Imaging of a fluid injection process using geophysical data - A didactic example

In many subsurface industrial applications, fluids are injected into or withdrawn from a geologic formation. It is of practical interest to quantify precisely where, when, and by how much the injected fluid alters the state of the subsurface. Routine geophysical monitoring of such processes attempts to image the way that geophysical properties, such as seismic velocities or electrical conductivity, change through time and space and to then make qualitative inferences as to where the injected fluid has migrated. The more rigorous formulation of the time-lapse geophysical inverse problem forecasts how the subsurface evolves during the course of a fluid-injection application. Using time-lapse geophysical signals as the data to be matched, the model unknowns to be estimated are the multiphysics forward-modeling parameters controlling the fluid-injection process. Properly reproducing the geophysical signature of the flow process, subsequent simulations can predict the fluid migration and alteration in the subsurface. The dynamic nature of fluid-injection processes renders imaging problems more complex than conventional geophysical imaging for static targets. This work intents to clarify the related hydrogeophysical parameter estimation concepts

Computation of 3D Frequency-Domain Waveform Kernals for c(x,y,z) Media

Seismic tomography, as typically practiced on both the exploration, crustal, and global scales, considers only the arrival times of selected sets of phases and relies primarily on WKBJ theory during inversion. Since the mid 1980’s, researchers have explored, largely on a theoretical level, the possibility of inverting the entire seismic record. Due to the ongoing advances in CPU performance, full waveform inversion is finally becoming feasible on select problems with promising results emerging from frequency-domain methods. However, frequency-domain techniques using sparse direct solvers are currently constrained by memory limitations in 3D where they exhibit a O(n4) worst-case bound on memory usage. We sidestep this limitation by using a hybrid approach, calculating frequency domain Green’s functions for the scalar wave equation by driving a high-order, time-domain, finite-difference (FDTD) code to steady state using a periodic source. The frequency-domain response is extracted using the phase sensitive detection (PSD) method recently developed by Nihei and Li (2006). The resulting algorithm has an O(n3) memory footprint and is amenable to parallelization in the space, shot, or frequency domains. We demonstrate this approach by generating waveform inversion kernels for fully c(x,y,z) models. Our test examples include a realistic VSP experiment using the geometry and velocity models obtained from a site in Western Wyoming, and a deep crustal reflection/refraction profile based on the LARSE II geometry and the SCEC community velocity model. We believe that our 3D solutions to the scalar Helmholtz equation, for models with upwards of 100 million degrees of freedom, are the largest examples documented in the open geophysical literature. Such results suggest that iterative 3D waveform inversion is an achievable goal in the near future.Shell GameChangerMassachusetts Institute of Technology. Earth Resources Laborator

A new phase space method for recovering index of refraction from travel times

We develop a new phase space method for reconstructing the index of refraction of a medium from travel time measurements. The method is based on the so-called Stefanov–Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The new algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for the index of refraction allows us to compute the solution in physical space. Numerical examples including isotropic metrics and the Marmousi synthetic model are shown to validate the new method

Bayesian Variational Time-lapse Full-waveform Inversion

Time-lapse seismic full-waveform inversion (FWI) provides estimates of dynamic changes in the subsurface by performing multiple seismic surveys at different times. Since FWI problems are highly non-linear and non-unique, it is important to quantify uncertainties in such estimates to allow robust decision making. Markov chain Monte Carlo (McMC) methods have been used for this purpose, but due to their high computational cost, those studies often require an accurate baseline model and estimates of the locations of potential velocity changes, and neglect uncertainty in the baseline velocity model. Such detailed and accurate prior information is not always available in practice. In this study we use an efficient optimization method called stochastic Stein variational gradient descent (sSVGD) to solve time-lapse FWI problems without assuming such prior knowledge, and to estimate uncertainty both in the baseline velocity model and the velocity change. We test two Bayesian strategies: separate Bayesian inversions for each seismic survey, and a single join inversion for baseline and repeat surveys, and compare the methods with the standard linearised double difference inversion. The results demonstrate that all three methods can produce accurate velocity change estimates in the case of having fixed (exactly repeatable) acquisition geometries, but that the two Bayesian methods generate more accurate results when the acquisition geometry changes between surveys. Furthermore the joint inversion provides the most accurate velocity change and uncertainty estimates in all cases. We therefore conclude that Bayesian time-lapse inversion, especially adopting a joint inversion strategy, may be useful to image and monitor the subsurface changes, in particular where uncertainty in the results might lead to significantly different decisions

Binary recovery via phase field regularization for first-arrival traveltime tomography

We propose a double obstacle phase field methodology for binary recovery of the slowness function of an Eikonal equation found in first-arrival traveltime tomography. We treat the inverse problem as an optimization problem with quadratic misfit functional added to a phase field relaxation of the perimeter penalization functional. Our approach yields solutions as we account for well posedness of the forward problem by choosing regular priors. We obtain a convergent finite difference and mixed finite element based discretization and a well defined descent scheme by accounting for the non-differentiability of the forward problem. We validate the phase field technique with a Γ—convergence result and numerically by conducting parameter studies for the scheme, and by applying it to a variety of test problems with different geometries, boundary conditions, and source—receiver locations