Mathematical Methods For Mechanical Sciences Pdf

Numerical Methods for Local Models

T.I. Lakoba , in Encyclopedia of Ecology, 2008

Numerical methods commonly used for solving ordinary differential equations, such as Euler and trapezoidal (both explicit and implicit), Runge–Kutta, and multistep methods are presented and compared with each other in terms of their accuracy and stability. The stability of any given method is shown to be a critical factor determining whether the method is useful. It is emphasized that numerical methods for conservative and nonconservative models must possess different stability features, and it is further shown that Runge–Kutta methods possess both of these features. The idea of methods with an adaptive step size is described. The phenomenon of numerical stiffness is explained. Suitable built-in commands of MATLAB are mentioned, and an example of their usage is given.

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Numerical Methods for Distributed Models☆

T.I. Lakoba , in Reference Module in Earth Systems and Environmental Sciences, 2013

Abstract

Numerical methods commonly used for solving partial differential equations of the reaction-advection–diffusion type are presented. The use of implicit (e.g., Crank-Nicolson) and semi-implicit methods is emphasized as being more time-efficient than that of explicit methods. The main kinds of boundary conditions are considered. Briefly discussed are operator-splitting and spectral methods, as well as methods for equations with two spatial dimensions and for hyperbolic problems. Relevant built-in commands of Matlab are presented and their counterparts in other programming languages are briefly mentioned.

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Particle Tracking

Mary P. Anderson , ... Randall J. Hunt , in Applied Groundwater Modeling (Second Edition), 2015

8.3.2 Numerical Methods

Numerical methods are well-suited for a wide variety of hydrogeological problems; Euler integration is the simplest tracking method. To illustrate the mathematics, we again use particle movement in the x-direction as an example, where dx = Δx = x _p − x ₀:

(8.8) $x_{p} = x_{0} + {(v_{x})}_{0} Δ t$

where x ₀ is the initial position of the particle and x _p is the position after tracking for a time period of Δt. Analogous equations are written for the y- and z-coordinates. Numerical errors tend to be large unless small tracking steps (Δt) are used.

In a Taylor Series expansion, the new position of the particle, x _p, is calculated from

(8.9) $x_{p} = x_{0} + (d x / d t) Δ t + (d^{2} x / d t^{2}) (Δ t^{2} / 2)$

analogous equations are written for the y- and z-coordinates. Equation (8.9) is essentially the Euler formula (Eqn (8.8)) with an additional higher order term that represents the time rate of change of velocity or the acceleration. Additional details are given by Kincaid (1988) and Zheng and Bennett (2002).

The fourth-order Runge-Kutta method is widely used in PT. The method calculates the velocity of the particle at four points for each tracking step: at the initial position of the particle (p ₁), at two intermediate points (p ₂ and p ₃), and at a trial end point (p ₄) (Fig. 8.11) where

(8.10a) $x_{p_{2}} = x_{p_{1}} + v_{x p_{1}} \frac{Δ t}{2}$

(8.10b) $x_{p_{3}} = x_{p_{1}} + v_{x p_{2}} \frac{Δ t}{2}$

(8.10c) $x_{p_{4}} = x_{p_{1}} + v_{x p_{3}} Δ t$

Analogous equations are written for the y- and z-coordinates. See Zheng and Bennett (2002, Chapter 6) for details. The final x-coordinate position of the particle (x _n+1) is calculated using an average of velocities at all four points:

(8.11) $x_{n + 1} = x_{n} + \frac{Δ t}{6} (v_{x p_{1}} + 2 v_{x p_{2}} + 2 v_{x p_{3}} + v_{x p_{4}})$

Accuracy of results from the Runge-Kutta and the Euler methods depends on the size of the tracking step (Zheng and Bennett, 2002). Smaller steps give better results but require more computational time. Zheng and Bennett (2002) describe methods to adjust the size of the time step during PT by using step-doubling or inverse distance methods (Franz and Guiguer, 1990) (Fig. 8.12). In step doubling, particles are advanced using a designated tracking step and then tracking is repeated using two half-tracking steps (Fig. 8.12(a)). Alternatively, reverse tracking from the computed particle location is performed (Fig. 8.12(b)). The difference in particle locations is determined and if the difference (i.e., the error, Δs) is considered unacceptable, the tracking step size is reduced and the process repeated until an acceptable error is obtained (see Zheng and Bennett, 2002, Chapter 6, for details).

