Search for Model

Intuitively, we know that the creep behavior of the samples must reach a maximum that we can call compressed. The lack of an upper bound is troubling. Fortunately similar behavior has been seen in other phenomena and gives us a clue. Vialov and Zaretsky (1973) showed exponential creep deformation of clays. Gross (1947) showed that this exponential form is equivalent to a Voigt-type material model that includes a very broad Gaussian range of parameters. Juarez-Badillo (1985) showed a Cole-Cole type deformation model that will be explored here to unite all of these observations.

The empirical relation developed by Juarez-Badillo is of the form

 

$$\epsilon(t) = \frac{\epsilon_{final}}{1 + (\tau/t)^d} \,.$$ (2)

Knowing that we need to find a power-law form to fit the observations, we can analyze this equation under the limit where the time of the experiment, t, is much less than the characteristic compaction time, $\tau$, defined as when the sample has undergone exactly half of the final strain limit. This seems appropriate as we are making an effort to do lab experiments at much less than the time that we imagine these processing happening in the field.

Equation 2 then becomes

$$\epsilon(t) = \frac{\epsilon_{final}}{(\tau/t)^d},$$

$$\epsilon(t) = \frac{\epsilon_{final}}{\tau^d} t^d.$$

We notice now that the strain at time 1

$$\epsilon(1) = \frac{\epsilon_{final}}{\tau^d}$$

and therefore

 

$$\epsilon(t) = \epsilon_{1}t^d \; .$$ (3)

Not only does this equation fit well with the observed data, but considering only progressive quartiles of the data, constant and stable values for the regressed parameters $\epsilon_1$ and d are obtained. This provides further justification in the selection of this model as this was one of the significant problems with use of the other models.

Now, assuming that our adoption of the Juarez-Badillo creep mechanism is correct, we have a model that helps explain our data. This fit implies several things:

• All tests are operating well in accordance with the assumption that $t \ll \tau$. Even the data shown in Figure 4 fit the exponential model nicely.
• Under these time constraints we cannot hope to solve for both $\epsilon_{final}$ and $\tau$, but only for their quotient.
• Further research is needed to understand how a sample can achieve such self-select its mode of operation.

 

year Figure 4 One year hold uniaxial creep test. Exponential function still fits meaning $t \ll \tau$ and implying that reservoir material must have a characteristic creep time on the order of decades.Dudley and Myers (1994)