A numerical method for the calculation of wave properties is presented. It involves expanding the velocity components and the equation of the profile in Fourier series and determining the Fourier coefficients numerically by the method of least squares from the Bernoulli equation and from the equation insuring that the particle motion at the surface matches the profile motion. An iterative procedure is used, since the velocity coefficients depend upon the profile coefficients, and vice versa. The calculations seem always to converge to an answer, although slowly. To judge the accuracy obtained, a comparison has been made with the Stokes waves. In every case the direct method is less in error.

The physical problem is to calculate the profile and velocity components of a wave, subject to the conditions that the effects of viscosity and surface tension can be neglected. The flow is required to be steady, irrotational, and two dimensional, and the depth of the water is assumed to be small and constant. The shallow‐water approximation, first proposed by Friedrichs, is employed, and two additional orders of approximation, beyond the three calculated by Keller, are obtained. The formulas describing the solitary wave are obtained as a special case, and the velocity of the solitary wave so obtained is found to agree almost exactly with the Boussinesq‐Rayleigh formula c=gd(1+H/d) , where c is the wave velocity, d is the depth, and H is the wave height. This velocity differs by no more than 2 per cent from the experimental values of Stephan and Dailey.

A model for oil-water coning in a partially perforated well has been developed and tested by perforated well has been developed and tested by comparison with numerical simulations. The effect of oil-water coning, including down-coning of oil, on field production is demonstrated by studying a small water drive reservoir whose complete production data arc known. production data arc known.The coning model is derived by assuming vertical equilibrium and segregated flow. A necessary correction for departure from vertical equilibrium in the immediate neighborhood of the well is developed The coning model is suitable for single-well studies or for inclusion in a reservoir simulator for two-dimensional, areal studies. Introduction The objective of this investigation of oil-water coning was to develop tools to evaluate operational problems for reservoirs with bottom water. Although problems for reservoirs with bottom water. Although any specific question can be answered (a least in principle) by finite-difference simulation, a practical principle) by finite-difference simulation, a practical problem occurs. Great detail may be necessary for problem occurs. Great detail may be necessary for a reservoir-wide simulation of problems involving coning. Two approaches are possible. One can use more accurate finite-difference equations (such as those derived by some type of Galerkin procedure) to solve the problem of insufficient accuracy. Or one can include in his simulator a "well model" that accurately predicts coning on the basis of near-well properties. The well model could be either another finite-difference subsystem or a formula theoretical or empirical (or both) in character. Our approach is to develop a theoretical model that can be installed in a finite-difference reservoir simulator. We feel that such a model, particularly if it is simple and widely applicable, has several advantages:(1)the assumptions made in the derivation aid in understanding coning;(2)the formula guides the engineer by indicating significant parameters and their relationships;(3)the existence parameters and their relationships;(3)the existence of a simple formula permits preliminary studies without a full simulation; and(4)the simple formula is easy to install in a reservoir simulator. This model for oil-water coning differs from others presented previously in two respects. First, presented previously in two respects. First, partial completion that does not necessarily extend partial completion that does not necessarily extend to the top of the formation is treated. Second, an effective radius that allows for vertical flow resistance is introduced. DESCRIPTION OF MODEL ASSUMPTIONS The geometric configuration for the coning model is a radially symmetric, homogeneous, anisotropic system with inflow at the outer boundary and with a partially perforated well. The fluid distribution is shown in Fig. 1. The presence of initial bottom water at 100-percent water saturation is considered. The perforated interval is assumed to be within the original oil column. The fluids are assumed to be incompressible. The model will be developed in steady-state flow. It is shown in Ref. 6 that the transient time for the start of flow is short for most practical problems and, thus, the rise of the cone can be represented as a succession of steady states. The fluids are assumed to flow in segregated regions, as shown in Fig. 1. The fractional flow into the perforated interval is assumed to be only a function of the fraction of the interval covered by each fluid and of the mobility ratio. The fluids are assumed to be in vertical equilibrium everywhere except near the wellbore. The departure from vertical equilibrium near the well caused by the vertical flow resistance is represented by an "effective radius." The expression for the effective radius represents the anisotropy through the vertical-to-horizontal permeability ratio. permeability ratio.The fluid flow equations are linearized by assuming that the average oil-column thickness over the drainage area can be used to compute the vertically averaged relative-permeability functions for the entire drainage area. SPEJ P. 65

Eleven companies participated in the Second Comparative Solution Project. The problem to be solved is a three-phase coning problem that can be described as a radial cross section with one central producing well. The oil and water densities are nearly equal, so the oil/water capillary transition zone extends high up into the oil column. Wide variations in rates occur, and the solution GOR is unusually high for oil with such high density. These problem characteristics make the problem difficult to solve, thus increasing its value as a test of simulation techniques. Various aspects of the numerical solutions obtained are compared in this paper. In general, the solutions agree reasonably well.

A reservoir simulation system uses an analytical model to represent flow within a grid block as it enters or leaves a well. This model is called a well model. We give a description here of the theoretical background of a well model, including how the sandface pressure and saturation boundary conditions can be calculated and how the well boundary itself can be replaced (approximately) by a source function. This paper and the following companion paper, SPE 9770, present a unified viewpoint of material, some of which may be already familiar to simulator developers. Introduction Our concern in this paper is the theory of representation of wells and the well boundary condition in a reservoir simulator.It frequently has been noted that, except in the case of a central well in a problem involving cylindrical coordinates, it is impractical to represent a well with an internal boundary. The ratio of well radius to desired grid-block length can be of order 0.001 or less. In such cases, an alternative procedure has evolved in which the well is represented by a source. The relationship between the source strength, the wellbore flow, and the flow in the surrounding grid blocks composes an essential part of the well model. Even when the grid around a well is sufficiently fine to represent the well as an internal boundary, other features such as partial perforation, partial penetration, or skin may be important to the local flow but extend over a "small" interval in relation to the appropriate grid-block dimension. Here also, a suitable source representation is advantageous. We shall develop the source representation of a well for a variety of circumstances.The well boundary condition generally involves the sandface pressure and flow rate. However, these quantities also must be consistent with the requirements of wellbore flow - i.e., reservoir and wellbore flows are coupled, and a wellbore flow model is required. We describe a means of treating a wide variety of wellbore flows without creating a numerically cumbersome simulator. We hope that this paper may provide a basis for further work and discussion of this essential topic. Review of Literature The source representation of a well can be described as a local, approximate, steady, singular solution of the flow equations. The idea of separating a singularity of this type for special treatment is an old idea in applied mathematics. In series solutions to certain elliptic and parabolic equations, it was found that the convergence of the series could be improved considerably by first extracting the singular part. In these cases the singular solution extended through the entire domain. The analogous approach using numerical methods in place of the series solution is also well known. The use of singular solutions in a purely local role in numerical solutions was introduced before the general use of digital computers. Woods' use of a local logarithmic expression in a solution of Poisson's equation by relaxation methods corresponds closely to the source representation of a well recently proposed by Peaceman. SPEJ P. 323＾