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Numerical modelling of faults

Andreas Henk , in Understanding Faults, 2020

4.1 Introduction

Numerical methods have been applied successfully to problems in various geoscience disciplines, including geodynamics, structural geology and rock mechanics (e.g., Gerya, 2009; Ramsay and Lisle, 2000; Jing and Hudson, 2002). Such numerical simulations improve not only the quantitative understanding of the underlying physical processes, but also allow for scenario testing and forecasting. With respect to faults, numerical models presented hitherto have addressed aspects of fault integrity, fault fluid flow and earthquake risk assessment, among others. For example, numerical techniques have been used to assess quantitatively the reactivation potential of faults in different tectonic regimes (e.g., McLellan et al., 2004; Buchmann and Connolly, 2009). Fluid flow issues along fault zones have been studied by Cappa (2009), Cappa and Rutqvist (2011) and Schuite et al. (2017), among others. This approach has been expanded also to address earthquake processes (e.g.,Rice et al., 2009; Cappa, 2011). More recently, modelling the effect of man-made pore pressure changes on fault stability due to fluid injection (pore pressure increase) and fluid production (pore pressure decrease) has attracted a lot of attention (Rutqvist et al. 2016; Zhang et al., 2016; Haug et al., 2018).

Any comprehensive numerical analysis of a fault requires at least a coupled hydromechanical (HM) modelling approach. Special applications may even ask for incorporation of thermal and chemical processes leading to thermohydromechanical (THM) and thermohydromechanical-chemical (THM-C) simulations, respectively. Different numerical techniques are available to account for the coupling and the mutual dependence of pore pressure, effective stress, volumetric strain, porosity and permeability (e.g., Minkoff et al., 2003; Rutqvist and Stephansson, 2003). Each technique has its pros and cons, but common to all numerical approaches is the challenge how to describe adequately the complexities of natural fault zones with respect to architecture and petrophysical property distribution. Depending on host rock lithology, deformation history and a possible later hydrothermal overprint, the dimensions and internal structure of damage zone and fault core as well as their hydromechanical characteristics can be quite variable (e.g., Evans et al., 1997; Wibberley et al., 2008; Childs et al., 2009; Faulkner et al., 2010). In particular, fracturing in the damage zone and connectivity of the fractures will determine whether the fault zone acts as barrier or conduit for fluid flow (Chapter 7).

In the following subchapters, an overview of the different numerical techniques commonly used for fault modelling is given. Thereby the focus is on modelling of existing faults rather than formation of new faults, i.e., fracturing and fault propagation (see Chapters 2 and 6 Chapter 2 Chapter 6 ). The numerical models have to be populated with specific hydromechanical parameters for each fault zone component. Hence, an overview of the typical parameter range describing host rock, damage zone and fault core is given. Finally, a generic numerical model is presented to illustrate the workflow and potential of fault zone modelling. Parameter studies allow to systematically vary model features like, fault zone architecture, hydromechanical fault rock properties, depth, fluid pressure and strain rate in order to study their impact on fault zone processes like, for example, stress perturbation, strain localisation, fault reactivation and fluid transfer. Such numerical simulations do not only help to improve our quantitative understanding of faults from a scientific point of view, but also serve as an important tool for a safe utilization of the subsurface.

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Energy harvesting in water supply systems

Armando Carravetta , ... Helena M. Ramos , in Sustainable Water Engineering, 2020

13.4.1 Localization for fresh water supply

Numerical methods make it possible to predict the effect of replacing one or more pressure reducing valves (PRVs) with PATs in a network [40]. The sustainable management of WSSs can be achieved by monitoring and controlling losses, which requires a deep knowledge of the network, in particular its characteristics and operating mode [33]. In recent years, significant research effort has been committed to proposing a general numerical model for the optimal location of the energy recovery node using different criteria, including leakage reduction and power plant production [16,32,48]. A complete analysis of these contributions is out of the scope of this paper, but a comprehensive review can be found in Fecarotta and McNabola [24]. At the present state of knowledge, these general models can handle only small hydraulic networks. As a result, in most of the real-life situations more simplified methods for PAT localization and sizing are implemented.

