Semi-infinite line

FIGURE 10.6 Electric circuit of a slice of a semi-infinite line. 

The final result is the expression of the impedance operator of a semi-infinite line (short-circuited coaxial cable) in terms of the power one half on the time integration... 

The reason for not having directly modeled the semi-infinite line with this Formal Graph approach is that the demonstrative virtue of the comparison would have been lost. 

In contrast to Chapter 1, we have explicitly introduced q(x,y,x,t), representing the local source volume flow rate per unit volume produced by any infinitesimal element of a general well. It is a three-dimensional, point singularity that applies to both injector and producer applications. For example, when q is a semi-infinite line, cylindrical radial flow is obtained over most of the source distribution, while spherical flow effects apply at the tip. In other words, partial penetration and spherical flow are modeled exactly. In this section, subscripts are used in three different contexts. First, they represent partial derivatives for example, Px is the partial derivative of p(x,y,z,t) with respect to the spatial coordinate x. Second, they are used as directional markers for example, ky (x,y,z) is the anisotropic permeability in the y direction. Finally, subscript indexes (i,j,k) in pijrepresent the centers of grid block volumes used in our finite difference discretizations. As usual. Ax, Ay, Az, and At are used to denote grid sizes for the independent variables x, y, z, and t. 

Here, we consider the semi-infinite line shown in Figure 1.6. The AC constant voltage source is connected to the sending end (x = 0), and the line extends infinitely to the right-hand side (x = +oo). 

From the general solutions in Equations 1.47 and 1.51, the solutions of voltages and currents on the semi-infinite line in Figure 1.6 are obtained by using the following boundary conditions ... 

Substituting constants A and B into the general solutions, that is, Equations 1.47 and 1.51, voltages and currents on a semi-infinite line are given in the following form ... 

The voltage on a semi-infinite line is expressed by the following equation ... 

Let us look at the meaning of the attenuation constant using the semi-infinite line case as an example. From Equation 1.62 and the boundary conditions ... 

Using the second part of Equation 1.40, the voltage in a semi-infinite line can also be found as follows, since Ij, = 0 ... 

For a semi-infinite line, substituting the first equation of Equation 1.112 into Equation 1.118 gives... 

The frequency dependence of the propagation constant appears as a wave deformation in the time domain. This is measured as a voltage waveform at distance x when a step (or impulse) function voltage is applied to the sending end of a semi-infinite line. The voltage waveform. 

It should be noted that the definition of Equation 1.202 proposed by the author in 1973 is effective only for a semi-infinite line or for a time period of 2t, where x is the traveling time of a line [20,21]. Also, the defiiution requires further study in conjunction with the wave equation in the time domain because it has not been proved that this definition expresses the physical behavior of the time-dependent characteristic impedance. 

FIGURE 1.33 (a) A semi-infinite line, (b) An equivalent circuit. 

This equation is the same as Ohm s law for a lumped-parameter circuit with resistance R. Thus, the semi-infinite line is equivalent to that in Figure 1.33b. 

Resistance-terminated line A resistance is equivalent to a semi-infinite line whose surge impedance is the same as the resistance as explained in Section 1.6.2.1 and in Figure 1.33. If the surge impedance of the semi-infinite line is taken to be the same as that of the line to which the resistance is connected, then a backward traveling wave + ct) = ej, is to be placed on the semi-infinite line ... 

Capacitance-terminated line When a semi-infinite line Zq is terminated by a capacitance C as shown in Figure 1.39, node voltage V and current I are calculated in the following... 

Equation 1.112 shows that the proportion of current to voltage at any point in a semi-infinite line, that is, the characteristic admittance matrix, is defined as follows ... 

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