Study of mixed mode fracture toughness and fracture characteristic in gypsum rock under brine saturation

Abstract

In the deep geological situation, the gypsum is almost always associated with high temperature and the other mineral substances such as sodium chloride. And, hot saline brine will greatly affect the strength and deformability of gypsum by influencing the petrographic characteristics such as microstructure, granular minerals, composition content and morphology. To quantify the effect of brine temperature and concentration on fracture toughness and fracture characteristics of gypsum, a serious of tests, including cracked straight through Brazilian disc (CSTBD) tests and scanning electron microscopy (SEM), were carried out. The results demonstrated that, when at a given crack inclination angle, the fracture peak of gypsum decreases with the increase of liquid temperature. When at a given liquid temperature, due to common-ion effect, the fracture peak first decrease then increase with the increase of brine concentration. In all these cases, due to the effect of friction coefficient, the fracture peak first decreases then increases with the increase of prefabricated crack angle. From SEM morphology results, the microcrack density increases with liquid temperature increasing, whereas first increase then decrease with brine concentration increasing. Subsequently, by adopting the approach of back-propagation neural network, the effective FPZ at different conditions were obtained. It was found that the size of effective FPZ is closely dependent on brine temperature and concentration.

Appendices

Appendix 1

A unified approach combining the interaction integral method and the finite element method is provided to evaluate the crack parameters. The new method has larger application scale than the conventional method. Then, an initial plane model is meshed with 2164 nodes and 693 elements (see in Figs. 22, 23, 24). The calculation results are shown in Fig. 25 and Table 9.

Fig. 22

Schematic of refined mesh and integral domains

Fig. 23

Schematic of integral strategy

Fig. 24

Schematic of refined mesh, integral domains and gauss integral point at various inclination angles. a 0°, b 7°, c 15°, d 22°

Fig. 25

The dimensionless parameters of SIFs and T stress for the CSTBD specimen at various inclination angles

Table 9 The dimensionless parameters of SIFs and T stress for the CSTBD specimen

Appendix 2

The method of neural network inversion is used to invert the parameter of r_c. We construct the training samples and test samples of the BP neural network model. Then the goodness of fit (i.e., the predicated diagrams of (K_I/K_IC)/(K_II/K_IC) and the experimental fracture envelopes) is used as the input sample, and the values of r_c are the output samples, and the neural network is trained until the training results coincide with the target. We assume that the non-linear mapping between the goodness of fit and r_c is the following formula.

$$\delta = f(r_{\text{c}} )$$

When the input value is $\delta$, then the output value of BP neural network is the parameter r_c. The detailed procedures for inversion of r_c using BP neural network are shown below.

Construct the learning samples of parameter values (r_c) and obtain the training samples of neural network (goodness of fitting). The value of learning samples range from 2.00 to 3.20. The number of test is 120. Then, following the step 2), these values (120 numbers) are substituted into the MMTS criterion to obtain the (K_I/K_IC)/(K_II/K_IC). Next, determine the points corresponding to each pair of (K_I/K_IC)/(K_II/K_IC) in a diagram of (K_I/K _IC )/(K _II /K _IC ). Finally, the predicated fracture envelopes and the experimental fracture envelopes are compared to obtain the goodness of fitting. Thus, the training samples (120 samples) of BP neural network are obtained.

Obtain the number of hidden nodes of BP neural network. The number of hidden nodes is constructed by the following formula.

$$n_{i} = \sqrt {a + b} + m$$

$$n_{i} \ge \log_{2}^{a}$$

where, $n_{i}$ is the hidden nodes; a is the number of input nodes; b is the output node M; m is the integer ranging from 1 to 10.

Obtain the transfer function of BP neural network. The commonly used transfer function for BP neural networks is the S-type function, which is expressed as follows.

$$y = f(x) = \frac{1}{{1 + e^{{ - (x + \theta )/h_{0} }} }}$$

where, $\theta$ is the neuron threshold; h₀ is the parameter that modifying the shape of the output function.

The neural network is used to construct the network and train the learning samples obtained in the first step. After several iterations, the goodness of fit, the threshold vector and the weight matrix are obtained. Then the parameter of r_c can be obtained by inversion based on these known quantities. Note that, in the course of neural network training, there are two neurons in the input part of BP neural network, in which the hyperbolic tangent S-type transfer function is used between the implicit part and the input part. There are three neurons in the output part, of which the linear transfer function is used between the implicit part and the output part, and the training function is the gradient descent function. We assume that the number of neurons in the implicit part is 10 and the expected error is set to 0.005.