Despite bold steps taken worldwide for the replacement or the reduction of the world’s dependence on fossil fuels, economic and societal realities suggest that a transition to alternative energy forms will be, at best, gradual. It also appears that exploration for new reserves is becoming increasingly more difficult both from a technical and an economic point of view, despite the advent of new technologies. These trends place renewed emphasis on maximizing oil recovery from known fields. In this sense, low-cost and reliable enhanced oil recovery (EOR) methods have a strong role to play. The goal of this dissertation is to explore, using computational simulations, the feasibility of the, so-called, seismic or elastic-wave EOR method, and to provide the mathematical/computational framework under which the method can be systematically assessed, and its feasibility evaluated, on a reservoir-specific basis. A central question is whether elastic waves can generate sufficient motion to increase oil mobility in previously bypassed reservoir zones, and thus lead to increased production rates, and to the recovery of otherwise unexploited oil. To address the many questions surrounding the feasibility of the elastic-wave EOR method, we formulate an inverse source problem, whereby we seek to determine the excitations (wave sources) one needs to prescribe in order to induce an a priori selected maximization mobility outcome to a previously well-characterized reservoir. In the industry’s parlance, we attempt to address questions of the form: how does one shake a reservoir?, or what is the “resonance” frequency of a reservoir?. We discuss first the case of wellbore wave sources, but conclude that surface sources have a better chance of focusing energy to a given reservoir. We, then, discuss a partial-differential-equation-constrained optimization approach for resolving the inverse source problem associated with surface sources, and present a numerical algorithm that robustly provides the necessary excitations that maximize a mobility metric in the reservoir. To this end, we form a Lagrangian encompassing the maximization goal and the underlying physics of the problem, expressed through the side imposition of the governing partial differential equations. We seek to satisfy the first-order optimality conditions, whose vanishing gives rise to a systematic process that, in turn, leads to the prescription of the wave source signals. We explore different (indirect) mobility metrics (kinetic energy or acceleration field maximization), and report numerical experiments under three different settings: (a) targeted formations within one-dimensional multi-layered elastic solids system of semi-infinite extent; (b) targeted formations embedded in a two-dimensional semi-infinite heterogeneous elastic solid medium; and (c) targeted poroelastic formations embedded within elastic heterogeneous surroundings in one dimension. The numerical experiments, employing hypothetical subsurface formation models subjected to, initially unknown, ground surface wave sources, demonstrate that the numerical optimizer leads robustly to optimal loading signals and the illumination of the target formations. Thus, we demonstrate that the theoretical framework for the elastic wave EOR method developed in this dissertation can systematically address the application of the method on a reservoir-specific basis. From an application point of view and based on the numerical experiments reported herein, for shallow reservoirs there is strong promise for increased production. The case of deeper reservoirs can only be addressed with further research that builds on the findings of this work, as we report in the last chapter. / text

The inverse source problem for the Helmholtz equation is studied. An unknown source is to be identified from the knowledge of its radiated wave. The focus is placed on the effect that multi-frequency data has on establishing uniqueness. In particular, we prove that data obtained from finitely many frequencies is not sufficient. On the other hand, if the frequency varies within an open interval of the positive real line, then the source is determined uniquely. An algorithm is based on an incomplete Fourier transform of the measured data and we establish an error estimate under certain regularity assumptions on the source function. We conclude that multi-frequency data not only leads to uniqueness for the inverse source problem, but in fact it contributes with a stability result for the reconstruction of an unknown source.

This paper studies the inverse source problem for the Helmholtz equation with a point source in a two dimensional domain. Given complete boundary data and appropriate discretization Tikhonov regularization is established to be an effective method at finding the point source. Furthermore, it was found that Tikhonov regularization can locate point sources even given significant noise, as well as incomplete boundary data in complicated domains.

