Three-dimensional thermal sensitivity analysis of cooling of a magma chamber in the Los Azufres geothermal field, Michoacán, Mexico
- Surendra P Verma^{1}Email author and
- Efraín Gómez-Arias^{2}
https://doi.org/10.1186/2195-9706-1-5
© Verma and Gomez-Arias; licensee Springer. 2013
Received: 24 July 2013
Accepted: 24 September 2013
Published: 5 November 2013
Abstract
Background
We present three-dimensional simulation of cooling of 63 models of a magma chamber in the Los Azufres geothermal field by varying the top of the chamber depth between 5 and 9 km (centroid from about 7 to 13 km) and its volume between 300 and 600 km^{3}.
Methods
Nineteen new best-fit cubic equations are presented to represent the temperature field in the geothermal reservoir in terms of the chamber centroid depth, its volume, or both.
Results
These equations clearly show that the thermal regime is much more sensitive to chamber depth than to its volume. These simulation results imply that, for a better estimation of the energy budget of a volcanic area, the depth parameter should be better constrained than the chamber volume.
Conclusion
Geoscientists are, therefore, encouraged to obtain more reliable estimates of magma chamber depths for active volcanoes and potential geothermal areas. Furthermore, the smallest discretization time and mesh size should be used for solving the heat flow equations in three-dimensions.
Keywords
Thermal modeling Subsurface temperatures Geothermal system Magma chamber Three-dimensional simulationBackground
The sensitivity of two magma chamber parameters - chamber depth and volume - was also evaluated by Verma et al. (2011) at the top of the magma chamber as well as at its sides, which indicated that the chamber depth is more sensitive than the chamber volume. Similarly, the influence of discretization time and mesh size was estimated from three-dimensional temperature field simulation in the LHGF and LPGF (Verma and Gómez-Arias 2013b).
For the Los Azufres geothermal field (LAGF in Figure 1) located in the central part of the Mexican Volcanic Belt, only two-dimensional thermal modeling was carried out long ago by Verma and Andaverde (1996). We present the first three-dimensional simulation study of this field and document the temperature regime in the underlying geothermal reservoir as a function of the chamber depth and volume. The reservoir temperatures are inferred to be much more sensitive to chamber depth than to its volume. This is the first study that evaluates the sensitivity of these two parameters for a geothermal reservoir. Six additional runs for the last thermal model are presented to predict the thermal regime of this geothermal area and to understand the influence of discretization time and mesh size.
Geological synthesis
The LAGF is located in the state of Michoacán, about 200 km NW of Mexico City, between approximately 100°38′ and 100°43′ W and 19°50′ and 19°45′ N, and covers an area of about 72 km^{2} (Dobson and Mahood 1985). The geology and geochemistry of the area were reported by Campos-Enríquez et al. (2005), Cathelineau et al. (1987), Dobson and Mahood (1985), Pandarinath (2011), Pandarinath et al. (2008), Pradal and Robin (1994), Verma (1985), Verma and Andaverde (1996), and Verma et al. (2005), among others. The pre-volcanic basement consists of shales, sandstones, and conglomerates of Eocene to Oligocene age. The oldest volcanic rocks are andesites of Miocene age (about 18 to 6 Ma) followed by the eruption of rhyolites at 1.6 to 0.84 Ma and andesites at about 0.86 Ma.
Voluminous eruption of dacites (about 19.3 km^{3}) took place at about 0.36 to 0.33 Ma. This major event was simulated in our thermal modeling, which was carried out for about 0.40 Ma to the present. This large eruption was followed by about 12.2 km^{3} of rhyolites during about 0.30 to 0.14 Ma and about 4.6 km^{3} of the youngest basalt, considered to have erupted during 0.15 Ma to the present. From geochemical modeling, Verma (1985) suggested that the volume of the magma chamber beneath the LAGF was at least about 400 km^{3}.
