Abstract. We present a formulation for mechanical modeling of geological processes in the seismogenic crust using damage rheology. The seismogenic layer is treated as an elastic medium where distributed damage, modifying the elastic stiffness, evolves as a function of the deformation history. The model damage rheology is based on thermodynamic principles and fundamental observations of rock deformation. The theoretical analysis leads to a kinetic equation for damage evolution having two principal coefficients. The first is a criterion for the transition between strength degradation and recovering (healing), and is related to friction. The second is a rate coefficient of damage evolution which can have different values or functional forms for positive (degradation) and negative (healing) evolution. We constrain these coefficients by fitting model predictions to laboratory data, including coefficient of friction in sawcut setting, intact strength in fracture experiments, first yielding in faulting experiments under three-dimensional strain, onset and evolution of acoustic emission, and dynamic instability. The model damage rheology accounts for many realistic features of three-dimensional deformation fields associated with an earthquake cycle. These include aseismic deformation, gradual strength degradation, development of process zones and branching faults around highdamage areas, strain localization, brittle failure, and state dependent friction. Some properties of the model damage rheology (e.g., cyclic stick-slip behavior with possible accompanying creep) are illustrated with simplified analytical results. The developments of the paper provide an internally consistent framework for simulating long histories of crustal deformation, and studying the coupled evolution of regional earthquakes and faults. This is done in a follow up work.

S U M M A R YWe perform analytical and numerical studies of aftershock sequences following abrupt steps of strain in a rheologically layered model of the lithosphere. The model consists of a weak sedimentary layer, over a seismogenic zone governed by a viscoelastic damage rheology, underlain by a viscoelastic upper mantle. The damage rheology accounts for fundamental irreversible aspects of brittle rock deformation and is constrained by laboratory data of fracture and friction experiments. A 1-D version of the viscoelastic damage rheology leads to an exponential analytical solution for aftershock rates. The corresponding solution for a 3-D volume is expected to be sum of exponentials. The exponential solution depends primarily on a material parameter R given by the ratio of timescale for damage increase to timescale for accumulation of gradual inelastic deformation, and to a lesser extent on the initial damage and a threshold strain state for material degradation. The parameter R is also inversely proportional to the degree of seismic coupling across the fault. Simplifying the governing equations leads to a solution following the modified Omori power-law decay with an analytical exponent p = 1. In addition, the results associated with the general exponential expression can be fitted for various values of R with the modified Omori law. The same holds for the decay rates of aftershocks simulated numerically using the 3-D layered lithospheric model. The results indicate that low R values (e.g. R ≤ 1) corresponding to cold brittle material produce long Omori-type aftershock sequences with high event productivity, while high R values (e.g. R ≥ 5) corresponding to hot viscous material produce short diffuse response with low event productivity. The frequency-size statistics of aftershocks simulated in 3-D cases with low R values follow the Gutenberg-Richter power law relation, while events simulated for high R values are concentrated in a narrow magnitude range. Increasing thickness of the weak sedimentary cover produces results that are similar to those associated with higher R values. Increasing the assumed geothermal gradient reduces the depth extent of the simulated earthquakes. The magnitude of the largest simulated aftershocks is compatible with the Båth law for a range of values of a dynamic damage-weakening parameter. The results provide a physical basis for interpreting the main observed features of aftershock sequences in terms of basic structural and material properties.

S U M M A R YA viscoelastic damage rheology model is presented that provides a generalization of Maxwell viscoelasticity to a non-linear continuum mechanics framework incorporating material degradation and recovery, transition from stable to unstable fracturing and gradual accumulation of non-reversible deformation. The model is a further development of the damage rheology framework of Lyakhovsky et al. for evolving effective elasticity. The framework provides a quantitative treatment for macroscopic effects of evolving distributed cracking with local density represented by an intensive state variable. The formulation, based on thermodynamic principles, leads to a system of kinetic equations for the evolution of damage. An effective viscosity inversely proportional to the rate of damage increase is introduced to account for gradual accumulation of irreversible deformation due to dissipative processes. A power-law relation between the damage variable and elastic moduli leads to a non-linear coupling between the rate of damage evolution and the damage variable itself. This allows the model to reproduce a transition from stable to unstable fracturing of brittle rocks and the Kaiser effect. 3-D numerical simulations based on the model formulation for homogeneous and heterogeneous materials account for the main features of rock behaviour under large strain. The model coefficients are constrained, using triaxial laboratory experiments with low-porosity Westerly granite and high-porosity Berea sandstone samples.

[1] The Dead Sea is a hypersaline terminal lake experiencing a water level drop of about 1 m/yr over the last decade. The existing estimations for the water balance of the lake are widely variable, reflecting the unknown subsurface water inflow, the rate of evaporation, and the rate of salt accumulation at the lake bottom. To estimate these we calculate the energy and mass balances for the Dead Sea utilizing measured meteorological and hydrographical data from 1996 to 2001, taking into account the impact of lowered surface water activity on the evaporation rate. Salt precipitation during this period was about 0.1 m/yr. The average annual inflow is 265-325 Â 10 6 m 3 /yr, corresponding to an evaporation rate of 1.1-1.2 m/yr. Higher inflows, suggested in previous studies, call for increased evaporation rate and are therefore not in line with the energy balance.

Abstract. We study the coupled evolution of earthquakes and faults in a model consisting of a seismogenic upper crust governed by damage rheology over a viscoelastic substrate. The damage rheology has two types of functional coefficients: (1) a "generalized internal friction" separating states associated with material degradation and healing and (2) damage rate coefficients for positive (degradation) and negative (healing) changes. The evolving damage modifies the effective elastic properties of material in the upper crust as a function of the ongoing deformation. This simulates the creation and healing of fault systems in the upper seismogenic zone. In addition to the vertically averaged thin sheet approximation we introduce a Green function for three-dimensional elastic half-space for the instantaneous component of deformation. The formulation accounts in an internally consistent manner for evolving deformation fields, evolving fault structures, aseismic energy release, and spatiotemporal seismicity patterns. These developments allow us to simulate long histories of crustal deformation and to study the simultaneous evolution of regional earthquakes and faults for various model realizations. To focus on basic features of a large strike-slip fault system, we first consider a simplified geometry of the seismogenic crust by prescribing initial conditions consisting of a narrow damage zone in an otherwise damage-free plate. For this configuration, the model generates an earthquake cycle with distinct interseismic, preseismic, coseismic, and postseismic periods. Model evolution during each period is controlled by a subset of physical properties, which may be constrained by geophysical, geodetic, rock mechanics, and seismological data. In the more generic case with a random initial damage distribution, the model generates large crustal faults and subsidiary branches with complex geometries. The simulated statistics depend on the space-time window of the observational domain. The results indicate that long healing timescale, •h, describing systems with relatively long memory, leads to the development of geometrically regular fault systems and the characteristic frequency-size earthquake distribution. Conversely, short zh (relatively short memory) leads to the development of a network of disordered fault systems and the Gutenberg-Richter earthquake statistics. For intermediate values of zh the results exhibit alternating overall switching of response from periods of intense seismic activity and the characteristic earthquake distribution to periods of low seismic activity and Gutenberg-Richter statistics.