In recent years, revolutionary "hybrid" or "multi-physics" methods of medical imaging have emerged. By combining two or three different types of waves these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information. Thermoacoustic (and photoacoustic) tomography is the most developed multi-physics imaging modality. Thermo- and photo-acoustic tomography require reconstructing initial acoustic pressure in a body from time series of pressure measured on a surface surrounding the body. For the classical case of free space wave propagation, various reconstruction techniques are well known. However, some novel measurement schemes place the object of interest between reflecting walls that form a de facto resonant cavity. In this case, known methods cannot be used. In chapter 2 we present a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity with a constant speed of sound. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations. In chapter 3 we consider the more general problem of an arbitrarily shaped resonant cavity with a non constant speed of sound and present the gradual time reversal method for computing solutions to the inverse source problem. It consists in solving back in time on the interval [0, T] the initial/boundary value problem for the wave equation, with the Dirichlet boundary data multiplied by a smooth cutoff function. If T is sufficiently large one obtains a good approximation to the initial pressure; in the limit of large T such an approximation converges (under certain conditions) to the exact solution.

Cavitation is a phenomenon present during many ultrasound therapies, including the thermal ablation of malignant tissue using high intensity focused ultrasound (HIFU). Inertial cavitation, in particular, has been previously shown to result in increased heat deposition and to be associated with broadband noise emissions that can be readily monitored using a passive receiver without interference from the main ultrasound signal. The present work demonstrates how an array of passive receivers can be used to generate maps of cavitation distribution during HIFU exposure, uncovering a new potential method of monitoring HIFU treatment. Using a commercially available ultrasound system (z.one, Zonare, USA), pulse transmission can be switched off and data from 64 elements of an array can be simultaneously acquired to generate passive maps of acoustic source power. For the present work, a 38 mm aperture 5-10 MHz linear array was used, with the 64 elements chosen to span the entire aperture. Theory and simulations were used to show the spatial resolution of the system, the latter showing that the broadband nature of inertial cavitation makes passive maps robust to interference between cavitating bubbles. Passive source mapping was first applied to wire scatterers, demonstrating the ability of the system to resolve broadband sources. With the array transversely placed to the HIFU axis, high-resolution passive maps are generated, and emissions from several cavitating bubbles are resolved. The sensitivity of passive mapping during HIFU exposure is compared with that of an active cavitation detector following exposure. The array was then placed within a rectangular opening in the centre of the HIFU transducer, providing a geometric setup that could be used clinically to monitor HIFU treatment. Cavitation was instigated in continuous and disjoint regions in agar tissue mimicking gel, with the expected regions of cavitation validating the passive maps obtained. Finally, passive maps were generated for samples of ox liver exposed to HIFU. The onset of inertial cavitation as detected by the passive mapping approach was found to provide a much more robust indicator of lesioning than post-exposure B-mode hyperecho, which is in current clinical use. Passive maps based on the broadband component of the received signal were able to localize the lesions both transversely and axially, however cavitation is generally indicated 5 mm prefocal to the lesions. Further work is needed to establish the source of this discrepancy. It is believed that with use of an appropriately designed cavitation detection array, passive mapping will represent a major advance in ultrasound-guided HIFU therapy. Not only can it be utilized in real-time during HIFU exposure, without the need to turn the therapeutic ultrasound field off, but it has also been shown in the context of the present work to provide a strong indicator of successful lesioning and high signal-to-noise compared to conventional B-mode ultrasound techniques.

