Mogi (1958): FEM:

1. Quantitative analysis of volcano deformation FEM domain parameters – A guide for modelers (V11B-2756) Jonathan Stone1 & Timothy Masterlark21jstone@crimson.ua.edu, Department of Geological Sciences, The University of Alabama, Tuscaloosa, AL 35487, USA 2timothy.masterlark@sdsmt.edu, Department of Geology and Geological Engineering, South Dakota School of Mines & Technology, Rapid City, SD 57701, USA Abstract Models – Analytical & Finite Element Mogi (1958): FEM: Finite Element Models (FEMs) can accurately simulate ground deformation due to migration of subsurface magma. FEMs are becoming an increasingly important tool for volcano modelers. However, FEMs are computationally expensive. Thus, it is highly beneficial for modelers to minimize computation costs, especially in iterative model processing or in models for which restrictions of number of elements becomes a limiting factor. To the authors’ knowledge, there has not been a systematic study of model domain parameters, such as vertical and lateral model extent and mesh fineness, that can serve as a guide for producing the most efficient FEMs. As a result, modelers often choose domain parameters arbitrarily, by trial-and-error, or based on earlier researchers’ models, which may also be inefficient. Therefore, we present a systematic study of model parameters, including how mesh fineness, geometry, and extent interact. We apply statistical analyses to our results in order to determine ranges of acceptable parameters. It is our hope that this study can serve as a valuable guide for other modelers in determining optimum volcanic model parameters. where ur is the displacement in the radial direction on the surface uz is the vertical displacement on the surface a is the radius of the spherical magma source ΔP is the change in magma source pressure Gis the shear modulus r is the radial distance on the surface from a point A f is the depth of the center of the spherical magma source (z = -f) Results Conclusions Goodness-of-fit tests (Pearson & Spearman correlations) likely produce irrelevant results, so RMSE was used to quantify similarity between the Mogi analytical model and FEMs. This value was also expressed in percentage of maximum analytical displacement. The maximum absolute difference in displacement between analytical and numerical models was also given in terms of percentage of maximum analytical displacement. This allows researchers to evaluate errors on the basis of their results and to determine acceptable limits with respect to the precision of their data acquisition method. Results of the FEMs were far more sensitive to limits on lateral domain extent than domain depth. Therefore, the boundary conditions of a 50-km3 domain, as described by Charco & del Sastre (2011) may produce an unsuitable amount of error for some applications. The FEM systematically underestimates the displacement due to the circular chamber being approximated by secant lines, resulting in a smaller effective radius. The underestimation can be reduced by increasing the number of nodes along the chamber. Alternately, it can be compensated by increasing the size of the chamber such that its effective radius is equal to the desired radius. If the latter method is chosen, note that the effective radius is a function of the number, size, and orientation of element faces that form the spherical chamber in three dimensions. In our models, with 16 elements on the magma chamber half-circle, there is little difference in the results between radius and effective radius. With 8 elements, the difference is still not great, but the model is already invalid. The triangular elements of the model result in a reduction of the chamber radius, which, in the axisymmetric case (assuming the formation of a regular n-gon), is given by: where rE is the effective radius, rC is the modeled radius and n is the number of sides of the n-gon. Analytical solutions for Mogi: z = 3000 m a = 500 m ΔP = 50 MPa G(shear modulus) = 2.4 × 1010 ν(Poisson’s ratio) = 0.25 E (Young’s modulus) = 6.0 × 1010 Circles show corresponding FEM solutions Domain depth: 50 km Domain radius: 50 km Chamber elements: 160 Domain depth: 10 km Domain radius: 50 km Chamber elements: 160 uz uz Displacement (m) ur ur For all the above models, chamber depth = 5000 m, chamber radius = 500 m, effective chamber radius = 499.9839 m, the number of elements on the half-circle representing the chamber wall = 160 (with the exception of the 5000-deep model, wherein the chamber is cut in half and, thus, contains 80 elements). Domain depth: 50 km Domain radius: 50 km Chamber elements: 16 Domain depth: 50 km Domain radius: 35 km Chamber elements: 160 References Charco, M. & del Sastre, P. G. (2011), Finite element numerical solution for modelling ground deformation in volcanic areas, In: Pardo, L., Balakrishnan, N., & Gil, M. A., (Eds.) Modern Mathematical Tools and Teqhniques in Capturing Complexity, Springer, Berlin, pp. 223-238. Masterlark, T. (2007), Magma intrusion and deformation predictions: Sensitivities to the Mogi assumptions. Journal of Geophysical Research 112, B06419, 1-17. doi:10.1029/2006JB004860. Masterlark, T. & Stone, J. (2011), Simulating volcano deformation with Abaqus FEM software, University of Iceland short course, Reykavik, Iceland. Masterlark, T., Feigl, K. L., Haney, M., Stone, J., Thurber, C., & Ronchin, E. (2012), Nonlinear estimation of geometric parameters in FEMs of volcano deformation: Integrating tomography models and geodetic data for Okmok volcano, Alaska, Journal of Geophysical Research, 117, B02407, doi:10.1029/2011JB008811. Mogi, K. (1958). Relations between the Eruptions of Various Volcanoes and the Deformations of the Ground surface around them. Bulletin of the Earthquake Research Institute 36, 99-134. uz uz Displacement (m) ur ur Distance from axis of symmetry (m) Distance from axis of symmetry (m) For all the above models, model domain depth = 50 km, model domain radius = 50 km, chamber depth = 5000 m, chamber radius = 500 m.