Methods
Conceptual models and methods
Emplacement conditions of magma chamber and geological properties for thermal models of the Los Azufres geothermal field (LAGF)
Physical property (units) | Emplacement of magma chamber |
---|---|
Emplacement conditions | |
Depth of the top the chamber (d) (km) | 5.0-9.0 |
Depth of the chamber centroid (dc) (km) | 7.000 to 12.875 |
Volume (V) (km^{3}) | 300 to 600 |
Thickness of the magma chamber (E) (km) | 4.00 to 7.75 |
Radius (r) (km) | 5.0 |
Magma emplacement temperature (T_{cham}) (°C) | 1,350 |
Boundary conditions | |
Surface temperature (T_{s}) (°C) | 25 |
Geothermal gradient (ΔT_{g}) (°C/km) | 30 |
Geological strata (strata 1–4 rock type)^{a} | |
Granite-granodiorite | |
Width (km) | 16.00 |
Thermal conductivity (W/mK) | 2.80 |
Specific heat (J/kg K) | 1,073 |
Density (kg/m^{3}) | 2,680 |
Metamorphic rocks | |
Width (km) | 1.00 |
Thermal conductivity (W/mK) | 2.73 |
Specific heat (J/kg K) | 1,050 |
Density (kg/m^{3}) | 2,280 |
Andesite | |
Width (km) | 2.75 |
Thermal conductivity (W/mK) | 1.72 |
Specific heat (J/kg K) | 1,151 |
Density (kg/m^{3}) | 2,180 |
Rhyolite | |
Width (km) | 0.25 |
Thermal conductivity (W/mK) | 3.44 |
Specific heat (J/kg K) | 1,074 |
Density (kg/m^{3}) | 2,460 |
Specifications of the 63 models of the Los Azufres geothermal field (LAGF) simulated in the present work
Model | Subsurface depth | Chamber volume V(km^{3}) | Thickness of the magma chamber E (km) | |
---|---|---|---|---|
Top of the chamber d (km) | Chamber centroid dc (km) | |||
M1 | 5.0 | 7.000 | 300 | 4.00 |
M2 | 5.5 | 7.500 | 300 | 4.00 |
M3 | 6.0 | 8.000 | 300 | 4.00 |
M4 | 6.5 | 8.500 | 300 | 4.00 |
M5 | 7.0 | 9.000 | 300 | 4.00 |
M6 | 7.5 | 9.500 | 300 | 4.00 |
M7 | 8.0 | 10.000 | 300 | 4.00 |
M8 | 8.5 | 10.500 | 300 | 4.00 |
M9 | 9.0 | 11.000 | 300 | 4.00 |
M10 | 5.0 | 7.250 | 350 | 4.50 |
M11 | 5.5 | 7.750 | 350 | 4.50 |
M12 | 6.0 | 8.500 | 350 | 4.50 |
M13 | 6.5 | 8.750 | 350 | 4.50 |
M14 | 7.0 | 9.250 | 350 | 4.50 |
M15 | 7.5 | 9.750 | 350 | 4.50 |
M16 | 8.0 | 10.250 | 350 | 4.50 |
M17 | 8.5 | 10.75 | 350 | 4.50 |
M18 | 9.0 | 11.250 | 350 | 4.50 |
