Modeling quasi-static poroelastic propagation using an asymptotic approach

Since the formulation of poroelasticity (Biot(1941)) and its reformulation (Rice & Cleary(1976)), there have been many efforts to solve the coupled system of equations. Perhaps because of the complexity of the governing equations, most of the work has been directed towards finding numerical solutions. For example, Lewis and co-workers published early papers (Lewis & Schrefler(1978); Lewis et al.(1991)Lewis, Schrefler, & Simoni) concerned with finite-element methods for computing consolidation, subsidence, and examining the importance of coupling. Other early work dealt with flow in a deformable fractured medium (Narasimhan & Witherspoon 1976); Noorishad et al.(1984)Noorishad, Tsang, & Witherspoon. This effort eventually evolved into a general numerical approach for modeling fluid flow and deformation (Rutqvist et al.(2002)Rutqvist, Wu, Tsang, & Bodvarsson). As a result of this and other work, numerous coupled, computer-based algorithms have emerged, typically falling into one of three categories: one-way coupling, loose coupling, and full coupling (Minkoff et al.(2003)Minkoff, Stone, Bryant, Peszynska, & Wheeler). In one-way coupling the fluid flow is modeled using a conventional numerical simulator and the resulting change in fluid pressures simply drives the deformation. In loosely coupled modeling distinct geomechanical and fluid flow simulators are run for a sequence of time steps and at the conclusion of each step information is passed between the simulators. In full coupling, the fluid flow and geomechanics equations are solved simultaneously at each time step (Lewis & Sukirman(1993); Lewis & Ghafouri(1997); Gutierrez & Lewis(2002)). One disadvantage of a purely numerical approach to solving the governing equations of poroelasticity is that it is not clear how the various parameters interact and influence the solution. Analytic solutions have an advantage in that respect; the relationship between the medium and fluid properties is clear from the form of the solution. Unfortunately, analytic solutions are only available for highly idealized conditions, such as a uniform (Rudnicki(1986)) or one-dimensional (Simon et al.(1984)Simon, Zienkiewicz, & Paul; Gajo & Mongiovi(1995); Wang & Kumpel(2003)) medium. In this paper I derive an asymptotic, semi-analytic solution for coupled deformation and flow. The approach is similar to trajectory- or ray-based methods used to model elastic and electromagnetic wave propagation (Aki & Richards(1980); Kline & Kay(1979); Kravtsov & Orlov(1990); Keller & Lewis(1995)) and, more recently, diffusive propagation (Virieux et al.(1994)Virieux, Flores-Luna, & Gibert; Vasco et al.(2000)Vasco, Karasaki, & Keers; Shapiro et al.(2002)Shapiro, Rothert, Rath, & Rindschwentner; Vasco(2007)). The asymptotic solution is valid in the presence of smoothly-varying, heterogeneous flow properties. The situation I am modeling is that of a formation with heterogeneous flow properties and uniform mechanical properties. The boundaries of the layer may vary arbitrary and can define discontinuities in both flow and mechanical properties. Thus, using the techniques presented here, it is possible to model a stack of irregular layers with differing mechanical properties. Within each layer the hydraulic conductivity and porosity can vary smoothly but with an arbitrarily large magnitude. The advantages of this approach are that it produces explicit, semi-analytic expressions for the arrival time and amplitude of the Biot slow and fast waves, expressions which are valid in a medium with heterogeneous properties. As shown here, the semi-analytic expressions provide insight into the nature of pressure and deformation signals recorded at an observation point. Finally, the technique requires considerably fewer computer resources than does a fully numerical treatment

Reservoir monitoring and characterization using satellite geodetic data: Interferometric Synthetic Aperture Radar observations from the Krechba field, Algeria

Deformation in the material overlying an active reservoir is used to monitor pressure change at depth. A sequence of pressure field estimates, eleven in all, allow us to construct a measure of diffusive travel time throughout the reservoir. The dense distribution of travel time values means that we can construct an exactly linear inverse problem for reservoir flow properties. Application to Interferometric Synthetic Aperture Radar (InSAR) data gathered over a CO{sub 2} injection in Algeria reveals pressure propagation along two northwest trending corridors. An inversion of the travel times indicates the existence of two northwest-trending high permeability zones. The high permeability features trend in the same direction as the regional fault and fracture zones. Model parameter resolution estimates indicate that the features are well resolved

A full field simulation of the in Salah gas production and CO2 storage project using a coupled geo-mechanical and thermal fluid flow simulator

AbstractWe report results from a full field simulation model of the combined production and injection reservoirs, extending from a depth of 3.5 km to ground level, and with a lateral, x and y, extent of approximately 50 km. The model couples geomechanical calculations to fluid flow with an energy transport equation. It simulates two-phase immiscible flow with four components (CH4, CO2, NaCl and H2O) in the gas and aqueous phases. CO2 may dissolve into the aqueous phase. Fractures are modelled explicitly in the grid

Centrality evolution of the charged-particle pseudorapidity density over a broad pseudorapidity range in Pb-Pb collisions at root s(NN)=2.76TeV

Peer reviewe

Invariance, groups, and non-uniqueness: The discrete case

Lie group methods provide a valuable tool for examining invariance and non-uniqueness associated with geophysical inverse problems. The techniques are particularly well suited for the study of non-linear inverse problems. Using the infinitesimal generators of the group it is possible to move within the null space in an iterative fashion. The key computational step in determining the symmetry groups associated with an inverse problem is the singular value decomposition (SVD) of a sparse matrix. I apply the methodology to the eikonal equation and examine the possible solutions associated with a crosswell tomographic experiment. Results from a synthetic test indicate that it is possible to vary the velocity model significantly and still fit the reference arrival times. the approach is also applied to data from corosswell surveys conducted before and after a CO2 injection at the Lost Hills field in California. The results highlight the fact that a fault cross-cutting the region between the wells may act as a conduit for the flow of water and CO2