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Modelling creep in shales and chalks

Most shales and chalks show a rate dependent behaviour when subjected to a load (e.g. Dahou et al. 1995, Chang and Zoback, 2009). For example, in a lab geomechanical experiment, if a compressional load is applied to the sample at relatively fast rates, the sample initially compacts only a fraction of the total expected compaction for the applied load magnitude. If then the load is maintained over time the sample creeps and leads to further compaction until a steady state is achieved.


We have recently implemented two models in ParaGeo that can be combined with critical state plasticity to account for creep in shale and chalk sediments. In those models, for fast loading rates, the stress path can cross the “true” yield surface (the one that is linked to the hardening / strain state of the material) and the stress state then sits on an “apparent” yield surface (which is an expanded/larger version of the “true” yield surface). Then if the stress load is maintained further plastic volumetric strain will develop and the “true” yield surface will grow until it reaches the size of the apparent yield surface indicating that the steady state / long term solution is achieved.


Schematic examples of the true and apparent yield surafces in the creep constitutive model after a fast load is applied following an arbitrary loading path
Schematic example of the true and apparent yield surfaces after a fast load following a given loading path


As can be observed in the figures below the implemented models can reproduce the typical creep behaviour observed in hydrostatic compression tests (e.g. see Dahou et al. 1995). The simulation consisted of an hct test with three loading stages with an hydrostatic load increment applied at a relatively fast rate, each followed by a holding period during which the load is kept constant. Due to the rate-dependent behaviour, initially the path deviates from the steady state solution with the states at the end of each load increment showing lower volumetric strains and higher void ratios that should be expected for the load magnitude in the long term. During the holding periods further volumetric plastic strain develops due to creep leading to a reduction in void ratio and the path moves towards the steady state / long term curves.

We show the numerical results obtained from a ParaGeo simulation of an hydrostatic compression test performed on a material that accounts for creep
ParaGeo results from a simulated hydrostatic compression test (HCT) experiment capturing creep in the sample. Vertical lines in the left figure indicate start and end of an step load. Dotted lines in the middle and right figures indicate the steady state (rate-independent) path
 

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