M19 | 5.0 | 7.500 | 400 | 5.00 |
M20 | 5.5 | 8.000 | 400 | 5.00 |
M21 | 6.0 | 8.500 | 400 | 5.00 |
M22 | 6.5 | 9.000 | 400 | 5.00 |
M23 | 7.0 | 9.500 | 400 | 5.00 |
M24 | 7.5 | 10.000 | 400 | 5.00 |
M25 | 8.0 | 10.500 | 400 | 5.00 |
M26 | 8.5 | 11.000 | 400 | 5.00 |
M27 | 9.0 | 11.500 | 400 | 5.00 |
M28 | 5.0 | 7.875 | 450 | 5.75 |
M29 | 5.5 | 8.375 | 450 | 5.75 |
M30 | 6.0 | 8.875 | 450 | 5.75 |
M31 | 6.5 | 9.375 | 450 | 5.75 |
M32 | 7.0 | 9.875 | 450 | 5.75 |
M33 | 7.5 | 10.375 | 450 | 5.75 |
M34 | 8.0 | 10.875 | 450 | 5.75 |
M35 | 8.5 | 11.375 | 450 | 5.75 |
M36 | 9.0 | 11.875 | 450 | 5.75 |
M37 | 5.0 | 8.250 | 500 | 6.50 |
M38 | 5.5 | 8.750 | 500 | 6.50 |
M39 | 6.0 | 9.250 | 500 | 6.50 |
M40 | 6.5 | 9.750 | 500 | 6.50 |
M41 | 7.0 | 10.250 | 500 | 6.50 |
M42 | 7.5 | 10.750 | 500 | 6.50 |
M43 | 8.0 | 11.250 | 500 | 6.50 |
M44 | 8.5 | 11.750 | 500 | 6.50 |
M45 | 9.0 | 12.250 | 500 | 6.50 |
M46 | 5.0 | 8.500 | 550 | 7.00 |
M47 | 5.5 | 9.000 | 550 | 7.00 |
M48 | 6.0 | 9.500 | 550 | 7.00 |
M49 | 6.5 | 10.000 | 550 | 7.00 |
M50 | 7.0 | 10.500 | 550 | 7.00 |
M51 | 7.5 | 11.000 | 550 | 7.00 |
M52 | 8.0 | 11.500 | 550 | 7.00 |
M53 | 8.5 | 12.000 | 550 | 7.00 |
M54 | 9.0 | 12.500 | 550 | 7.00 |
M55 | 5.0 | 8.875 | 600 | 7.75 |
M56 | 5.5 | 9.375 | 600 | 7.75 |
M57 | 6.0 | 9.875 | 600 | 7.75 |
M58 | 6.5 | 10.375 | 600 | 7.75 |
M59 | 7.0 | 10.875 | 600 | 7.75 |
M60 | 7.5 | 11.375 | 600 | 7.75 |
M61 | 8.0 | 11.875 | 600 | 7.75 |
M62 | 8.5 | 12.375 | 600 | 7.75 |
M63 | 9.0 | 12.875 | 600 | 7.75 |
Similarly, six runs were carried out for discretization time of 20, 10, and 1 year, mesh size of 0.20 and 0.10 km, and total simulation time of 0.40 million years representing the entire eruption history of the main volcanic events (Dobson and Mahood 1985; Verma and Andaverde 1996). These runs were obtained for 5 km depth of the top of magma chamber, 600 km^{3} of chamber volume, and three magma recharge events at 0.34 Ma (20 km^{3} of magma), 0.22 Ma (12 km^{3}), and 0.026 Ma (5 km^{3}). The magma chamber depth and volume for these runs correspond to the model M55 (Figure 3).