Nevertheless, the problem of the optimal location of PATs within a water distribution network is still an unresolved problem. Thus, the purpose of this case study is to analyse the energy recovery potential of the water distribution system of Funchal (Portugal) through the replacement of PRVs by PATs, ensuring both an adequate pressure management and valuable energy savings and showing the hydropower potential of a real distribution system. Only a section of the Funchal water distribution system was studied – the pilot zone selected for the study of leakage reduction carried out by the Funchal water industry – which comprises of roughly 40 percent of the entire municipality of Funchal and corresponds to the area of influence of the reservoirs of Terça, S. Martinho, Penteada, Ribeira Grande and Nazaré.

In fig. 13.13A, the pipeline and the terrain elevation distribution are reported, showing very high ground slopes in large parts of the network. The water demand distribution is reported in fig. 13.13B. In order to reduce water leakage, a pressure reducing strategy is in place, by the use of a large number of PRVs. The current pressure head distribution in the network at 13.00 H is plotted in Fig. 13.14A.

At the time of the study, almost 70 percent of the total water entering the water network of Funchal was lost within the system, mostly as a result of inadequate pressure regulation. This poses a serious threat to the environment and presents a significant economic impact for the involved water utilities. However, the Funchal water industry estimated these losses may be reduced to 15 percent by 2033 if the correct measures are taken, particularly the creation of district metered areas (DMAs) in the water distribution network and the correct placement of PRVs. After these modifications, the installation of micro hydro power plants within the system through the use of PATs can further improve the efficiency of the system.

The first required step to evaluate the possibility of recovering energy within the network was to investigate the best possible locations for the implementation of PATs, in order to make sure these would provide the optimal conditions for energy production. Since RSS proposed the implementation of 50 different PRVs, there are 50 available locations. Despite the considerable number of possible locations, only 10 of these PRVs were selected for the implementation of PATs since that would provide sufficient data to conclude the network's potential for energy recovery. This required a selection process which consisted of a basic preassessment of each PRVs potential for the implementation of a PAT, wherein the head drop defined for the PRV was multiplied by the flow rate, and the 10 PRVs which provided the highest values (Q ×H) were selected for further analysis. The 10 selected PRVs are displayed in Table 13.3 (values correspond to peak discharge conditions).

Table 13.3. Hydraulic characteristics of the pressure reducing valves within the Funchal network.

Valve (–)	Q (L/s)	Upstream pressurehead (m)	Downstream pressurehead (m)	Head drop (m)
1	32.20	44.75	23.71	21.04
2	20.66	52.27	22.35	29.92
3	25.68	40.10	17.80	22.30
4	43.12	53.36	29.80	23.56
5	24.65	55.66	31.10	24.56
6	152.41	44.26	25.55	18.71
7	24.12	49.70	25.25	24.45
8	22.47	48.60	24.30	24.30
9	26.89	47.37	24.28	23.09
10	82.28	44.15	21.60	22.55

Considering the predictable reduction of water losses which results from the leakage control program, the PRV head drop values display a tendency to increase and the flow rate values tend to decrease throughout the years. For this reason, to be able to perform an energy recovery analysis up to 2033, a network mathematical model was required to calculate these values for each PRV and each year. This implied the creation of a specific model for each year, which was accomplished using the Funchal water network model provided by the network manager as a starting point. This provided model corresponded to the 2018 situation and included every water loss control measure previously defined, such as the implementation of newly installed PRVs and the establishment of new DMAs. After concluding the PRV selection process and the calculation of hydraulic models, specific PATs had to be selected for each PRV. The PAT has been chosen among a database of 29 machines, for which the hydraulic behaviour was known. By associating the PAT curves to the selected PRVs in the network model, the energy production could be calculated for every year from 2018 to 2033.

There are several different PAT configurations which can be applied to generate power, namely: hydraulic regulation (HR); electric regulation (ER); hydraulic and electric regulation (HER); no regulation (NR). All these configurations can provide interesting energetic results depending on the water distribution network conditions, presenting a great economic benefit for water utilities if implemented correctly. However, in the case study of the Funchal water network, only ER operating modes were tested and analysed.

