The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in geophysical electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid spectral-finite element approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings.
Motivated by the need for improved forward modeling and inversion capabilities of geophysical response in geologic settings whose fine--scale features demand accountability, this project describes two novel approaches which advance the current state of the art. First is a hierarchical material properties representation for finite element analysis whereby material properties can be perscribed on volumetric elements, in addition to their facets and edges. Hence, thin or fine--scaled features can be economically represented by small numbers of connected edges or facets, rather than 10's of millions of very small volumetric elements. Examples of this approach are drawn from oilfield and near--surface geophysics where, for example, electrostatic response of metallic infastructure or fracture swarms is easily calculable on a laptop computer with an estimated reduction in resource allocation by 4 orders of magnitude over traditional methods. Second is a first-ever solution method for the space--fractional Helmholtz equation in geophysical electromagnetics, accompanied by newly--found magnetotelluric evidence supporting a fractional calculus representation of multi-scale geomaterials. Whereas these two achievements are significant in themselves, a clear understanding the intermediate length scale where these two endmember viewpoints must converge remains unresolved and is a natural direction for future research. Additionally, an explicit mapping from a known multi-scale geomaterial model to its equivalent fractional calculus representation proved beyond the scope of the present research and, similarly, remains fertile ground for future exploration.
The feasibility of Neumann-series expansion of Maxwell's equations in the electrostatic limit is investigated for potentially rapid and approximate subsurface imaging of geologic features proximal to metallic infrastructure in an oilfield environment. Although generally useful for efficient modeling of mild conductivity perturbations in uncluttered settings, we have raised the question of its suitability for situations such as oilfields, in which metallic artifacts are pervasive and, in some cases, in direct electrical contact with the conductivity perturbation on which the Neumann series is computed. Convergence of the Neumann series and its residual error are computed using the hierarchical finite-element framework for a canonical oilfield model consisting of an L-shaped, steel-cased well, energized by a steady-state electrode, and penetrating a small set of mildly conducting fractures near the heel of the well. For a given node spacing h in the finite-element mesh, we find that the Neumann series is ultimately convergent if the conductivity is small enough - a result consistent with previous presumptions on the necessity of small conductivity perturbations. However, we also determine that the spectral radius of the Neumann series operator grows as approximately 1/h, thus suggesting that in the limit of the continuous problem h→0, the Neumann series is intrinsically divergent for all conductivity perturbations, regardless of their smallness. The hierarchical finite-element methodology itself is critically analyzed and shown to possess the h2 error convergence of traditional linear finite elements, thereby supporting the conclusion of an inescapably divergent Neumann series for this benchmark example. Application of the Neumann series to oilfield problems with metallic clutter should therefore be done with careful consideration to the coupling between infrastructure and geology. The methods used here are demonstrably useful in such circumstances.
The project objective was to detail better ways to assess and exploit intelligent oil and gas field information through improved modeling, sensor technology, and process control to increase ultimate recovery of domestic hydrocarbons. To meet this objective we investigated the use of permanent downhole sensors systems (Smart Wells) whose data is fed real-time into computational reservoir models that are integrated with optimized production control systems. The project utilized a three-pronged approach (1) a value of information analysis to address the economic advantages, (2) reservoir simulation modeling and control optimization to prove the capability, and (3) evaluation of new generation sensor packaging to survive the borehole environment for long periods of time. The Value of Information (VOI) decision tree method was developed and used to assess the economic advantage of using the proposed technology; the VOI demonstrated the increased subsurface resolution through additional sensor data. Our findings show that the VOI studies are a practical means of ascertaining the value associated with a technology, in this case application of sensors to production. The procedure acknowledges the uncertainty in predictions but nevertheless assigns monetary value to the predictions. The best aspect of the procedure is that it builds consensus within interdisciplinary teams The reservoir simulation and modeling aspect of the project was developed to show the capability of exploiting sensor information both for reservoir characterization and to optimize control of the production system. Our findings indicate history matching is improved as more information is added to the objective function, clearly indicating that sensor information can help in reducing the uncertainty associated with reservoir characterization. Additional findings and approaches used are described in detail within the report. The next generation sensors aspect of the project evaluated sensors and packaging survivability issues. Our findings indicate that packaging represents the most significant technical challenge associated with application of sensors in the downhole environment for long periods (5+ years) of time. These issues are described in detail within the report. The impact of successful reservoir monitoring programs and coincident improved reservoir management is measured by the production of additional oil and gas volumes from existing reservoirs, revitalization of nearly depleted reservoirs, possible re-establishment of already abandoned reservoirs, and improved economics for all cases. Smart Well monitoring provides the means to understand how a reservoir process is developing and to provide active reservoir management. At the same time it also provides data for developing high-fidelity simulation models. This work has been a joint effort with Sandia National Laboratories and UT-Austin's Bureau of Economic Geology, Department of Petroleum and Geosystems Engineering, and the Institute of Computational and Engineering Mathematics.
Electromagnetic induction is a classic geophysical exploration method designed for subsurface characterization--in particular, sensing the presence of geologic heterogeneities and fluids such as groundwater and hydrocarbons. Several approaches to the computational problems associated with predicting and interpreting electromagnetic phenomena in and around the earth are addressed herein. Publications resulting from the project include [31]. To obtain accurate and physically meaningful numerical simulations of natural phenomena, computational algorithms should operate in discrete settings that reflect the structure of governing mathematical models. In section 2, the extension of algebraic multigrid methods for the time domain eddy current equations to the frequency domain problem is discussed. Software was developed and is available in Trilinos ML package. In section 3 we consider finite element approximations of De Rham's complex. We describe how to develop a family of finite element spaces that forms an exact sequence on hexahedral grids. The ensuing family of non-affine finite elements is called a van Welij complex, after the work [37] of van Welij who first proposed a general method for developing tangentially and normally continuous vector fields on hexahedral elements. The use of this complex is illustrated for the eddy current equations and a conservation law problem. Software was developed and is available in the Ptenos finite element package. The more popular methods of geophysical inversion seek solutions to an unconstrained optimization problem by imposing stabilizing constraints in the form of smoothing operators on some enormous set of model parameters (i.e. ''over-parametrize and regularize''). In contrast we investigate an alternative approach whereby sharp jumps in material properties are preserved in the solution by choosing as model parameters a modest set of variables which describe an interface between adjacent regions in physical space. While still over-parametrized, this choice of model space contains far fewer parameters than before, thus easing the computational burden, in some cases, of the optimization problem. And most importantly, the associated finite element discretization is aligned with the abrupt changes in material properties associated with lithologic boundaries as well as the interface between buried cultural artifacts and the surrounding Earth. In section 4, algorithms and tools are described that associate a smooth interface surface to a given triangulation. In particular, the tools support surface refinement and coarsening. Section 5 describes some preliminary results on the application of interface identification methods to some model problems in geophysical inversion. Due to time constraints, the results described here use the GNU Triangulated Surface Library for the manipulation of surface meshes and the TetGen software library for the generation of tetrahedral meshes.