Equations describing

Nearly all reservoirs are water bearing prior to hydrocarbon charge. As hydrocarbons migrate into a trap they displace the water from the reservoir, but not completely. Water remains trapped in small pore throats and pore spaces. In 1942 Arch/ e developed an equation describing the relationship between the electrical conductivity of reservoir rock and the properties of its pore system and pore fluids. 

A novel optimization approach based on the Newton-Kantorovich iterative scheme applied to the Riccati equation describing the reflection from the inhomogeneous half-space was proposed recently [7]. The method works well with complicated highly contrasted dielectric profiles and retains stability with respect to the noise in the input data. However, this algorithm like others needs the measurement data to be given in a broad frequency band. In this work, the method is improved to be valid for the input data obtained in an essentially restricted frequency band, i.e. when both low and high frequency data are not available. This... 

The equation describes the manner in which the nuclear magnetization, M, at position r and time t processes about the magnetic flux density, B, in which it is found. The constant y is the magnetogyric ratio of the nuclides under study. The precessional frequency, co, is given by the Lannor equation. 

This equation describes the Fourier transfonn of the scattering potential V r). It should be noted that, in the Bom approximation the scattering amplitude/(0) is a real quantity and the additional phase shift q(9) is zero. For atoms with high atomic number this is no longer tme. For a rigorous discussion on the effects of the different approximations see [2] or [5]. 

This equation describes the additional amount of gas adsorbed into the pores due to capillary action. In this case, V is the molar volume of the gas, y its surface tension, R the gas constant, T absolute temperature and r the Kelvin radius. The distribution in the sizes of micropores may be detenninated using the Horvath-Kawazoe method [19]. If the sample has both micropores and mesopores, then the J-plot calculation may be used [20]. The J-plot is obtained by plotting the volume adsorbed against the statistical thickness of adsorbate. This thickness is derived from the surface area of a non-porous sample, and the volume of the liquified gas. 

This basic equation describes waves, whose properties are related as follows ... 

The rate equation describing the kinetics of this reaction is... 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schrodinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes. 

Using the BO approximation, the Schrddinger equation describing the time evolution of the nuclear wave function, can be written... 

The problem of the theoretical description of biopolymer water adsorption isotherms has drawn the attention of researchers for a long time. In the works [19], [20] a rigorous statistical basis for equations describing the isotherms for the case of homogeneous adsorption surfaces and noninteracting adsorption sites of N different types has been suggested. The general equation is ... 

Th is last equation is the nuclear Schriidinger equation describing the motion of nuclei, Th e electron ic energy computed from solving the electronic Schrbdinger equation (3) on page 163 plus tfie nuclear-nuclear interactions y, (R,R) provide a potential for nuclear motion, a Potential Knergy Surface (PHS). 

To obtain equations describing the dusty gas model, equations (3.1) must be applied to a pseudo mixture of (n+1) species, in which the extra species, numbered n+1, represents the dust. We must also require... 

This equation describes the motion of a particle of mass m, along one coordinate (x,) with F. . being the force on the particle in that direction. 

After the substitution from Equations (3.82), (3.83) and (3.84) into Equation (3.81) and in turn substituting from the resultant relationship into Equation (3.80) and rearranging the following equation describing the trajectory of fluid particles is found... 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

The described continuous penaltyf) time-stepping scheme may yield unstable results in some problems. Therefore we consider an alternative scheme which provides better numerical stability under a wide range of conditions. This scheme is based on the U-V-P method for the slightly compressible continuity equation, described in Chapter 3, Section 1.2, in conjunction with the Taylor-Galerkin time-stepping (see Chapter 2, Section 2.5). The governing equations used in this scheme are as follows... 

The shear-dependent viscosity of the compound is found using the temperature-dependent form of the Carrean equation, described in Chapter 1, given as... 

Because the masses are the same, these equations describe independent oscillators with identical frequencies, as in Fig. 5-la. 

The energy expression consists of the sum of simple classical equations. These equations describe various aspects of the molecule, such as bond... 

The Poisson equation describes the electrostatic interaction between an arbitrary charge density p(r) and a continuum dielectric. It states that the electrostatic potential ([) is related to the charge density and the dielectric permitivity z by... 

Write a series of equations describing a synthesis of 1 bromo 2 Tietny rop ol from tert butyl bromide J... 

Write equations describing two different ways in which benzyl ethyl ether could be prepared by a Williamson ether synthesis J... 

Write equations describing the addition-elimination mechanism for the reaction of hexafluorobenzene with sodium methoxide clearly showing the structure of the rate determining intermediate j... 

To compute molecular orbitals, you must give them mathematical form. The usual approach is to expand them as a linear combination of known functions, such as the atomic orbitals of the constituent atoms of the molecular system. If the atomic orbitals, (Is, 2s, 2px, 2py, 2pz, etc.) are denoted as then this equation describes the molecular orbitals as linear combination of atomic orbitals (MO-LCAO) ... 

In the previous section we considered the amount of sample needed to minimize the sampling variance. Another important consideration is the number of samples required to achieve a desired maximum sampling error. If samples drawn from the target population are normally distributed, then the following equation describes the confidence interval for the sampling error... 

Mathematically, two factors are independent if they do not appear in the same term in the algebraic equation describing the response surface. For example, factors A and B are independent when the response, R, is given as... 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

Empirical Models of the Response Surface In many cases the underlying theoretical relationship between the response and its factors is unknown, making impossible a theoretical model of the response surface. A model can still be developed if we make some reasonable assumptions about the equation describing the response surface. For example, a response surface for two factors, A and B, might be represented by an equation that is first-order in both factors... 

A mechanism is a series of simple reaction steps which, when added together, account for the overall reaction. The rate law for the individual steps of the mechanism may be written by inspection of the mechanistic steps. The coefficients of the reactants in the chemical equation describing the step become the exponents of these concentrations in the rate law for... 

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