In order to calculate the variation of the hydraulic characteristics with the rotational speed for each PAT for both ER modes, the affinity law of turbomachines was applied to the respective nominal rotational speed curve.

After defining the configuration for both modes of operation, the most appropriate PATs had to be selected for each PRV location. However, having in consideration that there were 29 different PAT models to choose from and apply to the 10 PRV locations, a digital algorithm has been developed to simulate every possible scenario throughout the years, enabling the development of a fast and detailed testing analysis. In fact, with the aid of this program, all the previously mentioned PATs could be tested for every PRV location and every rotational speed, not to mention the different modes of operation. The range of rotational speeds analysed by the digital algorithm correspond to the minimum and maximum rotational speed limits at which the considered PATs could operate, from 770 rpm to 3500 rpm. Summed up briefly, the nominal speed curves of each PAT have been used as reference curves and alternative rotational speed curves have been obtained through dimensional analysis, treating them as distinct PATs. Finally, the energy production results relative to each possible scenario for every year along with the total accumulated energy from 2018 to 2033 has been calculated.

To study the feasibility of the energy recovery solutions proposed in the WDS of Funchal, an economic analysis was performed for each PAT application separately, considering each mode of operation and storage method. This was essential to accurately determine the most advantageous investment and exclude unprofitable PAT applications which could undermine the economic potential of the project. To evaluate the economic benefit, two scenarios have been considered, differing for the use of energy, i.e. local use and grid connection. For each solution the adopted electricity selling prices are 0.057 €/kWh for the grid connection and 0.098 €/kWh for the local use, and the discount rate considered is 7.5 percent. Table 13.4 shows the figures of the optimal investment for the water distribution system, such as where the investment costs along with the total produced energy in 15 years, the internal return rate (IRR), the 15 years net present value (NPV), the ratio between the benefits and the costs (B/C) and the payback period.

Table 13.4. Economic analysis of the optimal hydropower investment in Madeira WSS for grid or stand-alone energy use. The total energy refers to a period of 15 years.

Energy connection	Grid	Local
Number of PATs	2	3
Total investment (€)	24,648	38,851
Total energy production (MWh)	1716	2063
IRR (percent)	23 percent	34 percent
NPV (percent)	24,855	69,203
B/C (–)	2.0	2.8
Payback period [years]	5	4

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POLLEN METHODS AND STUDIES | Numerical Analysis Methods

H.J.B. Birks , in Encyclopedia of Quaternary Science (Second Edition), 2013

Software and Computing

Some numerical methods require the use of software to implement them, and new and important numerical methods relevant to Quaternary palynological data analysis are constantly being developed by applied statisticians and quantitative paleoecologists. The rate of development of new methods greatly exceeds the production of user-friendly dedicated software such as TILIA, C2, or PSIMPOLL. Virtually all the new methods are available as packages and scripts for public use free of charge within the R software development package and environment (www.r-project.org) ( Table 3 ). Given the wide array of freely available R packages (e.g., vegan, rioja, analogue, paleoSig) and scripts relevant to pollen analysts and the ever-increasing power of personal computers, there is no excuse for palynologists to use outdated, suboptimal numerical techniques in the analysis of their hard-earned data. There is inevitably a steep learning curve in the initial stages of using R, just as there is in any other programming environment (e.g., GenStat, MATLAB), but the final rewards are well worth the initial effort. Using R and new specially developed techniques is a challenge for the Quaternary palynological community now and in the immediate future.

Table 3. Software for paleoenvironmental analysis

Package	Web address	Use
TILIA	http://www.neotomadb.org/data/category/tilia	Plotting of palynological data
PSIMPOLL	http://www.chrono.qub.ac.uk/psimpoll/psimpoll.html/	Plotting and analysis of palynological data
rioja	www.staff.ncl.ac.uk/staff/stephen.juggins	Quaternary science data
analogue	http://analogue.r-forge.r-project.org	Analog and weighted averaging
vegan	http://cran.r-project.org	Community ecology analysis
vegan	http://vegan.r-forge.r-project.org	Community ecology analysis
paleoMAS	http://cran.r-project.org	Paleoecological analysis
fossil	http://cran.r-project.org/web/packages/fossil/index.html	Paleoecological and paleogeographical analysis
simba	www.r-project.org	Similarity coefficients
palaeoSig	http://cran.r-project.org/web/packages/palaeoSig/index.html	Significance tests of reconstructions

Birks HJB, Lotter AF, Juggins S, and Smol JP (eds.) (2012) Tracking Environmental Change Using Lake Sediments, vol. 5: Data Handling and Numerical Techniques. Dordrecht: Springer.

