Results

**1 - 6**of**6**### Noname manuscript No. (will be inserted by the editor) Simulations of 3D DC Borehole Resistivity Measurements with a Goal-Oriented hp Finite-Element Method. Part II: Through-Casing Resistivity Instruments

"... The date of receipt and acceptance will be inserted by the editor Abstract We simulate measurements acquired in steel-cased deviated wells at zero frequency (DC) for the as-sessment of rock formation properties. The assumed data acquisition configuration considers one transmitter and three receiver ..."

Abstract
- Add to MetaCart

The date of receipt and acceptance will be inserted by the editor Abstract We simulate measurements acquired in steel-cased deviated wells at zero frequency (DC) for the as-sessment of rock formation properties. The assumed data acquisition configuration considers one transmitter and three receiver electrodes that are utilized to measure the second (vertical) difference of the electric potential. We assume a homogeneous 1.27 cm thick steel-casing with resistivity equal to 10−5Ω · m. Simulations are performed with two different numer-ical methodologies. The first one is based on transferring two-dimensional (2D) axi-symmetric optimal grids to a three-dimensional (3D) simulation software. The second one produces automatically optimal 3D grids yielded by a 3D self-adaptive goal-oriented algorithm. Both method-

### -o or

, 2007

"... ing hydrocarbon reservoirs worldwide. Despite the success of well-logging measurements, the planning and drilling of a single well may cost several millions of dollars, and the interpretation of subsurface material properties, enabling real-time modifications on the well trajectory in the case of lo ..."

Abstract
- Add to MetaCart

ing hydrocarbon reservoirs worldwide. Despite the success of well-logging measurements, the planning and drilling of a single well may cost several millions of dollars, and the interpretation of subsurface material properties, enabling real-time modifications on the well trajectory in the case of logging-while-drilling (LWD) instruments. Despite the fact that 1D results are typically ‘‘corrected ” (modified) using semi-analytical formulas to account for modeling of 2D and 3D geometries, their accuracy is compromised in

### s p

"... F epe e sing D zek isotropy behind casing, but otherwise highly sensitive to fre-h c ..."

Abstract
- Add to MetaCart

F epe e sing D zek isotropy behind casing, but otherwise highly sensitive to fre-h c

### Efficient Sequential and Parallel Solvers for hp Finite Element Method

"... Abstract We present a sequential and parallel direct solver designed for hp Finite Element Method (FEM) applied to solve numerous problems, including non-stationary heat transfer problem, the Stokes problem, and the resistivity logging measurement simulations. The hp FEM incorporates a self-adaptive ..."

Abstract
- Add to MetaCart

(Show Context)
Abstract We present a sequential and parallel direct solver designed for hp Finite Element Method (FEM) applied to solve numerous problems, including non-stationary heat transfer problem, the Stokes problem, and the resistivity logging measurement simulations. The hp FEM incorporates a self-adaptive strategy that generates a sequence of hp refined meshes, delivering exponential convergence of the numerical error with respect to the number of degrees of freedom (mesh size or CPU time). The hp meshes generated by the self-adaptive strategy are obtained by multiple h or p refinements of the initial mesh. The self-adaptive mesh, generated in this way, is stored as refinement trees growing down from nodes of the initial mesh. First, we eliminate degrees of freedom starting from leaves of refinement trees, and then we eliminate common degrees of freedom traveling up the refinement trees. The solver is parallelized by utilizing the domain decomposition paradigm. In other words, the solver generates Schur complements of local sub-systems, from bottom of refinement trees, through initial mesh elements and sub-domains. Then, the global problem reduces to relatively small one common &quot;interface &quot; problem, and finally the backward substitution must be executed to propagate the solution from the common interface, through sub-domains, initial mesh elements, down to leafs of refinement trees. The LU factorizations computed at different levels of elimination trees are stored at tree nodes to be reutilized by the solver after the computational mesh is

### Parallel hp-Finite Element Simulations of 3D Resistivity Logging Instruments

"... We simulate electromagnetic (EM) measurements obtained by using one transmitter and one receiver antenna in a borehole environment. The measurements are used to assess electrical properties of rock formations. We have started from a problem where logging instruments as well as rock formation propert ..."

Abstract
- Add to MetaCart

(Show Context)
We simulate electromagnetic (EM) measurements obtained by using one transmitter and one receiver antenna in a borehole environment. The measurements are used to assess electrical properties of rock formations. We have started from a problem where logging instruments as well as rock formation properties are assumed to exhibit axial symmetry around the axis of a vertical borehole. The initial computations are performed with a 2D self-adaptive goal-oriented hp-Finite Element Method (FEM) that delivers exponential convergence rates in terms of the quantity of interest against the CPU time. The 2D mesh obtained from the code delivers very accurate solution at the receiver antenna with minimal number of degrees of freedom in the mesh. The 3D hp finite element mesh is obtained by revolution of obtained 2D mesh. The layers of formation in the 3D mesh are then deviated by 30, 45 and 60 degrees. The 3D parallel hp finite element code is utilized to solve fully 3D problem in deviated wells. The 3D code has been verified by comparison of numerical results with the known exact solution of the problem of antenna radiating into a homogenous space, and by comparison of results of 2D and 3D code for the problem with axially symmetric layers in formation.