Hybrid Computer Implementation of ADI Procedure

Abstract

This paper was prepared for the 44th Annual Fall Meeting of the of the Society of Petroleum Engineers of AIME, to be held in Denver, Colo., Sept. 28-Oct. 1, 1969. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal, provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Abstract Single-phase gas flow in two space dimensions in a reservoir is described by a non-linear partial differential equation. In order to solve this equation for meaningful reservoir problems, recourse is generally taken to a digital computer and numerical techniques of solution. The alternating direction implicit procedure (ADIP), combined with iteration procedure (ADIP), combined with iteration because of the non-linearity of the equation, is applicable. This paper describes the implementation of the ADI procedure on a hybrid computer. In this approach the time derivitive in the equation is integrated continuously over finite time steps using the analog portion of the hybrid. Basically, this integration replaces the Thomas or other algorithm required for the solution of the matrix generated in the formation of finite difference equations. By this new method the non-linear coefficients in the equation can be included directly in the analog integration, thus eliminating the necessity of iteration as required in digital algorithms. In this initial implementation of the procedure, the reservoir is assumed procedure, the reservoir is assumed homogeneous and gas properties (viscosity, compressibility factor) are assumed to be constant. However, the technique as described is quite general and can be extended to non-ideal gases and non-constant coefficient equations. Introduction Mathematical description of single-phase gas flow in a reservoir gives rise to a second order non-linear partial differential equation. The solution of this equation provides the engineer with useful information provides the engineer with useful information for analyses of gas field well test, the design of gas field drilling programs, gas storage reservoir design, etc. Since the gas flow equation is non-linear, analytical solutions are not readily available. However, many approximate solutions for the case of one-dimensional radial flow have been presented. Such papers have been especially useful for the design and interpretation of flow tests. For systems of more complicated geometry or heterogeneous systems it has been necessary to resort to numerical techniques, applied using a digital computer. In an early paper Bruce et. al. solved the gas flow equation numerically in one space dimension.

AES quantification of intergranular film thickness in the Ni–Bi system with respect to the liquid metal embrittlement phenomenon

“…15,16 This area was shown to induce strong intergranular room temperature brittleness in at least two systems: Cu-Bi 17 and Ni-Bi. 18 Mechanical testing was used to investigate the kinetics of depth embrittlement, which is frequently associated with the kinetics of intergranular penetration, i.e. describes the extension of this area with time.…”

“…The kinetics of its formation is not clearly established and was found to be either linear in Cu-Bi 17 and Ni-Bi 19 or parabolic in Ni-Bi. 18 Although it is surprising that the same system at the same temperature exhibits two different regimes, it is still true that the penetration kinetics are: very different from typical grain boundary diffusion with a quarter exponent; 20 are very fast; 17 and lead to a quick extension of the area ahead of the interfacial grooves 17 or micrometric film tip. 21 Consequently, the characteristics of this area are of major interest with respect to the LME mechanisms.…”