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This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.

This chapter is concerned with transport in the vicinity of electrodes, and hence on the coupling between Faradaic electrode processes and mass transport. It covers transport in stationary and transient condition, planar and spherical geometries, presence and absence of supporting electrolytes, as well as convective transport in hydrodynamic electrodes. Some common electrochemical techniques are also discussed, and the solutions of the corresponding transient transport problems are worked out in detail.

This chapter shows how both Fourier transformation and inverse Laplace transformation can be used in conjunction with pulsed gradient spin echo NMR, as a basis for a multidimensional characterisation, whether as separation, correlation, or exchange experiments in which diffusion and relaxation effects provide additional dimensions. In a porous medium, exchange experiments reveal changes in local relaxation rates and diffusion coefficients caused by molecular migration, and when measured as a function of mixing time can reveal exchange times. At zero mixing time, separate properties such as diffusion, local field, local field gradient, or relaxation time may be correlated. The chapter explains the details of inverse Laplace transformation, and takes the reader through the various multiplexed Fourier and Laplace dimensions in which molecular translation plays a role. Finally, measurement of the diffusion tensor is discussed as an example of multidimensional signal acquisition and processing.

Transformation, convolution, and correlation are used over and over again in nuclear magnetic resonance (NMR) spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one or more dimensions. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra. These are related to nonlinear correlation functions of excitation and response. This chapter discusses convolution in linear and nonlinear systems, along with the convolution theorem, linear system analysis, nonlinear cross-correlation, correlation theorem, Laplace transformation, Hankel transformation, Abel transformation, z transformation, Hadamard transformation, and wavelet transformation.

So far our treatment of fluid motions has not emphasized the behavior of fluids residing within porous geological materials. Let us now turn to this topic and, in doing so, make use of our insight regarding purely fluid flows. The general topic of fluid behavior within porous geological materials is an extensive one, forming the heart of such fields as groundwater hydrology, soils physics, and petroleum-reservoir dynamics. In addition, this topic is an essential ingredient in studies concerning the physical and chemical evolution of sedimentary basins, and the dynamics of accretionary prisms at convergent plate margins. In view of the breadth of these topics, the objective of this chapter is to introduce essential ingredients of fluid flow and transport within porous materials that are common to these topics. Our first task is to examine the physical basis of Darcy’s law, and to generalize this law to a form that can be used with an arbitrary orientation of the working coordinate system relative to the intrinsic coordinates of a geological unit that are associated with its anisotropic properties. We will likewise examine the basis of transport of solutes and heat in porous materials. We will then develop the equations of motion for the general case of saturated flow in a deformable medium. In this regard, several of the Example Problems highlight interactions between flow and strain of geological materials during loading, because this interaction bears on many geological processes. Examples include consolidation of sediments during loading, and responses of aquifers to loading by oceanic and Earth tides, and seismic stresses. We will concentrate on the description of diffuse flows within the interstitial pores of granular materials, as opposed to flows within materials containing dual, or multiple, pore systems such as karstic media, or media containing both interstitial and fracture porosities. We will consider unsaturated, as well as saturated, conditions. For simplicity, the subscript h is omitted from the notation of quantities such as specific discharge q and hydraulic conductivity K.