There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial differential equations in n variables, the so called U /K-system. Let G m,n denote the Grassmannian of n-dimensional linear subspaces in R m+n , and G 1 m,n the Grassmannian of space-like m-dimensional linear subspaces in the Lorentzian space R m+n,1 . In this paper, we use techniques from soliton theory to study submanifolds in space forms whose Gauss-Codazzi equations are gauge equivalent to the G m,n -system or the G 1 m,n -system. These include submanifolds with constant sectional curvatures, isothermic surfaces, and submanifolds admitting principal curvature coordinates. The dressing actions of simple elements on the space of solutions of the G m,n and G 1 m,n systems correspond to Bäcklund, Darboux and Ribaucour transformations for submanifolds. * Research supported in part by NSF Grant DMS 9626130Since the unit sphere is totally umbilic, the fundamental forms for S 2 in (x, y) coordinates via the parametrization e 3 (x, y) are I 2 = sin 2 u dx 2 + cos 2 u dy 2 , II 2 = −(sin 2 u dx 2 + cos 2 u dy 2 ).The Gauss-Codazzi equations for M (curvature −1) and e 3 (curvature 1) are the same sine-Gordon equationand the tangent plane of M at (x, y) is the same as the tangent plane of the sphere at e 3 (x, y). A direct computation shows that such u gives rise to a solution of A direct computation gives

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A rigorous method is presented to obtain-directly in the time domain and without the need for an integral transform over frequencies-the seismic response of horizontally layered, viscoelastic half-spaces due to arbitrarily distributed (or concentrated) sources. The method is based on exact expressions for the response of the underlying half-space cast in the wavenumber-time domain, and a complete modal solution in that domain for the layers. The proposed method is highly effective and avoids the problems of the usual propagator matrix method that arise when numerical integrations over both frequencies and wavenumbers are carried out, especially the waviness of the kernels due to resonances and reverberations in the layers when damping is light or even absent. Part I presents the basic methodology for the relatively simple case of antiplane SH sources in two-dimensional space, while Part II generalizes the concept to two-dimensional SV-P line sources and to threedimensional point sources, including seismic couples.

Fracture networks inside geological CO 2 storage reservoirs can serve as primary fluid flow conduit, particularly in low-permeability formations. While some experiments focused on the geophysical properties of brine-and CO 2 -saturated rocks during matrix flow, geophysical monitoring of fracture flow when CO 2 displaces brine inside the fracture seems to be overlooked.We have conducted laboratory geophysical monitoring of fluid flow in a naturally fractured tight sandstone during brine and liquid CO 2 injection. For the experiment, the low-porosity, lowpermeability naturally fractured core sample from the Triassic De Geerdalen Formation was acquired from the Longyearbyen CO 2 storage pilot at Svalbard, Norway. Stress-dependence, hysteresis and the influence of fluid-rock interactions on fracture permeability were investigated.The results suggest that in addition to stress level and pore pressure, mobility and fluid type can affect fracture permeability during loading and unloading cycles. Moreover, the fluid-rock interaction may impact volumetric strain and consequently fracture permeability through swelling and dry out during water and CO 2 injection, respectively. Acoustic velocity and electrical resistivity were measured continuously in the axial direction and three radial levels.Geophysical monitoring of fracture flow revealed that the axial P-wave velocity and axial electrical resistivity are more sensitive to saturation change than the axial S-wave, radial P-wave, and radial resistivity measurements when CO 2 was displacing brine, and the matrix flow was negligible. The marginal decreases of acoustic velocity (maximum 1.6% for axial V p ) compared to 11% increase in axial electrical resistivity suggest that in the case of dominant fracture flow within the fractured tight reservoirs, the use of electrical resistivity methods have a clear advantage compared to seismic methods to monitor CO 2 plume. The knowledge learned from

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In Part I of this set of two companion articles we presented a new, rigorous method to obtain the seismic response of horizontally layered, viscoelastic (or elastic) half-spaces caused by antiplane (SH) sources anywhere. This method is formulated directly in the time domain, and can be applied even in the absence of attenuation. This article (Part II) generalizes the concept to SV-P line sources and to three-dimensional point sources, including seismic moments.

Models of reduced dimensionality have been found to be particularly attractive in simulating the fate of injected CO 2 in supercritical state in the context of carbon capture and storage. This is motivated by the confluence of three aspects: the strong buoyant segregation of the lighter CO 2 phase above water, the relatively long time scales associated with storage, and finally the large aspect ratios that characterize the geometry of typical storage aquifers. However, to date, these models have been confined to considering only the flow problem, as the coupling between reduced dimensionality models for flow and models for geomechanical response has previously not been developed. Herein, we develop a fully coupled, reduced dimension, model for multiphase flow and geomechanics. It is characterized by the aquifer(s) being of lower dimension(s), while the surrounding overburden and underburden being of full dimension. The model allows for general constitutive functions for fluid flow (relative permeability and capillary pressure) and uses the standard Biot coupling between the flow and mechanical equations. The coupled model retains all the simplicities of reduced-dimensional models for flow, including less stiff nonlinear systems of equations (since the upscaled constitutive functions are closer to linear), longer time steps (since the high grid resolution in the vertical direction can be avoided), and less degrees of freedom. We illustrate the applicability of the new coupled model through both a validation study and a practical computational example.

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