A framework for the prediction of long-term time-dependent deformation of shale

The long-term creep behavior of shale has become a major concern in different engineering applications, which reduces oil and gas production of shale reservoirs and triggers devastating landslides. While indentation creep tests at the nanoscale can reveal creep responses of single-material phases within minutes, long-term experimental studies on the bulk shale samples that last months to years are still limited by their time-consuming nature. Meanwhile, numerical studies on this subject are restricted to empirical formulations, creating little connection to the microstructures of the material. In this work, a framework for predicting the long-term creep behavior of shale is proposed based on a novel constitutive formulation, which bridges the creep response of the bulk shale sample at the centimeter scale and the mechanical properties of single-material phases at the sub-micron scale. The constitutive law developed in this work incorporates three material properties: heterogeneity, anisotropy, and viscoplasticity. Based on experimental results, the constitutive law considers a simplified two-material mixture of a softer phase representing clay and organics, and a harder phase representing inorganic matter. This setting captures two levels of heterogeneity: (1) the two-material mixture at one point representing the volume-average of the sub-micron scale heterogeneity, and (2) the variation of volume fractions from point to point representing the mesoscale heterogeneity in a centimeter-scale bulk shale sample. At the same time, it naturally bridges the mechanical responses from nanoscale to centimeter scale. Under the framework of the modified Cam-Clay critical state model, transverse isotropy is incorporated into the harder material and assumed for both elastic and viscoplastic responses. Time-dependent behavior is addressed by the viscoplastic formulations of Duvaut-Lions and Perzyna, which exhibit excellent convergence properties. To demonstrate the features of the constitutive formulations, a finite element package called GeoScale is developed and applied to several numerical examples. A triaxial creep test on the organic-rich Barnett shale is first simulated. While transverse isotropy is only incorporated in the harder material phase, the solution successfully captures the anisotropic creep responses along different directions of the bulk material, which shows a strong agreement with the experimental results. It is worth noting that similar results cannot be replicated by a one-material model with or without anisotropy, which further justifies the necessity of the two-material model. Second, strain localization in the form of dilative shear band is observed in a set of 2D plane strain simulations under a strain-driven approach, in which the different shear band patterns, either conjugate or single-dominating, are shown to result from various strain rates. In this example, it is worth noting that no element enhancement is needed to accommodate the localized deformation, as viscoplasticity in the formulation acts as a form of regularization. Furthermore, a shale sample with spatially varying volume fractions of material phases is loaded under a constant axial stress. It is shown that the proposed model can trigger multiple creep-induced shear bands, which develop along the weaker direction of the sample and grow in width. GeoScale is further extended to simulate indentation tests on Woodford shale at the nanometer and micrometer scales using an incremental frictionless multi-body contact algorithm based on the Lagrange multipliers method, along with our constitutive framework. Simulation results suggest that creep of the sample is mostly attributed to the viscoplastic deformation of the material away from the indenter tip, and that such response is highly dependent on the stress rate during the loading stage. The time-related material properties are calibrated from the nano- and micro-indentation experiments. As the calibrated material properties become available, the long-term creep behavior of shale is predicted by simulating 3D triaxial creep tests, marking the completion of this work

Advisor: Borja Ronaldo I. (Ronaldo Israel)
Linder Christian (Engineering professor) 
Wren Jon