Potential Distribution Around Sources With Finite Length (Horizontal and Vertical Partially Penetrating wells and Fractures) Part I: Steady State Fluid Flow

Azar-Nejad, F.; Tortike, W.S.; S.M., Farouq

doi:10.2523/35270-ms

Cited by 3 publications

(2 citation statements)

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“…It is wellknown from Unified fracture design 14 that the dimensionless productivity index of such a fracture is J D = 0.467. In the DVS method The fracture has been divided into various number segments (1,2,4,8,16, and 32) and the J D was calculated. Table 4 shows the results.…”

Section: Development Of Solution For Finite Conductivity Source With mentioning

confidence: 99%

“…The response of the system to a continuous source (production or injection) is obtained by numerically integrating the response of instantaneous source over time as it has been discussed, for instance, by AzarNejad et al 2,3 , The aim of this work is to: 1. Develop the analytical solution of the box-in-box system to an instantaneous source.…”

Section: Introductionmentioning

confidence: 99%

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The Method of Distributed Volumetric Sources for Calculating the Transient and Pseudosteady-State Productivity of Complex Well-Fracture Configurations

Amini

2007

Proceedings of SPE Hydraulic Fracturing Technology Conference

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TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThe method of Distributed Volumetric Sources (DVS) is developed to solve problems of transient and pseudo steady state fluid flow. The basic building block of the method comprises calculation of the analytical response of a rectilinear reservoir with closed outer boundaries to an instantaneous volumetric source, also shaped as a rectilinear body. The solution also provides the well-testing derivative of the response to a continuous source in analytical form. This can be integrated over time to provide the pressure response to a continuous source. For production engineering applications, we cast the results into a transient/pseudo-steady productivity index form. The new method is validated through comparison to results of some of the well known well-testing solutions for simple configurations such as vertical wells with full and partial penetration, horizontal wells with unform flux and infinite conductivity, and fractured wells with uniform flux, finite or infinte conductivity. The results show a very good agreement with the existing models. The main advantage of the new solution is its applicability over the more complex fracture/well configurations, some of which is studied in the paper.The new method has shown to provide a fast, robust, and reliable way to pressure transient analysis, and well performance prediction whenever complex well/fracture configuration is considered.

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Section: Development Of Solution For Finite Conductivity Source With mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Method of Distributed Volumetric Sources for Calculating the Transient and Pseudosteady-State Productivity of Complex Well-Fracture Configurations

Amini

2007

Proceedings of SPE Hydraulic Fracturing Technology Conference

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Water and Gas Coning in the Oil Reservoirs With Steady State and Transient Flow (Part I: Vertical Wells)

Azar-Nejad

Tortike

Ali

1996

Permian Basin Oil and Gas Recovery Conference

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This paper examines fluid flow mechanism in an infinite rectilinear reservoir with bottom water and/or gas cap. The water-oil contact (WOC) is treated as a no-flow moving boundary while the initial WOC is kept at a constant pressure. The potential distribution around the partially penetrating wellbore is a solution of Laplace or diffusivity equation with uniform potential inner boundary condition (IBC). A new dimensionless variable is introduced. Dimensionless Density, as a function of reservoir geometry, reservoir anisotropy, fluid properties, production rate and the difference of the densities of two fluids. This paper reports the existence of a Critical Dimensionless Density, CDD, above which cone is stable. The results of this study show that for large values of Dimensionless Density the cone height is very small and the well pressure response is almost identical to that of the reservoirs without bottom water or a gas cap. Therefore for reservoirs with a high Dimensionless Density a solution of no-flow boundary at the initial WOC level seems more accurate with respect to solution with constant potential boundary. CDD indicates thatin anisotropic reservoirs, smaller the vertical permeability the larger is the critical production rate or time of breakthrough,Heavy oils, at equivalent conditions, have smaller critical production rate or time of the breakthrough due to higher viscosity. Thus thermal processes such as steam stimulation reduce the cone height and consequently increase the time of the breakthrough.

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Potential Distribution Around Sources With Finite Length Horizontal, Vertical Partially Penetrating Wells and Fractures Part II: Transient Flow

Azar-Nejad

Tortike

Ali

1996

SPE Mid-Continent Gas Symposium

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The potential distribution around a partially penetrating source is a solution of the diffusivity equation. Presenting a new method "Discrete Flux Element" (DFE) this paper provides the solution to the diffusivity equation with uniform potential as well as uniform flux inner boundary conditions (IBC). This solution provides the ability for modeling any sources with arbitrary geometry. The pressure derivative equations have been derived for uniform potential as well as uniform flux inner boundary conditions. The pressure derivative reflects the wellbore length and the distance of a horizontal well to the no-flow boundary. Introduction The potential distribution due to any sources is a solution of the diffusivity equation. A solution can easily be found for the sources that are fully penetrated and perpendicular to the no-flow boundaries. However, there are many cases where these conditions do not exist, such as horizontal wells, vertical partially penetrating wells and fractures. In this paper sources of this type are called Sources with Finite Length (SFL). A simple mathematical equation does not exist for SFL. Sources that are called horizontal or vertical do not always consist of straight line, rather they can have irregular geometry. Approximating the geometry of the sources with a straight line sometimes could cause error especially when dealing with pressure analysis. This paper presents a new analytical method, called here Discrete Flux Element (DFE). Because of the nature of the method it is possible to take the geometry of the sources into account without losing accuracy. The potential due to a source is to be obtained by integrating a point source solution over the length of the source. This integration is possible by the assumption of uniform flux IBC and straight configuration. Both uniform flux and uniform potential IBC cannot be satisfied at the same time. Muskat studied the potential distribution under steady state conditions due to a vertical partially penetrating well. He showed that a uniform flux IBC solution calculated at a certain point is identical to the solution of a uniform potential IBC. Ref. 3 for a fracture and Refs. 4 through 6 for a horizontal well, using a uniform potential solution at an equivalent pressure point. simulated a uniform potential IBC (infinite conductivity) solution under unsteady state conditions. In this paper we show that:the equivalent pressure point moves in time and only at late times this point will be stabilized,even at late times the location of the equivalent pressure point does not have a unique value for wellbores with different lengths,the pressure derivative with respect to time at this point does not represent the pressure derivative of the uniform potential IBC. In this study we use potential instead of pressure to include the gravity. DFE Method provides the ability to study the problems, such as coning, where gravity plays an important role. Azar-Nejad and Tortike and Farouq Ali applying DFE Method studied the reservoirs with bottom water or a gas cap. Refs. 7 and 8 showed the ability of the DEF Method for solving a moving boundary problem. P. 131

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Potential Distribution Around Sources With Finite Length (Horizontal and Vertical Partially Penetrating wells and Fractures) Part I: Steady State Fluid Flow

Abstract: TX 75083-3836, U.S. A., fax 01·214-952-9435. Abstract

Cited by 3 publications

References 4 publications

The Method of Distributed Volumetric Sources for Calculating the Transient and Pseudosteady-State Productivity of Complex Well-Fracture Configurations

The Method of Distributed Volumetric Sources for Calculating the Transient and Pseudosteady-State Productivity of Complex Well-Fracture Configurations

Water and Gas Coning in the Oil Reservoirs With Steady State and Transient Flow (Part I: Vertical Wells)

Potential Distribution Around Sources With Finite Length Horizontal, Vertical Partially Penetrating Wells and Fractures Part II: Transient Flow

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