There are also many books available now that provide excellent introductions to the use of R in the numerical and statistical analysis of data – see Table 4 .

Table 4. Books on the use of R for numerical and statistical analyses

General texts
Crawley MJ (2007) The R Book. Chichester: Wiley.
Dalgaard P (2008) Introductory Statistics with R. New York: Springer.
Fox J and Weisberg S (2011) An R Companion to Applied Regression. Sage, London.
Logan M (2010) Biostatistical Design and Analysis Using R. Chichester: Wiley-Blackwell.
Torgo L (2011) Data Mining and R. Learning with Case Studies. Boca Raton, FL: CRC Press.
Wright DB and London K (2009) Modern Regression Techniques Using R. A Practical Guide for Students and Researchers. London: Sage.
Zuur AF, Ieno EN, and Meesters EHWG (2009) A Beginner's Guide to R. New York: Springer.

Specialized texts in the UseR! series
Bivand RS, Pebesma EJ, and Gómez-Rubio V (2008) Applied Spatial Data with R. New York: Springer.
Borcard D, Gillet F, and Legendre P (2011) Numerical Ecology with R. New York: Springer.
Cowpertwait PSP and Metcalfe AV (2009) Introductory Time Series with R. New York: Springer.
Everitt BS and Hothorn T (2011) An Introduction to Applied Multivariate Analysis with R. New York: Springer.
Wehrens R (2011) Chemometrics with R. New York: Springer.
Williams G (2011) Data Mining with Rattle and R. New York: Springer.

Birks HJB, Lotter AF, Juggins S, and Smol JP (eds.) (2012) Tracking Environmental Change Using Lake Sediments, vol. 5: Data Handling and Numerical Techniques. Dordrecht: Springer.

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Stability, accuracy, and efficiency of numerical methods for coupled fluid flow in porous rocks

Richard Giot , Albert Giraud , in Porous Rock Fracture Mechanics, 2017

12.2.1 Review of the numerical methods for coupled fluid flow modeling in continuous porous rocks

Upon numerical methods developed for numerical modeling of coupled fluid flow in porous media, the most popular and widespread are FEM, finite volume method (FVM), and finite difference method (FDM). These are classical approaches to obtain numerical approximation of partial differential equations (PDEs) that describe the physical processes governing the coupled fluid flow in porous rocks. Some of these methods have been combined, which appears to be a sequential resolution of the set of PDEs, for example, FVM for the resolution of the fluid flow problem and FEM for the mechanical problem. Here, we do not want to present in details these different numerical methods, which were largely discussed in the literature. Instead, we propose an overview on the application of these methods to coupled fluid flow in porous rocks and present some advantages, drawbacks, and adaptation of the methods in this framework. Moreover, we chose to confine ourselves to the numerical approaches that explicitly account for mechanical and hydraulic behaviors and their couplings.

Due to the presence of gas and hydromechanical couplings in rocks, the coupled diffusion process in porous rocks in both fully and partially saturated states is highly nonlinear (Olivella et al., 1994). When considering the general case of partially saturated porous media, these equations are highly nonlinear due to the presence of gas and as an consequence numerical methods, essentially finite element or FVMs, are required to solve the direct problem (Thomas et al., 1994; Gawin et al., 1995; Eymard et al., 2000; Chavant et al., 2002; Bianco et al., 2003; Schrefler, 2004; Benard et al., 2006; Gowing et al., 2006; Morency et al., 2007; Dal Pizzol and Maliska, 2012).

