Estimation of aquifer dimensions from passive seismic signals with approximate wave propagation models

Abstract

Recently, it has been proposed that spontaneous seismic activity could be used in the estimation of hydrological parameters of aquifers such as permeability and storage. Approximate wave propagation models such as ray tracing, which are commonly used in hydrological parameter estimation with active sources and backscattering geometry, are not feasible with passive seismological imaging. With respect to full wave propagation models, the most accurate known model for aquifers is the poroelastic model while bedrock is usually modelled as an elastic medium. Using a poroelastic model in the forward model can be a computationally impractical choice. In this paper, we carry out a feasibility study in which we attempt to estimate the aquifer depth and water table using a highly approximate elastic model also for the aquifer. We adopt the Bayesian approximation error approach in which a statistical model is constructed for the errors that are induced by using model approximations such as sparse meshing and simplified physical models. We consider the problem in a simple two-dimensional geometry and show that straightforward adoption of approximate models leads to inconsistent parameter estimates, that is, the true parameters have essentially vanishing posterior density. On the other hand, using the Bayesian approximation error approach, the parameter estimates are consistent.

A discontinuous Galerkin method for poroelastic wave propagation: The two-dimensional case

“…In many applications to geophysics, one is interested in coupling elastic and poroelastic wave propagation; see [9,10]. In this section we outline the DG discretisation for two-dimensional elastic waves, again for a velocity/strain formulation.…”

“…In the high-frequency regime the viscodynamic effects are more complicated (see Section 4). In practice the existence of the slow P-wave puts a significant constraint on numerical schemes, making the task of solving related inverse problems very challenging, [9,10]. In these preliminary papers we addressed some restricted inverse problems and dealt mainly with the computational challenges associated with poroelastic inverse problems.…”

“…As noted above the most notable 70 conclusion of the elementary Biot theory is the existence of a secondary slow P-wave, where the solid and fluid wave amplitudes have opposite phase. This has the undesirable feature that it may set an inconvenient restriction on a numerical scheme, since the grid refinement is controlled by the shortest wavelength, see [9,10]. Biot also considered the situation of wave dissipation, which may occur when there is a relative motion between the solid and fluid, and identified a low-frequency (laminar) and high-frequency regime.…”

“…In any event, the slow P-wave can impose a significant restraint on the grid resolution or basis order, and hence the time step, in the poroelastic part. Usually this makes it impracticable to use a finely-resolved poroelastic model to invert 395 data, even in a synthetic setting [9,10], necessitating viscoelastic approximations [64].…”

“…In our previous studies [9,10], we have have shown in the content of full wave inversion that one cannot ignore the slow P-wave without introducing unacceptable error in the prediction (albeit for synthetic data), although its existence in real situations is debated, [43]. One of the problems is that with scattering in heterogeneous media, it is 410 very difficult to isolate the slow P-wave.…”