Interaction of a normally-incident plane wave with a nonlinear poroelastic fracture.

While it has been recognized that a large amplitude incident wave upon a dry fracture can exhibit nonlinear seismic wave scattering due to its stress-dependent mechanical compliance, the impact of pore fluid in the fracture and a fluid-filled poroelastic background medium-features common for fractures in the Earth-are not well understood. As a first step toward an understanding of the nonlinear poroelastic response of elastic waves in fractured media, analytical approximate formulas are used for the amplitude and phase of a normally incident plane wave using a perturbation method, assuming a fluid-filled, highly compliant nonlinear interface embedded in a linear poroelastic solid. The stress-closure behavior of the fracture is modeled by nonlinear, poroelastic displacement-discontinuity boundary conditions (a linear-slip interface). The theory predicts that the static ("Direct current," or DC) and higher-order-harmonic waves produced by the nonlinear scattering can be greatly reduced by the presence of fluid in the fracture. This, however, depends upon a number of parameters, including fracture compliance, fluid properties (compressibility and viscosity), and the permeability of the background medium, as well as environmental parameters such as the initial fluid pressure and stress acting on the fracture. The static effect produces low-frequency fluid pressure pulses when a finite-duration wave is incident upon the fracture-behavior unique to fluid-filled fractures within a poroelastic medium.

Dispersive transport and symmetry of the dispersion tensor in porous media.

The macroscopic laws controlling the advection and diffusion of solute at the scale of the porous continuum are derived in a general manner that does not place limitations on the geometry and time evolution of the pore space. Special focus is given to the definition and symmetry of the dispersion tensor that is controlling how a solute plume spreads out. We show that the dispersion tensor is not symmetric and that the asymmetry derives from the advective derivative in the pore-scale advection-diffusion equation. When flow is spatially variable across a voxel, such as in the presence of a permeability gradient, the amount of asymmetry can be large. As first shown by Auriault [J.-L. Auriault et al. Transp. Porous Med. 85, 771 (2010)TPMEEI0169-391310.1007/s11242-010-9591-y] in the limit of low Péclet number, we show that at any Péclet number, the dispersion tensor D_{ij} satisfies the flow-reversal symmetry D_{ij}(+q)=D_{ji}(-q) where q is the mean flow in the voxel under analysis; however, Reynold's number must be sufficiently small that the flow is reversible when the force driving the flow changes sign. We also demonstrate these symmetries using lattice-Boltzmann simulations and discuss some subtle aspects of how to measure the dispersion tensor numerically. In particular, the numerical experiments demonstrate that the off-diagonal components of the dispersion tensor are antisymmetric which is consistent with the analytical dependence on the average flow gradients that we propose for these off-diagonal components.

Onsager symmetry from mesoscopic time reversibility and the hydrodynamic dispersion tensor for coarse-grained systems.

Onsager reciprocity relations derive from the fundamental time reversibility of the underlying microscopic equations of motion. This gives rise to a large set of symmetric cross-coupling phenomena. We here demonstrate that different reciprocity relations may arise from the notion of mesoscopic time reversibility, i.e., reversibility of intrinsically coarse-grained equations of motion. We use Brownian dynamics as an example of such a dynamical description and show how it gives rise to reciprocity in the hydrodynamic dispersion tensor as long as the background flow velocity is reversed as well.

Lattice Boltzmann method for diffusion-limited partial dissolution of fluids.

A lattice Boltzmann model for two partially miscible fluids is developed. By partially miscible we mean that, although there is a definite interfacial region separating the two fluids with a surface tension force acting at all points of the transition region, each fluid can nonetheless accept molecules from the other fluid up to a set solubility limit. We allow each fluid to diffuse into the other with the solubility and diffusivity in each fluid being input parameters. The approach is to define two regions within the fluid: one interfacial region having finite width, across which most of the concentration change occurs, and in which a surface tension force and color separation step are allowed for and one miscible fluid region where the concentration of the binary fluids follows an advection-diffusion equation and the mixture as a whole obeys the Navier-Stokes incompressible flow equations. Numerical examples are presented in which the algorithm produces results that are quantitatively compared to exact analytical results as well as qualitatively examined for their reasonableness. The model has the ability to simulate how bubbles of one fluid flow through another while dissolving their contents as well as to simulate a range of practical invasion problems such as injecting supercritical CO(2) into a porous material saturated with water for sequestration purposes.

Acoustic attenuation in self-affine porous structures.