FEM is one of the most widespread numerical methods for the numerical modeling of Hydro-Mechanical (HM) couplings in porous rocks, accounting for fluid flow and rock deformation. It has been considered by several authors in the framework of nuclear waste storage accounting for different physical processes such as elastoplasticity, damage, viscosity, two-phase flow, and thermal loading (Jia et al., 2009; Charlier et al., 2013; Levasseur et al., 2013; Salehnia et al., 2015; Pardoen et al., 2015; Gerard et al., 2014; Giraud et al., 2009; Giot et al., 2011; Giot et al., 2012; Guillon et al., 2012). FEM was also applied in the context of CO₂ storage (Dieudonné et al., 2015; Saliya et al., 2015). Special HM coupled finite elements were developed, for example, two-dimensional (2D) second gradient finite elements to deal with localized shear zones in rocks (Collin et al., 2006) or coupled interface finite elements for contact problem (Cerfontaine et al., 2015). A special attention was also given to poroelastic anisotropy, which is of paramount importance when dealing with rocks. Cui et al. (1996) considered an anisotropic poroelastic model for finite element analysis of geomechanical problems. Noiret et al. (2011) applied a transverse isotropic poroelastic constitutive law implemented in a finite element code for the interpretation of œdometer tests on a claystone, whereas Giot et al. (2012, 2014) used a similar approach for the interpretation of axial and radial pulse tests for intrinsic permeability identification of a claystone.

Finite difference method was also considered for several application fields for porous rocks. Amongst many others, let us cite some recent works aiming at modeling coupled fluid flow in porous rock with FDM. Blanco-Martin et al. (2015) considered FDM in the framework of the long-term THM modeling of nuclear waste repository in salt. They used a sequential coupling between FLAC3D (Itasca), which is based on FDM scheme and solves the geomechanical problem, and TOUGH2, which solves the multiphase fluid and heat flow problem. Bian et al. (2012) also employed the FDM to model the THM response of a claystone to a thermal loading, considering a full coupling rather a staggered approach. In this work, the rock is supposed to be fully saturated. Hu et al. (2013) use an Integral Finite Difference Method for geothermal application.

Other methods are proposed by several authors, such as control-volume finite element (Mello et al., 2009) which can be seen as an hybrid method based on FEM and FVM, or a coupling of lattice Boltzmann with distinct element method (Boutt et al., 2007, 2011). This last method is based on a micromechanical approach for the consideration of solid–fluid couplings. The fluid flow is accounted for through a lattice Boltzmann method, while the mechanical problem is solved with a DEM. Cavalcanti and Telles (2003) used the BEM to solve 2D problems concerning fully saturated porous media. FVM is classically used for fluid flow problem but coupled to a FEM for the modeling of rock deformation. Authors such as Li et al. (2016) propose numerical methods to improve the data transfer between both schemes.

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Computational Aerodynamics

David A. Caughey , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

Shock capturing

Most numerical methods for solving compressible flow problems rely on the introduction of artificial (nonphysical) terms in the equations to smear out any shock waves that develop in the solution and allow shocks to be captured naturally by the numerical scheme without any special treatment at points near the shocks. This shock capturing approach is in contrast with that of fitting the shocks as surfaces of discontinuity, which then must be treated as internal boundaries in the flow calculation, across which the appropriate jump relations must be enforced as an internal boundary (or compatibility) condition. The artificial (or numerical) viscosity added in a shock-capturing scheme acts to smear out the discontinuities that the inviscid theory would predict in much the same manner as molecular viscosity smears out shock waves in the real world. The length scales of these phenomena are, or course, quite different. Under conditions typically found in the atmosphere, shock wave thicknesses are of the order of a few microns (10⁻⁶ m); under the action of the artificial viscosity of a numerical scheme, the shock thicknesses scale with the grid spacing, which might correspond to a physical distance of 10 cm for a reasonable mesh spacing on a full-scale wing. It is important to realize that even when the Navier-Stokes equations are solved, artificial viscosity usually is necessary when the solution contains shock waves, since it is impractical to use mesh spacings fine enough to resolve the physical shock structure defined by the molecular viscosity.