Results and discussion
Evaluation of sensitivity of chamber depth and volume
Cubic best-fit equations for the simulated thermal gradient in the geothermal reservoir as a function of the depth of the centroid of the magma chamber
Equation # | Depth of the top of magma chamber d (km) | Centroid of the magma chamber dc (km) [Model #] | Volume of the magma chamber V(km^{3}) | R ^{2} | Equation^{a} |
---|---|---|---|---|---|
1 | 5.0 to 9.0 | 7.0 to 11.0 [M1 to M9] | 300 | 0.999000 | δT = (1668 ± 96) - (498.9 ± 32.6) dc + (49.76 ± 3.66) × dc^{2} - (1.653 ± 0.135) dc^{3} |
2 | 5.0 to 9.0 | 7.25 to 11.25 [M10 to M18] | 350 | 0.998993 | δT = (1798 ± 105) - (524.9 ± 34.6) dc + (51.08 ± 3.77) dc^{2} - (1.656 ± 0.136) dc^{3} |
3 | 5.0 to 9.0 | 7.5 to 11.5 [M19 to M27] | 400 | 0.998991 | δT = (1933 ± 114) - (551.1 ± 36.5) dc + (52.35 ± 3.88) dc^{2} - (1.657 ± 0.136) dc^{3} |
4 | 5.0 to 9.0 | 7.875 to 11.875 [M28 to M36] | 450 | 0.998990 | δT = (2148 ± 128) - (591.1 ± 39.5) dc + (54.22 ± 4.04) dc^{2} - (1.657 ± 0.136) dc^{3} |
5 | 5.0 to 9.0 | 8.25 to 12.75 [M37 to M45] | 500 | 0.998990 | δT = (2377 ± 143) - (632.5 ± 42.6) dc + (56.09 ± 4.19) dc^{2} - (1.657 ± 0.136) dc^{3} |
6 | 5.0 to 9.0 | 8.5 to 12.5 [M46 to M54] | 550 | 0.998990 | δT = (2539 ± 154) - (661 ± 45) dc + (57.33 ± 4.29) dc^{2} - (1.657 ± 0.136) dc^{3} |
7 | 5.0 to 9.0 | 8.875 to 12.875 [M55 to M63] | 600 | 0.998990 | δT = (2795 ± 172) - (705 ± 48) dc + (59.2 ± 4.4) dc^{2} - (1.657 ± 0.136) dc^{3} |
The values of the coefficients and the respective errors of the first term (without dc) and the three other terms (dc, dc^{2}, and dc^{3}) in Equation 1 are included. The statistically significant fit, quantitatively expressed in the R^{2} parameter, is also indicated by relatively low errors of the coefficients in Equation 1. The values of the coefficients (-498.9, 49.76, and -1.653, respectively, for cd, dc^{2}, and dc^{3}; respective errors of 32.6, 3.66, and 0.135, equivalent to about 6.5%, 7.3%, and 8.2%, respectively) indicate the sensitivity of the dc variable. Similarly statistically valid results were obtained for the other equations (Equations 2 to 7; see R^{2} values of 0.998990 to 0.998993). Note that had we reported rounded R^{2} values to less number of decimal places, most of them will be indistinguishable from each other and from the maximum value of 1. The differences among the errors of the coefficients in Equations 1 to 7 could not then be explained from small differences in the R^{2} quality parameter (Table 3).
Cubic best-fit equations for the simulated thermal gradient in the geothermal reservoir as a function of the volume of the magma chamber
