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Keysight Technologies
Measuring Stress-Strain Curves
for Shale Rock by Dynamic
Instrumented Indentation



        Application Note




                   Abstract
                   Three samples of shale rock, two from the Eagle Ford play, and one
                   from the Haynesville play, were successfully tested by instrumented
                   indentation. Results were remarkably repeatable, and hardness and
                   Young's modulus were independent of force for test forces above
                   300mN. For the two samples from the Eagle Ford play, the reduced
                   moduli were 54.3GPa and 40.6GPa, and the hardness values were
                   1.55GPa and 1.12GPa. For the Haynesville sample, the modulus was
                   22.5GPa and the hardness was 0.51GPa. By assuming a Poisson's
                   ratio of 0.25 and negligible work hardening, stress-strain curves
                   were deduced from these indentation measurements. Finite-element
                   simulations of indentation experiments were conducted wherein
                   the simulated materials were assigned the deduced stress-strain
                   curves. Simulated force-displacement curves matched experimental
                   force-displacement curves reasonably well, thus lending credibility
                   to the material model and to the indentation method of determining
                   constitutive properties.
Introduction
    Shale formations host vast natural gas            of plasticity. For isotropic materials,
    and oil reserves which are accessed by            elasticity is fully described by the
    hydraulic fracturing. Experts in the oil          Young's modulus and the Poisson's
    and gas industry have analytical tools at         ratio, n. For stresses above the yield
    their disposal for optimizing fractures to        stress, the material deforms plastically,
    maximize the productivity of a well, and          exhibiting large strains for relatively
    these analytical tools require knowing            small increases in stress. If the material
    the stress-strain curve for the shale, as         has a capacity for work-hardening, then
    well as other mechanical properties.              the stress-strain curve has a positive
                                                      slope, F, beyond the yield point. If the
    The simplest elastic-plastic constitutive         material has no capacity for work-
    model is illustrated schematically in             hardening, then the stress-strain
    Figure 1 as a bi-linear stress-strain             curve is flat beyond the yield point (F
    curve. Materials for which this model             = 0). In summary, such materials are
    is appropriate experience elastic                 mechanically described by only four
    deformation so long as the principle              parameters: Young's modulus (E),
    stress remains below the yield stress,            Poisson's ratio (n), yield stress (Y), and
    Y. The primary characterization of the            hardening slope (F).
    elasticity of the material is the Young's
    modulus, E, which is the slope of the
    stress-strain curve prior to the onset




    Figure 1. Idealized bi-linear stress-strain curve which requires four material constants for
    full definition: E, v, Y, F.
03 | Keysight | Measuring Stress-Strain Curves for Shale Rock by Dynamic Instrumented Indentation - Application Note



Indeed, shale is a complex composite of                    Thus, the purpose of an instrumented                        For many materials, the hardness is
clay, minerals, and organic material. Yet                  indentation test is to cause a controlled                   simply proportional to the yield stress
there are reasons to hope that at large                    contact during which measurements                           of the material, Y, with the constant of
scales, relative to the microstructure, shale              are made to determine the contact                           proportionality being about 3 [6]:
might succumb to a simple mechanical                       stiffness, S, and contact area, A, so that
                                                                                                                                                                (8)
model like that illustrated in Figure 1, and               reduced modulus, and ultimately, the
that further, instrumented indentation might               Young's modulus of the test material
be used to obtain essential mechanical                     can be determined [3

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