As acoustic waves propagate through fluid-filled porous materials possessing heterogeneity in the elastic compressibility at scales less than wavelengths, the local wave-induced fluid-pressure response will also possess spatial heterogeneity that correlates with the compressibility structure. Such induced fluid-pressure gradients equilibrate via fluid-pressure diffusion causing wave energy to attenuate. This process is numerically simulated using finite-difference modeling. It is shown here, both numerically and analytically, that in the special case where the compressibility structure is a self-affine fractal characterized by a Hurst exponent H, the wave's quality factor Q (where Q(-1) is a measure of acoustic attenuation) is a power law in the wave's frequency omega given by Q proportional to omega(H) when /H/<<1, and given by Q proportional to omega(tanhH) in general.

Interacting damage models mapped onto Ising and percolation models.

We introduce a class of damage models on regular lattices with isotropic interactions between the broken cells of the lattice. Quasi-static fiber bundles are an example. The interactions are assumed to be weak, in the sense that the stress perturbation from a broken cell is much smaller than the mean stress in the system. The system starts intact with a surface-energy threshold required to break any cell sampled from an uncorrelated quenched-disorder distribution. The evolution of this heterogeneous system is ruled by Griffith's principle which states that a cell breaks when the release in potential (elastic) energy in the system exceeds the surface-energy barrier necessary to break the cell. By direct integration over all possible realizations of the quenched disorder, we obtain the probability distribution of each damage configuration at any level of the imposed external deformation. We demonstrate an isomorphism between the distributions so obtained and standard generalized Ising models, in which the coupling constants and effective temperature in the Ising model are functions of the nature of the quenched-disorder distribution and the extent of accumulated damage. In particular, we show that damage models with global load sharing are isomorphic to standard percolation theory and that damage models with a local load sharing rule are isomorphic to the standard Ising model, and draw consequences thereof for the universality class and behavior of the autocorrelation length of the breakdown transitions corresponding to these models. We also treat damage models having more general power-law interactions, and classify the breakdown process as a function of the power-law interaction exponent. Last, we also show that the probability distribution over configurations is a maximum of Shannon's entropy under some specific constraints related to the energetic balance of the fracture process, which firmly relates this type of quenched-disorder based damage model to standard statistical mechanics.

Dispersion of waves in porous cylinders with patchy saturation: formulation and torsional waves.

Laboratory experiments on wave propagation through saturated and partially saturated porous media have often been conducted on porous cylinders that were initially fully saturated and then allowed to dry while continuing to acquire data on the wave behavior. Since it is known that drying typically progresses from outside to inside, a sensible physical model of this process is concentric cylinders having different saturation levels-the simplest example being a fully dry outer cylindrical shell together with a fully wet inner cylinder. We use this model to formulate the equations for wave dispersion in porous cylinders for patchy saturation (i.e., drainage) conditions. In addition to multiple modes of propagation obtained numerically from these dispersion relations, we find two distinct analytical expressions for torsional wave modes. We solve the resulting torsional wave dispersion relation for two examples: Massillon sandstone and Sierra White granite. One essential fact that comes to light during the analysis is that the effective shear moduli of the gas- and liquid-saturated regions must differ, otherwise it is impossible to account for the laboratory torsional wave data. Furthermore, the drainage analysis appears to give improved qualitative and quantitative agreement with the data for both of the materials considered.

Linear dynamics of double-porosity dual-permeability materials. I. Governing equations and acoustic attenuation.

The equations governing the linear acoustics of composites with two isotropic porous constituents are derived from first principles using volume-averaging arguments. The theory is designed for modeling acoustic propagation through heterogeneous porous structures. The only restriction placed on the geometry of the two porous phases is that the overall composite remains isotropic. The theory determines the macroscopic fluid response in each porous phase in addition to the combined bulk response of the grains and fluid in the composite. The complex frequency-dependent macroscopic compressibility laws that are obtained allow for fluid transfer between the porous constituents. Such mesoscopic fluid transport between constituents within each averaging volume provides a distinct attenuation mechanism from the losses associated with the net Darcy flux within individual constituents as is quantified in the examples.

Linear dynamics of double-porosity dual-permeability materials. II. Fluid transport equations.

