The surface width, a global quantity that depends on time, is used to characterize the temporal evolution of growing surfaces. One of the most successful concepts for describing the property of the surface width is the famous Family-Vicsek scaling relation. We discuss an extended scaling relation that yields a complete description for various growth models. For two linear Langevin equations, namely the Edwards-Wilkinson equation and the Mullins-Herring equation, we furthermore study analytically the behavior of global quantities related to the surface width or to a quantity which is conjugated to the diffusion constant. The global quantities depend in a non-trivial way on two different times. We discuss the dynamical scaling forms of global correlation and response functions. For global functions related to the surface width, we show that the scaling behavior of the response can depend on how the system is perturbed. Different dynamic regimes, characterized by a power-law or by an exponential relaxation, are identified, and a dynamic phase diagram is constructed. We discuss global fluctuation-dissipation ratios and how to use them for the characterization of non-equilibrium growth processes. We also numerically study the same two-time quantities for the non-linear Kardar-Parisi-Zhang equation. For global functions related to the quantity which is conjugated to the diffusion constant of the linear Langevin equations, we show that the integrated response is proportional to the correlation in the linear response regime. In the aging regime, the autocorrelation and autoresponse exponents are identical and the aging exponent for the response is equal to the aging exponent for the correlation. We investigate the non-equilibrium fluctuation-dissipation theorem for non-equilibrium states based on this quantity. In the non-linear response regime a certain dissipation-fluctuation ratio approximates unity for small waiting times but approaches the ratio of perturbed and unperturbed diffusion constants for larger waiting times. / Ph. D.

The heterogeneous structure of fractured media can lead to complex spatiotemporal fluid invasion dynamics. It thus brings forward challenging fundamental questions in the context of out-of-equilibrium dynamical systems, but also relevant to many processes of interest. The goal of the Thesis is to study the spatio-temporal dynamics of the oil-air interface between displaced air and invading oil, in imbibition through a model open fracture. The research combines exhaustive experimental work with accurate data analysis based on methods of nonlinear statistical physics. The mean postion of the interface h(t) is studied in capillary rise experiments, giving rise to a new analytical solution for h(t). The fluctuations of the interface in forced-flow experiments are analysed in the context of kinetic roughening, characterizing a super-rough scaling scenario. Finally, the burst-like dynamics is studied by analysing the local and global velocities of the front, which are widely distributed and display complex spatio-temporal correlations. We define local and global avalanches whose sizes and durations are also widely distributed, with cutoffs that diverge with the capillary number. Intermittentcy of the global signal is quantified. The ensemble of results presented in this Thesis supports a very general picture of the nonequilibrium dynamics of slowly-driven fronts in open fractures: the lateral propagation of interfacial fluctuations is controlled by local mass conservation, through the lateral correlation length; and the advancement of the interface in the direction of propagation is controlled by the characteristic extent of the disorder d and by the mean front velocity.

Quand un fluide mouillant visqueux (comme une huile) pénètre un milieu hétérogène tel qu’une fracture, l’interface (entre l’air déplacé et l’huile) développe des corrélations à longue portée menant à une dynamique spatio-temporelle complexe. Dans cette Thèse, nous avons étudié expérimentalement et théoriquement ce processus de transport d’un fluide, appelé imbibition, dans un modèle de fracture ouverte, pertinent dans diverses situations. Notre travail a combiné une étude expérimentale détaillée, avec une analyse précise des données, basées sur des méthodes de physique statistique et non-linéaire. D’abord, la position moyenne de l'interface h(t) est étudiée lors d’expériences de montée capillaire donnant lieu à une nouvelle solution analytique pour h(t). Nous avons ensuite étudié les propriétés d’invariance d’échelle de l’interface et en particulier leur évolution pour des processus d’imbibition forcées, caractérisée par un scénario dit «super-rugueux». Enfin, nous avons étudié et quantifié la dynamique intermittente par avalanches des fronts d’imbibition à partir de l’analyse multi-échelle (spatiales et temporelles) de leurs vitesses. L'ensemble des résultats présentés dans cette Thèse propose une image très générale de la dynamique hors équilibre des fronts d’imbibition se propageant lentement dans des fractures ouvertes. La propagation latérale des fluctuations interfaciales est contrôlée par conservation de la masse locale. L'avancement de l'interface dans la direction de propagation est contrôlé par l’échelle caractéristique du désordre et la vitesse moyenne du front. / The heterogeneous structure of fractured media can lead to complex spatiotemporal fluid invasion dynamics. It thus brings forward challenging fundamental questions in the context of out-of-equilibrium dynamical systems, but also relevant to many processes of interest. The goal of the Thesis is to study the spatio-temporal dynamics of the oil-air interface between displaced air and invading oil, in imbibition through a model open fracture. The research combines exhaustive experimental work with accurate data analysis based on methods of nonlinear statistical physics. The mean postion of the interface h(t) is studied in capillary rise experiments, giving rise to a new analytical solution for h(t). The fluctuations of the interface in forced-flow experiments are analysed in the context of kinetic roughening, characterizing a super-rough scaling scenario. Finally, the burst-like dynamics is studied by analysing the local and global velocities of the front, which are widely distributed and display complex spatio-temporal correlations. We define local and global avalanches whose sizes and durations are also widely distributed, with cutoffs that diverge with the capillary number. Intermittentcy of the global signal is quantified. The ensemble of results presented in this Thesis supports a very general picture of the nonequilibrium dynamics of slowly-driven fronts in open fractures: the lateral propagation of interfacial fluctuations is controlled by local mass conservation, through the lateral correlation length; and the advancement of the interface in the direction of propagation is controlled by the characteristic extent of the disorder d and by the mean front velocity.