A quadratic element method for evaluating groundwater flow by the inversion of surface tilt with application to the Tono Area, Japan

Abstract

Summary By modifying a previous method with constant elements, we developed a quadratic element method for more accurately estimating groundwater flow by the inversion of tilt data. In this method: (1) a region of groundwater flow is divided into quadratic elements in which the change in groundwater volume per unit volume of rock (Δ v ) and the Skempton coefficient ( B ) vary in a quadratic manner with the coordinates, (2) the values of Δ v are set to zero at the boundaries of the region of groundwater flow and (3) the sum of the squared second derivatives of Δ v is adopted as a constraining condition that is weighted and added to the sum of the squared errors in tilt. First, analyses were performed for a flow model to determine the accuracy of this method for estimating groundwater flow and also to clarify the effect of the assumed size of a region of groundwater flow. These analyses showed that the quadratic element method proposed in this study gives a much better estimation of Δ v than the constant element method and that a large region of groundwater flow should be assumed, rather than a small region, since the values of Δ v at points outside of the actual region of groundwater flow are estimated to be nearly zero when a large region is assumed while these values are greatly overestimated when an excessively small region is assumed. Finally, the quadratic element method was applied to the site of the Mizunami Underground Research Laboratory in the Tono area, Japan. Inverse analyses were performed for tilt data measured by four tiltmeters with a resolution of 10 −9 radians during the excavation of two shafts under the assumption that the rock mass is an isotropic and homogeneous half- space. The results showed that the method proposed in this study reproduced the tilt data very accurately. Thus, the distribution of Δ v was estimated without sacrificing the reproducibility of the tilt data. The contour maps of B (1 + ν)Δ v ( ν : Poisson’s ratio) showed that the heterogeneous flow of groundwater occurred at the site and that groundwater volume decreased mainly in the area surrounded by two faults. The latter result is consistent with the finding obtained by previous investigations that these faults have low permeability in the direction perpendicular to the strike and may act as a flow barrier.