Microwave propagation and absorption and its thermo-mechanical consequences in heterogeneous rocks

Abstract

A numerical analysis in a two-component model rock is presented including the propagation and absorption of a microwave beam as well as the microwave-induced temperature and stress distributions in a consistent way. The analyses are two-dimensional and consider absorbing inclusions (discs) in a non-absorbing matrix representing the model of a heterogeneous rock. The microwave analysis (finite difference time domain - FDTD) is performed with values of the dielectric permittivity typical for hard rocks. Reflections at the discs/matrix interfaces and absorption in the discs lead to diffuse scattering with up to 20% changes of the intensity in the main beam compared to a homogeneous model rock. The subsequent thermo-mechanical finite element (FE) analysis indicates that the stresses become large enough to initiate damage. The results are supported by preliminary experiments on hard rock performed at 2.45 GHz.

Keywords: Microwave analysis; Microwave heating; Microwave irradiation experiments; Rock damage; Thermally induced stresses.

Fig. 1

Two-dimensional model of a block of rock with statistical distribution of discs in a matrix (details see Fig. 2). The microwave source (not shown) is positioned at x = 0 and z = − 1 cm. Its width in x direction is 8.6 cm.

Fig. 2

Enlarged section of the two-dimensional model rock. The grains in the matrix are represented by discs positioned on a square lattice with positional disorder. The crosses represent the lattice points, the dots the actual centers of the discs. δ_x and δ_z denote the deviation from the ideal position on a lattice point. After Meisels and Kuchar (2007).

Fig. 3

Results for heterogeneity in the real parts of the permittivity and no loss (κ_m.i = κ_d,i = 0). (a) Distribution of E_y² in the model rock and in front of it. κ_m = 5.4, κ_d = 9.4. Definition of E_y² see Section 2. (b, c) Difference ΔE_y² of the distributions of E_y² in the heterogeneous and homogeneous model. κ_m = 5.4, κ_d = 7.4 and 9.4, respectively. κ_eff calculated from Eq. (4). (d–f) Cuts along the z direction at x = 0 through the E_y² and ΔE_y² distributions of (a), (b), and (c). In (d) the solid line is for the heterogeneous model, the dashed one for the homogeneous model.

Fig. 4

Results for heterogeneity only due to absorption in the discs: κ_m,r = κ_d,r = 7.4, κ_d,i = 0.1, 0.88 and 2.0, κ_m,i = 0. (a–c) Distribution of E_y² in the model rock and in front of it. (d–f) Difference between the distributions of E_y² in the heterogeneous and homogeneous models. Note the change of the color scale.

Fig. 5

Results for differences in the real as well as in the imaginary part of the permittivity of the matrix and the discs, i.e. heterogeneity due to reflection at the discs/matrix interfaces and absorption in the discs. (a–c) Distributions of E_y² in the model rock and in front of it. (d–f) Distributions of ΔE_y². Permittivity values: κ_m,r = 7.1, κ_m,i = 0, and κ_eff,r = 7.4. Values of κ_eff,i (varied), and of κ_d,r and κ_d,i (calculated from the other values, Eq. (3)) are given in the figures.

Fig. 6

E_y² (a) and ΔE_y² (b) for a larger difference of the real parts (κ_m,r = 5.4, and κ_d,r = 9.4) than in Fig. 5; κ_d,i = 0.88. In this case κ_eff is calculated from the other values (Eq. (4)).

Fig. 7

Pattern of the absorbed power density within the model rock. (a) Large penetration depth. (b) Small penetration depth. Compare E_y² in Fig. 5(a) and (b). The color scale indicates the time-averaged absorbed power density in units of W/cm³ normalized to a time-averaged E_y² of 1 V²/cm² of the microwave source at x = 0 cm and z = − 1 cm. Note the difference in the values of the absorbed power. (c) Absorbed power density for a homogeneous reference case (cf. (b)).

Fig. 8

Temperature distribution in °C after 15 s of microwave irradiation (17.5 kW) in the 2D model rock corresponding to Fig. 7(b). Sample size as in Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7.

Fig. 9

Maximum principal stresses in Pa in the model rock after 15 s of microwave irradiation calculated from the temperature distribution of Fig. 8. Dark red indicates values larger than 9 MPa.

Fig. 10

Vector plot of the maximum principal stress (greater than tensile strength of 9 MPa) after 15 s of microwave irradiation in the magnified area shown in Fig. 9; the arrows indicate the direction of the normal to a potential crack plane.

Fig. 11

Comparison of disc arrangement (circles) and the ΔE_y² distribution (color pattern). Enlarged section of Fig. 5(e) near the microwave source.

Fig. 12

Comparison of density variations of the disc distribution (contour lines) and the ΔE_y² distribution as in Fig. 5(e) (color pattern). 0.34 is the average density (= filling factor). Permittivities of discs and matrix same as in Fig. 7. Near the boundary of the model rock (x = 0) the values of the density approach zero. Contours with values smaller than 0.3 are not shown here.

Fig. 13

Spot on basalt irradiated with 2.45 GHz microwaves for 15 s. The power of the microwave source is 25 kW. Losses are estimated to be 30% yielding 17.5 kW available for absorption in the rock. Spallation is clearly visible. The irradiated area has approximately the size of the circle.

Fig. 14

Three spots (indicated by red circles) on gabbro (a) and granite (b) after irradiation with 25 kW (power of the microwave source) for durations given in the figures. Cracks develop due to irradiation for more than 30 and 20 s, respectively, and extend far outside the irradiated area. Surface treated with penetration spray to provide better visibility of cracks.