Deliverable 2.1.
A grid generation algorithm with h-refinements
´
Angel Rodr´ıguez-Rozas (UPV/EHU and BCAM)
and David Pardo (UPV/EHU and BCAM)
June 13, 2016
Associated Work Package (WP). WP 2: High-order Galerkin Methods for Electromagnetic Exploration
Objectives. To develop an a priori adaptive mesh generator algorithm for Logging-While-Drilling (LWD) borehole resistivity measurement simulations for inversion.
State of the Art, Developed Work, Obtained Results, and Main Innovative Aspects. Introduction. Inversion techniques using numerical simulations involving LWD tools require solving many times a computational expensive forward problem, described by the three-dimensio-nal Maxwell equations. At every iteration in the inversion algorithm, several tool positions are considered in the tool trajectory of interest, say Np (typically several hundred), each one requiring
solving a full problem. Thus, if the whole inversion procedure consists of Ni iterations, NiNp
for-ward problems are to be solved. Clearly, alleviating the computational cost of the forfor-ward problem impacts drastically in the total computational time spent in the inversion. This computational cost may significantly be reduced if an h-adaptive grid generation method is considered instead of using fixed uniform mesh.
We have developed a simple a priori h-adaptive grid generation method for LWD resistivity measurement simulations. Based on the characteristics of LWD triaxial tools, an important amount of degrees of freedom are saved by just considering a finer mesh resolution around a certain area that covers the location of both transmitters and receivers. This fact allows for a faster simulation of the forward problem under consideration, impacting significantly on the savings in terms of computational cost for the overall inversion process.
Our simple a priori adaptive mesh algorithm can be applied to LWD forward simulations together with other a posteriori mesh adaptive algorithms [1, 2].
Mesh adaptation for LWD forward simulations. LWD tools incorporates three mutually orthogonal transmitter coils located at the same position along the tool axis, and three receivers coils all located at the same distance away from the transmitters. Combining every transmitter with every receiver, nine magnetic field components are measured, the so-called XX, XY, XZ, YX, YY, YZ, ZX, ZY and ZZ couplings.
In LWD simulations, a forward problem is considered on which the transmitters of the tool are located at a position xt given in a cartesian system of coordinates of reference. The mathematical
model can be described by the following time-harmonic Maxwell’s equations in a unbounded, anisotropic medium:
∇ × E = iωµH + iωµM, in Ω ∇ × H = σ(x) · E + iωεE, in Ω
where µ = µ0 = 4π10−7H/m is the free-space magnetic permeability, σ(x) = [σx(x)ˆx, σy(x)ˆy, σz(x)ˆz]
is the piecewise conductivity tensor of the anisotropic medium, ε = ε0 is the permittivity of the
medium. Each transmitter coil represents a magnetic dipole source which is characterized by a time-dependent factor eiωt, where ω = 2πf and f is the source’s natural frequency. These three magnetic sources are mathematically modeled by its vector moment M0 and location xt, so that
M = M0δ(xt).
We explain how we employ our simple a priori h-adaptive mesh generation algorithm, focusing on tensor product meshes. In LWD tools, the receivers are typically placed at a close distance away from the receivers, say at xr, also on the axis of the LWD tool. Despite the numerical solution
of the magnetic field is to be computed everywhere in Ω (with a proper truncation), the interest in LWD simulations is to have a good numerical accuracy only at the position of the receivers, suggesting the use a finer mesh resolution (higher density of points) around xr. On the other
hand, a finer mesh around the transmitter’s location, xt is required when modeling a Dirac delta
source. Fortunately, transmitters and receivers are closed one another. The region containing all them should be small compared to the rest of the truncated domain Ω.
Let d be the distance between the transmitters and the receivers (see Fig. 1). Consider the medium point between the transmitters and receivers, xm. Inspired by the fact that the solution
in an homogeneous media decays radially away from the source, a finer mesh resolution is imposed with constant step size hf ine within the following radial region:
Ωf ine = {x | dist(x − xm) ≤ αd} ,
where α is typically chosen to be a factor of two, or three, depending upon the prescribed accuracy to be achieved. Since our mesh is tensorial in cartesian coordinates, a convex (square in 2D or cube in 3D) hull containing Ωf ine is considered. Away from this region a constant step size
Figure 1: Sketch diagram for the mesh refinement algorithm: the darkest gray colored region containing all transmitters and receivers is discretized with constant step size hf ine; the white
(exterior) region, is discretized with a bigger constant step size, hcoarse; in the lightest gray colored,
transient region, we apply a geometrical progression between the step size hf ine and hcoarse.
Test examples. Consider an LWD tool in a purely vertical position, i.e., co-aligned with the z-axis. For simplicity of illustration, consider the plane x-z determined by y = 0. In Fig. 2 we show a portion of the mesh with a local coordinate system in the domain [0,16]x[0,100], where the tool position is centered at (x,z) = (8, 50). In this LWD configuration, we consider two transmitters located, respectively, at zt1 = 50.1 and zt2 = 49.9, and two receivers located, respectively, at zr1 = 50.6 and zr2 = 49.4. Thus, in this case and according to our previous definitions, xm = 50 and d = 0.6. With a factor of α = 3 and this particular scenario, we finally determine the finer region by Ωf ine|y=0 = [6.2,9.8]x[48.2,51.8], drawn in more detail in Fig. 3.
Impact for Science and Society. The developed algorithm made a piori enables the deploy-ment of adaptive meshes in space to be used in LWD scenarios, achieving a good compromise in terms of numerical accuracy and computational cost.
Dissemination and Transfer of Knowledge. This work has been exposed in several occasions: • Workshop on Advanced Subsurface Visualization Methods:“Exploring the Earth”, 26-27 May
2015, Pau, France.
0 2 4 6 8 10 12 14 16 x1 0 20 40 60 80 100 x2
Figure 2: Adaptive mesh for an LWD tool.
– The 2015 Formation Evaluation annual technical meeting, August 14-16, 2015, The University of Texas at Austin, Austin, Texas, U.S.A.
– ICCS 2016 International Conference, June 6 - 8 2016, San Diego, California, U.S.A. Main Participants. The algorithm was designed and developed by researches of the GEAGAM project at UPV/EHU and BCAM, in coordination with the PI and with the useful feedback provided after the secondments carried out in the University of Texas at Austin.
Intellectual Property Rights (IPR). Public.
References
[1] M. J. Nam, D. Pardo and C. Torres-Verdn, Simulation of borehole-eccentered tri-axial induc-tion measurements using a Fourier hp finite-element method: Geophysics, v. 78, 1 (2013), D41-D52.
[2] V. Darrigrand, D. Pardo, and I. Muga, Goal-oriented adaptivity using unconventional error representations for the 1D Helmholtz equation. Comput. Math. Appl. 69, 9 (2015), 964-979.
5 6 7 8 9 10 11 x1 46 48 50 52 54 56 x2
Figure 3: Detail of adaptive mesh for an LWD tool.
[3] ´A. Rodr´ıguez-Rozas, D. Pardo and C. Torres-Verd´ın, Development of a General Framework for the Rapid Simulation of 2D and 2.5D Borehole Resistivity Measurements, 2015 Formation Evaluation annual technical meeting, August 14-16, 2015, The University of Texas at Austin, Austin, Texas, U.S.A., 2015.
[4] ´A. Rodr´ıguez-Rozas and D. Pardo, A Priori Fourier Analysis for 2.5D Finite Elements Simula-tions of Logging-while-drilling (LWD) Resistivity Measurements, Procedia Computer Science, Volume 80, 2016, Pages 782-791, ISSN 1877-0509,
