PSEUDO-STEADY-STATE PRODUCTIVITY FORMULA FOR A PARTIALLY PENETRATING VERTICAL WELL IN A CIRCULAR CYLINDER RESERVOIR

This paper presents steady state productivity equations for a fully penetrating vertical well in the following three anisotropic systems: (a) sector fault, (b) channel, and (c) rectangular reservoir using a uniform line sink model. The new equations, which are based on conformal mapping method, are simple, accurate, and easy to use in field practice. If the well is in a sector fault reservoir, the productivity is a function of the angle of the sector, wellbore location angle, off-vertex distance, and drainage radius. If the well is in a channel reservoir with two parallel impermeable lateral boundaries, well flow rate reaches a maximum value when the well is located in the middle of the channel width. If the well is in a rectangular reservoir with constant pressure lateral boundaries, a new equation is provided to calculate the productivity of the well arbitrarily located in the anisotropic reservoir for the case where the flow rate of an off-center well is bigger than that of a centered well. It is concluded that, for a vertical well, different steady state productivity equations should be used in different reservoir geometries.

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A general method for solving steady-state internal gravity wave problems

Journal of Fluid Mechanics ◽

10.1017/s0022112072002629 ◽

1972 ◽

Vol 56 (4) ◽

pp. 721-740 ◽

Cited By ~ 13

Author(s):

D. G. Hurley

Keyword(s):

Steady State ◽

Circular Cylinder ◽

Gravity Waves ◽

Internal Gravity Wave ◽

Internal Gravity Waves ◽

Two Dimensional ◽

Wave Problem ◽

Wave Problems ◽

General Method ◽

Stratified Liquid

The paper describes a simple but general method for solving 'steady-state’ problems involving internal gravity waves in a stably stratified liquid. Under the assumption that the motion is two-dimensional and that the Brunt-Väisälä frequency is constant, the method is used to re-derive in a very simple way the solutions to problems where the boundary of the liquid is either a wedge or a circular cylinder. The method is then used to investigate the effect that a model of the continental shelf has on an incident train of internal gravity waves. The method involves analytic continuation in the frequency of the disturbance, and may well prove to be effective for other types of wave problem.

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Pseudo-Steady-State Productivity Formula for a Partially Penetrating Vertical Well in a Box-Shaped Reservoir

Mathematical Problems in Engineering ◽

10.1155/2010/907206 ◽

2010 ◽

Vol 2010 ◽

pp. 1-35 ◽

Cited By ~ 2

Author(s):

Jing Lu ◽

Djebbar Tiab

Keyword(s):

Steady State ◽

Dimensional Space ◽

Three Dimensional ◽

Neumann Boundary ◽

Vertical Well ◽

Homogeneous Neumann Boundary Condition ◽

Arbitrary Position ◽

Dirac Function ◽

Flow Boundaries ◽

Three Dimensional Space

For a bounded reservoir with no flow boundaries, the pseudo-steady-state flow regime is common at long-producing times. Taking a partially penetrating well as a uniform line sink in three dimensional space, by the orthogonal decomposition of Dirac function and using Green's function to three-dimensional Laplace equation with homogeneous Neumann boundary condition, this paper presents step-by-step derivations of a pseudo-steady-state productivity formula for a partially penetrating vertical well arbitrarily located in a closed anisotropic box-shaped drainage volume. A formula for calculating pseudo skin factor due to partial penetration is derived in detailed steps. A convenient expression is presented for calculating the shape factor of an isotropic rectangle reservoir with a single fully penetrating vertical well, for arbitrary aspect ratio of the rectangle, and for arbitrary position of the well within the rectangle.

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Pressure Drop Equations for a Partially Penetrating Vertical Well in a Circular Cylinder Drainage Volume

Mathematical Problems in Engineering ◽

10.1155/2009/267964 ◽

2009 ◽

Vol 2009 ◽

pp. 1-33

Author(s):

Jalal Farhan Owayed ◽

Jing Lu

Keyword(s):

Pressure Drop ◽

Circular Cylinder ◽

Constant Pressure ◽

Dimensional Space ◽

Outer Boundary ◽

Superposition Principle ◽

Vertical Well ◽

Drainage Volume ◽

Partial Penetration ◽

Line Sink

Taking a partially penetrating vertical well as a uniform line sink in three-dimensional space, by developing necessary mathematical analysis, this paper presents unsteady-state pressure drop equations for an off-center partially penetrating vertical well in a circular cylinder drainage volume with constant pressure at outer boundary. First, the point sink solution to the diffusivity equation is derived, then using superposition principle, pressure drop equations for a uniform line sink model are obtained. This paper also gives an equation to calculate pseudoskin factor due to partial penetration. The proposed equations provide fast analytical tools to evaluate the performance of a vertical well which is located arbitrarily in a circular cylinder drainage volume. It is concluded that the well off-center distance has significant effect on well pressure drop behavior, but it does not have any effect on pseudoskin factor due to partial penetration. Because the outer boundary is at constant pressure, when producing time is sufficiently long, steady-state is definitely reached. When well producing length is equal to payzone thickness, the pressure drop equations for a fully penetrating well are obtained.

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Steady-State Temperature Solution for a Heat-Generating Circular Cylinder Cooled by a Ring of Holes

Journal of Heat Transfer ◽

10.1115/1.3688737 ◽

1964 ◽

Vol 86 (4) ◽

pp. 531-536 ◽

Cited By ~ 3

Author(s):

J. C. Rowley ◽

J. B. Payne

Keyword(s):

Heat Conduction ◽

Steady State ◽

Circular Cylinder ◽

Heat Conduction Equation ◽

Series Solution ◽

Concentric Circle ◽

Optimum Location ◽

Shape Factors ◽

Steady State Temperature ◽

Temperature Solution

A series solution is presented to the heat-conduction equation for a heat-generating circular cylinder pierced axially by a ring of equal holes spaced uniformly on a concentric circle. The solution is based upon a class of potential functions previously defined by Howland. Typical nondimensional curves for peak and average temperatures and optimum location of the ring of holes for uniform convective cooling are presented. In addition, some shape factors or conductances for the geometry are given.

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The Effects of Reservoir Area Extent on the Performance of a Vertical Well in a Reservoir Subject to Bottom Water Drive

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.752-753.790 ◽

2015 ◽

Vol 752-753 ◽

pp. 790-795

Author(s):

I. Eiroboyi ◽

P.O. Obeta

Keyword(s):

Steady State ◽

Bottom Water ◽

Maximum Pressure ◽

Large Area ◽

Pressure Derivative ◽

Type Curves ◽

Reservoir Area ◽

Vertical Well ◽

Flow Equations ◽

Dimensionless Pressure

Reservoir performance can be understood from system type curves. The type curve gives vivid information about maximum pressure drops, magnitude of near wellbore effects, reservoir fluid and wellbore properties needed to ascertain the strength of available drive mechanism, maximum withdrawal rates and remaining fluid in real time. This paper investigates the effects of reservoir area extent on the performance of a reservoir, subject to active bottom water, when it is completed with a vertical well. Type curves of dimensionless pressures and dimensionless pressure derivatives were produced for various dimensionless values of area extent of the reservoir. These type curves were developed from solutions to flow equations using relevant source and Green’s functions. From the results, it can be observed that the larger the reservoir area extent, the larger the dimensionless pressure drop, the longer the time it takes to attain steady state. This is validated from the pressure derivative curve, which shows that reservoirs with large area extent are characterized by longer period of radial flow and subsequently delay in the attainment of steady state, thus prolonging the arrival of bottom water.

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