A 2D and 3D hp-Finite Element Method for Simulation of Through Casing Resistivity Logging Instruments D. Pardo*, C. Torres-Verdin, M. Paszynski, C. Michler, and L. Demkowicz The University of Texas at Austin, Austin TX, USA

Introduction In order to avoid collapse of wells in oil fields, the use of steel casing has been a common technique throughout the last decades. However, the casing of a well with a steel pipe has some undesired effects. For instance, the assessment of electrical properties of the rock formation becomes more challenging, since the casing highly attenuates the electromagnetic fields. More precisely, the main difficulties for simulating through casing resistivity tools (TCRT) in a borehole environment, for the assessment of rock formation properties, are the following. • The electrical conductivity contrast is high. • The computational domain that we need to consider is large (several miles) in the vertical direction. • The dynamic range -quotient between the solution at the transmitter antenna and the quantity of interest evaluated at the receiver antennas- is usually large (107 − 1013 ). • The aspect ratio dictated by the geometry of the casing is large, which makes extremely challenging the use of standard iterative solvers in combination with conforming finite elements (or finite differences) methods. All these difficulties may be overcome by utilizing a novel numerical method based on a self-adaptive, goal-oriented hp-Finite Element Method (FEM) that converges exponentially in terms of the quantity of interest vs. the problem size (as well as CPU time). The method is based on a purely numerical, highly accurate, and very reliable adaptive algorithm, which has been successfully applied to a variety of resistivity logging devices, including TCRT [6, 5], Induction [3], and Normal/Laterolog instruments [4]. This methodology provides a guaranteed numerical error threshold below 0.1%, which allows for accurate simulation of meaningful physical effects in cased wells, such as invasion of water and/or anisotropy.

The Self-Adaptive Goal-Oriented hp-Finite Element Method In here, we outline the main ideas of the numerical methodology used for simulating TCRT problems. The method is based on combining the use of hp-Finite Elements with a selfadaptive algorithm designed for approximating a user-prescribed quantity of interest (also called goal-oriented adaptivity).

A 2D and 3D hp-FEM. Our numerical method utilizes continuous and edge elements of variable order of approximation. Thus, it supports hp-FE discretizations of electromagnetic

(EM) problems. Here h stands for the element size, and p denotes the polynomial element order (degree) of approximation, both varying locally throughout the grid. The main motivation for using hp-FE is given by the following result: For an optimal sequence of grids, both in terms of element size h and polynomial order of approximation p, the corresponding sequence of solutions converges exponentially to the exact solution with respect to the number of unknowns (as well as the CPU time), independently of the number, intensity, and/or distribution of singularities in the solution. In the following, we present an algorithm intended to obtain the ‘optimal sequence of grids’ mentioned on the above result, so exponential convergence will be attained.

A Self-Adaptive hp Goal-Oriented Algorithm. The algorithm described in [4, 2] produces a sequence of optimally hp-refined meshes that delivers exponential convergence rates in terms of a user prescribed quantity of interest against the size of the discrete problem or CPU time. A given (coarse) conforming hp mesh is first refined globally in both h and p to yield a fine mesh, i.e. each element is broken into four new elements (eight in 3D), and the discretization order of approximation p is raised uniformly by one. Subsequently, the problem of interest on the fine mesh is solved. The next optimal coarse mesh is then determined as the one that maximizes the decrease of the projection based interpolation error [1] averaged by the added number of unknowns. Since the mesh optimization process is based on the minimization of the interpolation error rather than the residual, the algorithm is problem independent, and it can be applied to different physics (acoustics, elasticity, etc.), nonlinear and eigenvalue problems as well.

Numerical Results Fig. 1 describes the geometry of a TCRT model problem. If we consider a vertical well, the corresponding solution may be obtained by utilizing an axisymmetric 2D simulator. For a particular selection of the electrical properties of the formation materials, the final solution (log) at different frequencies is displayed in Fig. 2. Finally, Fig. 3 displays an automatically computer-generated optimal hp-grid for a particular position of a simple 3D resistivity logging instrument.

References [1] L. Demkowicz and A. Buffa. H 1 , H(curl), and H(div) conforming projection-based interpolation in three dimensions: quasi optimal p-interpolation estimates. Computer Methods in Applied Mechanics and Engineering, 194:267–296, 2005. [2] D. Pardo, L. Demkowicz, C. Torres-Verdin, and M. Paszynski. A goal oriented hpadaptive finite element strategy with electromagnetic applications. Part II: electrodynamics. Submitted to Computational Methods on Applied Mechanics and Engineering (CMAME). [3] D. Pardo, L. Demkowicz, C. Torres-Verdin, and M. Paszynski. Simulation of resistivity logging-while-drilling (LWD) measurements using a self-adaptive goal-oriented

MATERIAL I

BO RE LE

E ST

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EL

MATERIAL II

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N SI

CA

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MATERIAL III

ELECTRODES

MATERIAL III

MATERIAL IV

Figure 1: 2D cross section of the geometry of a TCRT problem. The model consists of one transmitter and three receiver electrodes, a conductive borehole, a metallic casing, and four layers in the formation material with varying resistivities First Difference of Magnetic Field (Normalized) 2

2 0.1 Hz 10 Hz 1 Khz 10 Khz 100 Khz 1 Mhz

1.5 Position of receiver antenna in the z−axis (in m)

Position of receiver antenna in the z−axis (in m)

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1

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0

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−1 −15 10

0.1 Hz 10 Hz 1 Khz 10 Khz 100 Khz 1 Mhz

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−12

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10 10 Amplitude of Magnetic Field (A/m)

−6

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−1 −180

−90

0 Phase (degrees)

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Figure 2: TCRT problem. First difference of the magnetic field against the position in the z-axis of the middle point between the two receiver antennas (in meters). We display amplitude (left panel) and phase (right panel). Results are shown for various operating frequencies. The numerical error threshold for this problem is below 1%.

Figure 3: Final hp-grid. Different colors indicate different polynomial degrees of approximation, ranging from 1 (dark blue) to 8 (pink). The dimensions of the displayed grid are 200m (horizontal) x 400m (vertical). hp-finite element method. SIAM J. on Appl. Math. (in press). Preprint available at: www.ices.utexas.edu/%7Epardo. [4] D. Pardo, L. Demkowicz, C. Torres-Verdin, and L. Tabarovsky. A goal-oriented hpadaptive finite element method with electromagnetic applications. Part I: electrostatics. Int. J. Numer. Methods Eng., 65:1269–1309, 2006. [5] D. Pardo, C. Torres-Verdin, and L. Demkowicz. Simulation of invasion and electrical anisotropy effects on axisymmetric through-casing borehole resistivity measurements using a self-adaptive goal-oriented hp-finite element method. To be submitted to Geophysics. Preprint available at: www.ices.utexas.edu/%7Epardo. [6] D. Pardo, C. Torres-Verdin, and L. Demkowicz. Simulation of multi-frequency borehole resistivity measurements through metal casing using a goal-oriented hp-finite element method. IEEE Transactions on Geosciences and Remote Sensing (in press). Preprint available at: www.ices.utexas.edu/%7Epardo.

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