For the purpose of understanding the acoustic attenuation of double-porosity composites, the key macroscopic equations are those controlling the fluid transport. Two types of fluid transport are present in double-porosity dual-permeability materials: (1) a scalar transport that occurs entirely within each averaging volume and that accounts for the rate at which fluid is exchanged between porous phase 1 and porous phase 2 when there is a difference in the average fluid pressure between the two phases and (2) a vector transport that accounts for fluid flux across an averaging region when there are macroscopic fluid-pressure gradients present. The scalar transport that occurs between the two phases can produce large amounts of wave-induced attenuation. The scalar transport equation is derived using volume-averaging arguments and the frequency dependence of the transport coefficient is obtained. The dual-permeability vector Darcy law that is obtained allows for fluid flux across each phase individually and is shown to have a symmetric permeability matrix. The nature of the cross coupling between the flow in each phase is also discussed.

Fracture of disordered solids in compression as a critical phenomenon. I. Statistical mechanics formalism.

This is the first of a series of three articles that treats fracture localization as a critical phenomenon. This first article establishes a statistical mechanics based on ensemble averages when fluctuations through time play no role in defining the ensemble. Ensembles are obtained by dividing a huge rock sample into many mesoscopic volumes. Because rocks are a disordered collection of grains in cohesive contact, we expect that once shear strain is applied and cracks begin to arrive in the system, the mesoscopic volumes will have a wide distribution of different crack states. These mesoscopic volumes are the members of our ensembles. We determine the probability of observing a mesoscopic volume to be in a given crack state by maximizing Shannon's measure of the emergent-crack disorder subject to constraints coming from the energy balance of brittle fracture. The laws of thermodynamics, the partition function, and the quantification of temperature are obtained for such cracking systems.

Fracture of disordered solids in compression as a critical phenomenon. II. Model Hamiltonian for a population of interacting cracks.

To obtain the probability distribution of two-dimensional crack patterns in mesoscopic regions of a disordered solid, the formalism of Paper I requires that a functional form associating the crack patterns (or states) to their formation energy be developed. The crack states are here defined by an order parameter field representing both the presence and orientation of cracks at each site on a discrete square network. The associated Hamiltonian represents the total work required to lead an uncracked mesovolume into that state as averaged over the initial quenched disorder. The effect of cracks is to create mesovolumes having internal heterogeneity in their elastic moduli. To model the Hamiltonian, the effective elastic moduli corresponding to a given crack distribution are determined that includes crack-to-crack interactions. The interaction terms are entirely responsible for the localization transition analyzed in Paper III. The crack-opening energies are related to these effective moduli via Griffith's criterion as established in Paper I.

Fracture of disordered solids in compression as a critical phenomenon. III. Analysis of the localization transition.

The properties of the Hamiltonian developed in Paper II are studied showing that at a particular strain level a "localization" phase transition occurs characterized by the emergence of conjugate bands of coherently oriented cracks. The functional integration that yields the partition function is then performed analytically using an approximation that employs only a subset of states in the functional neighborhood surrounding the most probable states. Such integration establishes the free energy of the system, and upon taking the derivatives of the free energy, the localization transition is shown to be continuous and to be distinct from peak stress. When the bulk modulus of the grain material is large, localization always occurs in the softening regime following peak stress, while for sufficiently small bulk moduli and at sufficiently low confining pressure, the localization occurs in the hardening regime prior to peak stress. In the approach to localization, the stress-strain relation for the whole rock remains analytic, as is observed both in experimental data and in simpler models. The correlation function of the crack fields is also obtained. It has a correlation length characterizing the aspect ratio of the crack clusters that diverges as xi approximately ( epsilon (c)- epsilon )(-2) at localization.

The role of Biot slow waves in electroseismic wave phenomena.

The electromagnetic fields that are generated as a spherical seismic wave (either P or S) traverses an interface separating two porous materials are numerically modeled both with and without the generation of Biot slow waves at the interface. In the case of an incident fast-P wave, the predicted electric-field amplitudes when slow waves are neglected can easily be off by as much as an order of magnitude. In the case of an incident S wave, the error is much smaller (typically on the order of 10% or less) because not much S-wave energy gets converted into slow waves. In neglecting the slow waves, only six plane waves (reflected and transmitted fast-P, S, and EM waves) are available with which to match the eight continuity conditions that hold at each interface. This overdetermined problem is solved by placing weights on the eight continuity conditions so that those conditions that are most important for obtaining the proper response are emphasized. It is demonstrated that when slow waves are neglected, it is best to also neglect the continuity of the Darcy flow and fluid pressure across an interface. The principal conclusion of this work is that to properly model the electromagnetic (EM) fields generated at an interface by an incident seismic wave, the full Biot theory that allows for generation of slow waves must be employed.