Equation # | Depth of the top of magma chamber (d, km) | Centroid of the magma chamber (dc, km) | Volume of the magma chamber ( V, km^{3}) [Model #] | R ^{2} | Equation^{a} |
---|---|---|---|---|---|
8 | 5.0 | 7.000 to 8.875 | 300 to 600 [M1, 10, 19, 28, 37, 46, 55] | 0.991220 | δT = (45.712 ± 0.129) + (5.80 × 10^{- 3} ± 0.90x 10^{- 3}) V - (11.67 × 10^{- 3} ± 2.05 × 10^{- 3}) V^{2} + (7.69 × 10^{- 3} ± 1.52 × 10^{- 3}) V^{3} |
9 | 5.5 | 7.500 to 9.375 | 300 to 600 [M2, 11, 20, 29, 38, 47, 56] | 0.988839 | δT = (26.113 ± 0.044) + (1.87 × 10^{- 3} ± 3.12 × 10^{- 4}) V - (37.6 × 10^{- 4} ± 7.1 × 10^{- 4}) V^{2} + (24.9 × 10^{- 4} ± 5.2 × 10^{- 4}) V^{3} |
10 | 6.0 | 8.000 to 9.875 | 300 to 600 [M3, 12, 21, 30, 39, 48, 57] | 0.986291 | δT = (13.7530 ± 0.0139) + (55.2 × 10^{- 5} ± 9.8 × 10^{- 5}) V - (11.14 × 10^{- 4} ± 2.23 × 10^{- 4}) V^{2} + (7.42 × 10^{- 4} ± 1.65 × 10^{- 4}) V^{3} |
11 | 6.5 | 8.500 to 10.375 | 300 to 600 [M4, 13, 22, 31, 40, 49, 58] | 0.983621 | δT = (6.69010 ± 4.03 × 10^{- 3}) + (15.04 × 10^{- 5} ± 2.83 × 10^{- 5}) V - (30.5 × 10^{- 5} ± 6.4 × 10^{- 5}) V^{2} + (20.4 × 10^{- 5} ± 4.8 × 10^{- 5}) V^{3} |
12 | 7.0 | 9.000 to 10.875 | 300 to 600 [M5, 14, 23, 32, 41, 50, 59] | 0.980872 | δT = (3.00922 ± 1.07 × 10^{- 3}) + (37.8 × 10^{- 6} ± 7.5 × 10^{- 6}) V - (7.70 × 10^{- 5} ± 1.70 × 10^{- 5}) V^{2} + (5.15 × 10^{- 5} ± 1.26 × 10^{- 5}) V^{3} |
13 | 7.5 | 9.500 to 11.375 | 300 to 600 [M6, 15, 24, 33, 42, 51, 60] | 0.978078 | δT = (1.252520 ± 2.59x 10^{- 4}) + (8.80 × 10^{- 6} ± 1.82 × 10^{- 6}) V - (17.96 × 10^{- 6} ± 4.13 × 10^{- 6}) V^{2} + (12.06 × 10^{- 6} ± 3.06 × 10^{- 6}) V^{3} |
14 | 8.0 | 10.000 to 11.875 | 300 to 600 [M7, 16, 25, 34, 43, 52, 61] | 0.975271 | δT = (0.172276 ± 1.19 × 10^{- 5}) + (37.8 × 10^{- 8} ± 8.4 × 10^{- 8}) V - (7.75 × 10^{- 7} ± 1.91 × 10^{- 7}) V^{2} + (5.22 × 10^{- 7} ± 1.41 × 10^{- 7}) V^{3} |
15 | 8.5 | 10.500 to 12.375 | 300 to 600 [M8, 17, 26, 35, 44, 53, 62] | 0.972478 | δT = (0.172276 ± 1.19 × 10^{- 5}) + (37.8 × 10^{- 8} ± 8.4 × 10^{- 8}) V - (7.75 × 10^{- 7} ± 1.91 × 10^{- 7}) V^{2} + (5.22 × 10^{- 7} ± 1.41 × 10^{- 7}) V^{3} |
16 | 9.0 | 11.000 to 12.875 | 300 to 600 [M9, 18, 27, 36, 45, 54, 63] | 0.969720 | δT = (0.05697113 ± 2.28 × 10^{. - 6}) + (69.8 × 10^{- 9} ± 1.6 × 10^{- 8}) V - (14.37 × 10^{- 8} ± 3.64 × 10^{- 8}) V^{2} + (9.68 × 10^{- 8} ± 2.69 × 10^{- 8}) V^{3} |
The coefficients of the linear dc term in Equations 1 to 7 range from about -500 to -700, whereas those for the V term vary from about 7.0 × 10^{-3} to 5.8 × 10^{-3}. Similar relationship is valid for the quadratic and cubic terms (Tables 3 and 4). The relatively large values of the coefficients for the dc as compared to the V parameter imply that for thermal gradient, the magma chamber depth is much more sensitive than the chamber volume.