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Laterolog Tools and Array Laterolog Tools

C. Richard Liu , in Theory of Electromagnetic Well Logging, 2017

15.5 Validation of the Computational Method

The numerical method discussed in this chapter is validated by the results from Anderson [2], where the frequency of the LLd is set to 35 Hz and the frequency of the LLs is set to 280 Hz. First is the comparison of tool coefficients. Tool coefficients are the most important variable to calculate apparent resistivity as $R_{a} = K_{L L d} \frac{(V_{M 1} + V_{M 1'}) / 2 - V_{N}}{I_{A 0}}$ (Eq. 15.11) and $R_{a} = K_{L L s} \frac{(V_{M 1} + V_{M 1'}) / 2 - V_{N}}{I_{A 0}}$ (Eq. 15.14). Tool coefficients are first calculated by the computational code to guarantee that the apparent resistivity calculated is equal to the true formation resistivity.

Table 15.2 lists the tool coefficients calculated by Anderson and this chapter. K _LLs is exactly the same and K _LLd is very close to each other. The slight difference is caused by the different setting of voltage electrodes M, where Anderson assumed that the electrodes M are thin rings and in the model of electrodes M in this chapter it still has actual geometry.

Table 15.2. Validation from tool coefficient

	K _LLs	K _LLd
Anderson	1.45	0.89
Author	1.45	0.87

Fig. 15.8 shows the comparison of dual laterologs computed from a benchmark formation which has 30 in. of invasion in some beds. The borehole has a radius of 4 in. and a resistivity of 0.5 ohm-m. The resistivity of the formation is 0.5, 5, and 50 ohm-m, and the resistivity of the invasion is 2.5 and 10 ohm-m. The LLs and LLd logs were not borehole corrected since correction is only necessary for large holes. In the uninvaded bed between 47 and 57 ft, LLd departs from R _t because the survey current flows preferentially in the conductive bed (squeeze effect), while much of the bucking current remains in the resistive shoulders as it flows to the remote return. Fig. 15.9 shows the result for an artificial testing formation [2].

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Deep Earth Seismology

J. Tromp , in Treatise on Geophysics (Second Edition), 2015

1.07.2.1 Benchmarking

Any numerical method aimed at simulating global seismic wave propagation must be carefully benchmarked against semianalytic methods, for example, normal-mode summation, as well as other 3D techniques. At the scale of the globe, such benchmarks are difficult and must be carefully executed. In this section, we briefly enumerate some of the main challenges associated with benchmarking.

First, rather obviously, the Earth model must be exactly the same in all techniques competing in the benchmark, which means among other things that all first- and second-order discontinuities must be honored. At a first-order discontinuity, a function – and possibly its first derivative – is discontinuous, whereas at a second-order discontinuity, a function is continuous but its first derivative is not. Most 1D Earth models, for example, PREM, have both types of discontinuities. Special attention must be paid to the implementation of the crust. Seemingly subtle model differences, for example, in crustal thickness or a division in terms of upper and lower crust, can have a profound effect on surface-wave dispersion and surface-reflected body-wave amplitudes. The presence of an ocean layer has a profound effect on the dispersion of Rayleigh waves and can affect the amplitudes of surface-reflected P–SV body waves. Most 1D Earth models incorporate radial models of attenuation, and one must ensure that the associated effects are properly implemented. As mentioned among the challenges, in an anelastic medium, the shear and compressional wave speeds become frequency-dependent, an effect that most normal-mode codes incorporate. So, besides reducing wave amplitudes, attenuation can have a strong effect on the phase of a broadband signal. The fact that we need to simulate seismograms over more than three decades in frequency, that is, periods from shorter than 1 s to longer than 1000 s, makes the implementation of such physical dispersion in numerical algorithms a necessity. Finally, one needs to decide whether or not effects due to self-gravitation and transverse isotropy, both readily accommodated in normal-mode algorithms, will form part of the benchmark.

The COSY project (Igel et al., 2000) brought together several research groups in an attempt to benchmark numerical algorithms for spherically symmetrical and 3D Earth models. The results are summarized in a special issue of Physics of the Earth and Planetary Interiors (volume 119, 2000). Even for 1D models, the COSY benchmarks are disappointing in terms of their relatively low level of agreement, in particular for surface waves. This serves as an illustration of the great difficulties associated with developing, implementing, and benchmarking numerical techniques for the simulation of global seismic wave propagation.

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