Mathematical Modeling of the Dynamics of Multi-scale Phenomena During Folding and Fracturing of Sedimentary Rocks

Project Summary

This proposal seeks support for collaborative research in mathematical geosciences that addresses three CMG theme areas: 1) Mathematical modeling of large, complex geosystems; 2) Analyzing large geoscience data sets; and 3) Modeling geosystems with a broad range of interacting scales. The team of principal investigators includes a geoscientist with expertise in structural geology, a mathematician with expertise in differential geometry, and a civil engineer with expertise in computational mechanics. Specifically we propose: a) to characterize the geometric shapes of the sedimentary layers within two well-exposed folds using Global Positioning System (GPS) and Light Detection and Ranging (LiDAR) data sets and the principles of differential geometry; b) to investigate the dynamics of the folding process using continuum mechanical principles and Finite Element Methods (FEM); and c) to investigate the physical interactions between kilometer-scale folds and the meter-scale fractures that propagated within them using fracture and damage mechanics principles. This study was motivated by the unprecedented opportunity to characterize the shapes of folded strata at the kilometer scale with decimeter precision using the new LiDAR technology. Our objectives necessitate the integration of mathematical, mechanical and geological concepts and principles, in order to formulate models constrained by the new geological data.

We have gathered preliminary fracture data from the Sheep Mountain Anticline, Wyoming, and preliminary GPS and fracture data from Raplee Ridge Monocline, Utah. Both folds are composed of apparently brittle layers (limestones and sandstones) interspersed among apparently ductile layers (siltstones and shales). Our underlying hypothesis is that the local 3D shape of the surfaces (as characterized with differential geometry) of a sandwich of several layers, will adequately constrain the internal deformation such that solutions to the appropriate boundary-value problem of continuum mechanics will give stress distributions from which the orientation and spatial density of meter-scale fractures can be predicted. The boundary-value problem is minimally a bonded coupling of brittle linear elastic layers with ductile elastoplastic layers, which we propose to analyze with a state-of-the-art multiscale nonlinear finite element technique based on the strong discontinuity concept. The technique accommodates a discontinuity in the displacement field and allows a mathematical capture of the small-scale resolution against the backdrop of the large-scale finite element approximation without resorting to severe mesh refinement. For the problem at hand, the small-scale resolution pertains to fracture kinematics, whereas the large-scale approximation pertains to the geometry of folds

Intellectual merit: 1) each co-PI is a proven leader in his field; 2) applications of differential geometry to geological problems are rare, yet have great promise; 3) strain localization by fracturing during folding is ripe for a new approach heralded by recent advances in computational mechanics; 4) the research involves innovative applications of technology (GPS and LiDAR) that promise unprecedented data quantities and precision.

Broader impacts: 1) the collaboration of here-to-fore unrelated research groups in three different departments at Stanford University; 2) the initiation of a new seminar series and field seminars; 3) the development of a community accessible website with field data and computer codes; 4) the interdisciplinary training of graduate students at the intersection of mathematical and geological sciences; 5) the dissemination of co-authored results in peer-reviewed journals in three disciplines; and 6) the potential applications of original concepts from this research to hydrocarbon production, groundwater management, and earthquake hazards mitigation.

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