Cubic best-fit equations for thermal gradient in the geothermal reservoir as a function of either chamber depth or volume, or both, from all 63 simulated models
Equation # | Depth of the top of magma chamber (d, km) | Centroid of the magma chamber (dc, km) | Volume of the magma chamber ( V, km^{3}) | R ^{2} | Equation^{a} |
---|---|---|---|---|---|
17 | 5.0 to 9.0 | 7.000 to 12.875 | ----------- | 0.691799 | δT = (299 ± 289) - (50 ± 89) dc + (2.0 ± 9.1) dc^{2} - (0.009 ± 0.306) dc^{3} |
18 | ----------- | ----------- | 300 to 600 | 0.000000 | δT = (11 ± 239) + (1 ± 1680) V - (2 ± 3820) V^{2} + (1 ± 2820) V^{3} |
19 | 5.0 to 9.0 | 7.000 to 12.875 | 300 to 600 | 0.838748 | $\begin{array}{l}\mathit{\delta T}=\left(677\pm 238\right)-\left(169\pm 68\right)\phantom{\rule{0.5em}{0ex}}\mathrm{dc}+\left(13.9\pm 7.0\right)\phantom{\rule{0.5em}{0ex}}{\mathrm{dc}}^{2}-\left(0.392\pm 0.234\right)\phantom{\rule{0.5em}{0ex}}{\mathrm{dc}}^{3}\\ \phantom{\rule{2.74em}{0ex}}-\left(0.069\pm 0.69\right)\phantom{\rule{0.5em}{0ex}}V+\left(0.37\pm 1.58\right)\phantom{\rule{0.5em}{0ex}}{V}^{2}-\left(0.31\pm 1.17\right)\phantom{\rule{0.5em}{0ex}}{V}^{3}\end{array}$ |
Therefore, using all 63 simulations, we present our best-fit Equation 19 for the thermal gradient as a function of both parameters (dc and V). The R^{2} value of 0.838748 was obtained for this equation, which is not as high as for Equations 1 to 16. In Equation 19, the coefficients for the dc terms are consistently much higher than the respective coefficients for the V terms; for example, 169 for dc as compared to 0.069 for V, equivalent to a factor of about 2,400. The coefficients of quadratic and cubic terms are also higher for dc than for V (Table 5). The final combined equation, therefore, clearly confirms that the chamber depth is more sensitive than the volume. From the energy point of view, geoscientists (in particular geologists, volcanologists, and geophysicists) are encouraged to better constrain the chamber depth than its volume. This recommendation is valid not only for geothermal areas but also for active volcanoes. We also note that although we have evaluated the thermal regime in terms of the centroid depth, the inference will not change had we used the top of the chamber depth, instead of the centroid depth.
Preliminary three-dimensional thermal runs for a model of the Los Azufres geothermal field (LAGF)
The simulated 'present-day’ temperatures of about 160°C at the middle of the geothermal reservoir (for δt = 1 year and mesh size = 0.10 km) are generally consistent with the actually measured temperatures (generally 120°C to 250°C) in the LAGF (Verma and Andaverde 1996). The simulated runs for the LAGF can be improved in the future by incorporating all geological processes, such as magma evolution (fractional crystallization, assimilation, and magma mixing), convection in the magma chamber and geothermal reservoir, and heat generation from radioactive elements. Similarly, the smallest possible discretization time and mesh size will be used.
Conclusions
The first three-dimensional thermal simulation study carried out for the Los Azufres geothermal field (LAGF) provided 19 best-fit cubic equations from 63 simulations to understand the influence of the depth and volume of the underlying magma chamber. The coefficients of the centroid depth terms were much higher than those of the volume terms, implying that the centroid depth is much more sensitive than the chamber volume. The chamber depth should therefore be better constrained than the chamber volume, not only in geothermal areas but also in active volcanoes. Preliminary thermal modeling of the LAGF also shows that the present-day mean simulated temperatures in the geothermal reservoir are around 160°C.
Declarations
Acknowledgements
The second author (EGA) is grateful to Conacyt (Mexico) for granting him a post-doctoral fellowship. Computing facilities were those from the DGAPA-UNAM PAPIIT project IN104813. We are grateful to Alfredo Quiroz-Ruiz for help in computer maintenance. We also thank three anonymous reviewers of the journal for their high appreciation of our work and for indicating minor errors in our two earlier versions, which were corrected in the final paper.
Authors’ Affiliations
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