experimental investigation of the spatiotemporal dynamics of stable imbibition fronts in a disordered medium, in the regime of capillary disorder, for a wide range of experimental conditions. We have used silicone oils of various viscosities µ and nearly identical oil-air surface tension, and forced them to slowly invade a model open fracture at very different flow rates v. In this second part of the study we have carried out a scale-dependent statistical analysis of the front dynamics. We have specifically analyzed the influence of µ and v on the statistical properties of the velocity V_{ℓ}, the spatial average of the local front velocities over a window of lateral size ℓ. We have varied ℓ from the local scale defined by our spatial resolution up to the lateral system size L. Even though the imposed flow rate is constant, the signals V_{ℓ}(t) present very strong fluctuations which evolve systematically with the parameters µ, v, and ℓ. We have verified that the non-Gaussian fluctuations of the global velocity V_{ℓ}(t) are very well described by a generalized Gumbel statistics. The asymmetric shape and the exponential tail of those distributions are controlled by the number of effective degrees of freedom of the imbibition fronts, given by N_{eff}=ℓ/ℓ_{c} (the ratio of the lateral size of the measuring window ℓ to the correlation length ℓ_{c}∼1/sqrt[µv]). The large correlated excursions of V_{ℓ}(t) correspond to global avalanches, which reflect extra displacements of the imbibition fronts. We show that global avalanches are power-law distributed, both in sizes and durations, with robustly defined exponents-independent of µ, v, and ℓ. Nevertheless, the exponential upper cutoffs of the distributions evolve systematically with those parameters. We have found, moreover, that maximum sizes ξ_{S} and maximum durations ξ_{T} of global avalanches are not controlled by the same mechanism. While ξ_{S} are also determined by ℓ/ℓ_{c}, like the amplitude fluctuations of V_{ℓ}(t), ξ_{T} and the temporal correlations of V_{ℓ}(t) evolve much more strongly with imposed flow rate v than with fluid viscosity µ.

Disorder-induced capillary bursts control intermittency in slow imbibition.

A multiscale analysis of the spatially averaged velocity of an imbibition front V_{ℓ}(t) measured at scale ℓ reveals that the slow front dynamics is intermittent: the distributions of ΔV_{ℓ}(τ)=V_{ℓ}(t+τ)-V_{ℓ}(t) evolve continuously through time scales τ, from heavy-tailed to Gaussian-reached at a time lag τ_{c} set by the extent of the medium heterogeneities. Intermittency results from capillary bursts triggered from the smallest scale of the disorder up to the scale ℓ_{c} at which viscous dissipation becomes dominant. The effective number of degrees of freedom of the front ℓ/ℓ_{c} controls its intensity.

Capillary rise in Hele-Shaw models of disordered media.

We study the capillary rise of a viscous liquid in large Hele-Shaw models of disordered media, both analytically and experimentally. Compared to the Fries-Dreyer and Lucas-Washburn solutions for capillary rise with and without gravity, our experimental data reveal a systematic deviation at short and intermediate times. The original pressure balance equation leading to Washburn's results is reformulated in order to include an additional resisting term, proportional to the mean velocity of the front h˙, which appears naturally as a result of the geometry of the cell. Analytical solutions h(t) are found for displacements with and without gravity. These new solutions reproduce the experimental results very accurately in Hele-Shaw cells of constant gap thickness, where the capillary pressure can be approximated by a constant. In cells of fluctuating gap thickness, where the capillary pressure fluctuates in space, a small additional pressure contribution is required. This correction that depends on h˙ is also studied.