Distortion and harmonic generation in the nearfield of a finite amplitude sound beam are considered, assuming time-periodic but otherwise arbitrary on-source conditions. The basic equations of motion for a lossy fluid are simplified by utilizing the parabolic approximation, and the solution is derived by seeking a Fourier series expansion for the sound pressure. The harmonics are governed by an infinite set of coupled differential equations in the amplitudes, which are truncated and solved numerically. Amplitude and phase of the fundamental and the first few harmonics are calculated along the beam axis, and across the beam at various ranges from the source. Two cases for the source are considered and compared: one with a uniformly excited circular piston, and one with a Gaussian distribution. Various source levels are used, and the calculations are carried out into the shock region. The on-axis results for the fundamental amplitude are compared with results derived using the linearized solution modified with various taper functions. Apart from a nonlinear tapering of the amplitude along and near the axis, the results are found to be very close to the linearized solution for the fundamental, and for the second harmonic close to what is obtained from a quasilinear theory. The wave profile is calculated at various ranges. An energy equation for each harmonic is obtained, and shown to be equivalent within our approximation to the three-dimensional version of Westervelt’s energy equation. Recent works on one-dimensional propagation are reviewed and compared.

Summary This paper discusses models for one- and two-phase injection-well tests where the injected fluid has a temperature different from that of the reservoir fluid. After a brief discussion of the relative significance of different nonisothermal effects, an integral form of the model equations is used to develop approximate analytical expressions describing temperature and pressure in the reservoir during injection. The expressions are valid for a general variation in viscosity with temperature, and different analytical viscosity functions are used to study the influence of this variation. Introduction In a water-injection well test, the injected fluid usually has a temperature different from the initial reservoir temperature, and during injection, both a saturation and a temperature front propagate into the reservoir. Still, most techniques for analyzing injection welltests are based on isothermal models1,2 and hence may give erroneous estimates of reservoir and fluid parameters. A one-phase theory describing nonisothermal effects has been applied by several authors3–7 in the investigation of pressure behavior in geothermal reservoirs. Mangold et al.4 present a comparison between the wellbore pressure described by a thermal simulator and that described by the Theis solution2 that is valid for isothermal conditions. The comparison indicates that the effects of temperature variations during injection are similar to effects produced by lateral permeability changes. This similarity is confirmed in a simulation study by Benson and Bodvarsson,7 where nonisothermal effects are found to behave as skin effects. Tsang and Tsang3 present an analytical solution describing reservoir pressure based on use of the Boltzmann transformation and a certain approximation of the variation in viscosity. This analytical result is in agreement with the previously mentioned numerical results, but the significance of the viscosity approximation is not discussed. Nonisothermal effects may occur during several different processes where two-phase theory must be used, such as steamflooding,8 in-situ combustion,9 and water injection.10,22 The mathematical models describing these situations generally will not be equal, and only water injection will be discussed here. Investigations of isothermal water injection have often applied the Verigin model,12–15 where the reservoir is assumed to consist of two distinct fluid zones separated by a moving discontinuity in saturation. Studying this Verigin problem, Woodward and Thambynayagam11 generalize analytical solutions to nonisothermal conditions, resulting in a skin factor similar to the one predicted by Benson and Bodvarsson.7 Woodward and Thambynayagam do not give any formal proof for the generalization, but the analytical expressions are in agreement with numerical results presented by them and by Weinstein.10 The purpose of this paper is to give a formal derivation of the results presented by Benson and Bodvarsson7 and by Woodward and Thambynayngam.11 The underlying models are presented, and the validity of the approximations needed to derive the analytical expressions are discussed. Mathematical Model for One-Phase Tests The reservoir is assumed to be homogeneous of uniform thickness, and of infinite extent. When effects of gravity, as well as heat losses to the over- and underburden, are neglected, a one-dimensional radial model can be used. Viscosity is assumed to be a function of temperature alone, whereas all other thermodynamic parameters - e.g., compressibility, heat capacity, etc. - are approximated by constants. In the equation expressing energy conservation, the term representing viscous dissipation usually is regarded as negligible16,19 and is left out here, although Smith and Steffensen20 claim that the term may be significant in the vicinity of the wellbore. Energy and mass conservation then are expressed by the equations Equations 1a and 1b A short derivation of the equations is given in Appendix A. All variables are dimensionless, pD=dimensionless pressure; TD=dimensionless temperature; µD=µD(TD)=scaled viscosity; NPe=Peclet number for water injection of order 10-4 to 10 -3D=diffusivity ratio of order 10-7 to 10-6; K=ratio between the fluid and effective heat capacities; and E=coefficient related to thermal expansion of order 102 to 103. The equations are coupled through the convection term in the energy/temperature equation and through the temperature dependency of density and viscosity in the mass/pressure equation. The order of magnitude between the two terms on the left side of the pressure equation is given by. Equation 2 The relationship of the two terms on the right side is also of order DE, and in the following equations, the two terms involving the thermal expansion factor E will be neglected. With the conservation equations, the following boundary and initial conditions are used. For rD=1, Equations 3a through 4b