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Cette thèse porte sur l’étude d’un problème inverse de sources pour un modèle bidimensionnel d’intrusion saline. Dans un premier temps, nous nous intéressons à la modélisation du phénomène d’intrusion saline dans un aquifère côtier non confiné. En tenant compte des hypothèses particulières, nous obtenons dans le cas stationnaire une équation elliptique de la charge hydraulique dont le second membre est constitué des sources ponctuelles. L’étude du problème direct consiste à analyser le modèle dérivé et à établir un résultat d’existence et d’unicité d’une solution. Ensuite, dans la partie problème inverse, il s’agit de l’identification de termes sources à partir des mesures locales. Nous traitons les trois questions relatives à ce problème inverse, l’identifiabilité, l’identification et la stabilité. Concernant l’identification, nous formulons le problème inverse comme un problème de contrôle avec une fonctionnelle coût qui calcule l’écart quadratique entre les mesures expérimentales et celles obtenues par la résolution du problème direct. L’optimisation de cette fonction nécessite le calcul de son gradient que nous obtenons par la méthode de sensibilités et par la méthode de l’état adjoint. Quant à la stabilité, nous établissons deux types d’estimations, logarithmiques et lipschitziennes, pour les positions et les intensités de sources dans le cas de l’équation elliptique obtenue et en considérant toujours des mesures intérieures. De plus, nous avons généralisé les résultats des estimations lipschitziennes pour l’équation elliptique de la forme –Δu+k2u=F. La dernière partie de la thèse est destinée à montrer les résultats de l’identification numérique en fonction des paramètres intervenant dans le modèle principal. / This thesis deals with the study of an inverse source problem for a two dimensional seawater intrusion model. First, we focus on the modeling of the seawater intrusion phenomenon in a costal unconfined aquifer. Then considering some specific assumptions, we obtain, in the steady state, an elliptic equation of the hydraulic head with a left hand side formed by point wise sources. The study of the direct problem aims to analyze the derived model and to establish a result of existence and uniqueness of solution. The inverse problem concerns the identification of sources from local measurements. We are interested in the study of uniqueness, identification and stability.Concerning the identification, we transform the inverse problem to a control problem with a cost functional computing the quadratic error between the experimental measures and those obtained by solving the direct problem. To optimize this function, we need to compute its gradient and this can be done by the sensibility and the adjoint methods. Moreover, regarding the stability, we establish two types of estimates, logarithmic and lipschitz, for sources positions and intensities in the case of the elliptic equation assuming interior observations. Furthermore, we have generalized the results of Lipschitz estimates for the elliptic equation –Δu+k2u=F. The last part of the thesis is intended to show the results of the numerical identification based on parameters involved in the main model.

Cette thèse porte sur l’étude de quelques questions liées à l’identifiabilité et l’identification d’un problème inverse non-linéaire de source. Il s’agit de l’identification d’une source ponctuelle dépendante du temps constituant le second membre d’une équation de type advection-dispersion-réaction à coefficients variables. Dans le cas monodimensionnel, la souplesse du modèle stationnaire nous a permis de développer des réponses théoriques concernant le nombre des capteurs nécessaires et leurs emplacements permettant d’identifier la source recherchée d’une façon unique. Ces résultats nous ont beaucoup aidés à définir la ligne de conduite à suivre afin d’apporter des réponses similaires pour le modèle transitoire. Quant au modèle bidimensionnel transitoire, en utilisant quelques résultats de nulle contrôlabilité frontière et des mesures de l’état sur la frontière sortie et de son flux sur la frontière entrée du domaine étudié, nous avons établi un théorème d’identifiabilité et une méthode d’identification permettant de localiser les deux coordonnées de la position de la source recherchée comme étant l’unique solution d’un système non-linéaire de deux équations, et de transformer l’identification de sa fonction de débit en la résolution d’un problème de déconvolution. La dernière partie de cette thèse discute la difficulté principale rencontrée dans ce genre de problèmes inverses à savoir la non identifiabilité d’une source dans sa forme abstraite, propose une alternative permettant de surmonter cette difficulté dans le cas particulier où le but est d’identifier le temps limite à partir duquel la source impliquée a cessé d’émettre, et donc ouvre la porte sur de nouveaux horizons. / The thesis deals with the two main issues identifiability and identification related to a nonlinear inverse source problem. This problem consists in the identification of a time-dependent point source occurring in the right hand-side of an advection-dispersion-reaction equation with spatially varying coefficients. Starting from the stationnary case in the one-dimensional model, we derived theoritical results defining the necessary number of sensors and their positions that enable to uniquely determine the sought source. Those results gave us a good visibility on how to proceed in order to obtain similar results for the time-dependent (evolution) case. As far as the two-dimensional evolution model is concerned, using some boundary null controllability results and the records of the generated state on the inflow boundary and its flux on the outflow boundary of the monitored domain, we established a constructive identifiability theorem as well as an identification method that localizes the two coordinates of the sought source position as the unique solution of a nonlinear system of two equations and transforms the identification of its time-dependent intensity function into solving a deconvolution problem. The last part of this thesis highlights the main difficulty encountred in such inverse problems namely the nonidentifiabilityof a source in its abstract form, proposes a method that enables to overcome this difficulty in the particular case where the aim is to identify the time active limit of the involved source. And thus, this last part opens doors on new horizons and prospects.