Nano-dynamic mechanical analysis (nano-DMA) of creep behavior of shales: Bakken case study

Abstract

Understanding the time-dependent mechanical behavior of rocks is important from various aspects and different scales such as predicting reservoir subsidence due to depletion or proppant embedment. Instead of using the conventional creep tests, nano-dynamic mechanical analysis (nano-DMA) was applied in this study to quantify the displacement and mechanical changes in shale samples over its creep time at a very fine scale. The results showed that the minerals with various mechanical properties exhibit different creep behavior. It was found that under the same constant load and time conditions, the creep displacement of hard minerals would be smaller than those that are softer. On the contrary, the changes in mechanical properties (storage modulus, loss modulus, complex modulus and hardness) of hard minerals are larger than soft minerals. The results from curve fitting showed that the changes in creep displacement, storage modulus, complex modulus and hardness over creep time follow a logarithmic function. We further analyzed the mechanical changes in every single phase during the creep time based on the deconvolution method to realize each phase’s response independently. Two distinct mechanical phases can be derived from the deconvolution histograms. As the creep time increases, the volume percentage of the hard mechanical phase decreases, while this shows an increase for soft phases. The results suggest that nano-DMA can be a strong advocate to study the creep behavior of rocks with complex mineralogy.

Appendix

Statistical deconvolution [8, 26, 36].

Let N be the number of the total indents performed on the shale sample. Then storage modulus {E′ _i } and can be driven. We assumed that shale is composed of J = 1, n material phases with sufficient contrast in mechanical properties. Then we can get the following equations:

$$ f_{J} = \frac{{N_{J} }}{N};\,\sum\limits_{J = 1}^{n} {N_{J} } = N $$

(9)

where N _J is the number of the indents on the mechanical material phase and f _J is the volume fraction of the material phase.

The appropriate probability density function of the single phase, which is assumed to fit a normal distribution, will be:

$$ P_{J} = \frac{2}{{\sqrt {2\pi (S_{J} )^{2} } }}\exp \left(\frac{{ - (x - (U_{J} ))^{2} }}{{2(S_{J} )^{2} }}\right) $$

(10)

where U _J and S _J are the mean value and the standard deviation of storage modulus of all N _J values of each phase.

To ensure that the phases have sufficient contrast in properties, the overlap of successive Gaussian curves representative of the two phases is constrained by the following criterion:

$$ U_{J} + S_{J} < U_{J + 1} + S_{J + 1} $$

(11)

Based on the assumption that each material phase obeys the normal distribution and do not mechanically interact with each other, then we can get the overall frequency distribution of the storage modulus using the following equation:

$$ P = \sum\limits_{J = 1}^{n} {f_{J} } P_{J} $$

(12)

where P _J is the normal distribution of the material phase and f _J is the volume fraction of material phase which subjects the constraint:

$$ \sum\limits_{J = 1}^{n} {f_{J} } = 1 $$

(13)

Here, from Eq. (6), we can get n × 3 unknowns {f _J , U _J , S _J }, J = 1, n. And these unknowns can be determined by minimizing the difference between the data from the weighted model-phase probability distribution function (PDF) and the experimental PDF using the following equation:

$$ \hbox{min} \left[ {\sum\limits_{k = 1}^{m} {\frac{{\sum\nolimits_{i = 1}^{N} {\left( {\sum\nolimits_{J}^{n} {f_{J} P_{J} - P^{i} } } \right)^{2} } }}{m}} } \right] $$

(14)

Here, P ⁱ is the observed value of the experimental frequency density of the storage modulus and m is the number of the intervals (bins).