Constant-Rate Drawdown Solutions Derived for Multiple Arbitrarily-Oriented Uniform-Flux, Infinite-Conductivity, or Finite-Conductivity Fractures in an Infinite-Slab Reservoir

2006 ◽  
Author(s):  
David P. Craig ◽  
Thomas Alwin Blasingame
Keyword(s):  
Finite Conductivity ◽  
Constant Rate ◽  
Infinite Slab ◽  
Infinite Conductivity

RESPONSE OF THIN DYKE TO OSCILLATING DIPOLE

Geophysics ◽  
1958 ◽  
Vol 23 (1) ◽  
pp. 134-143 ◽  
Author(s):  
James Paul Wesley
Keyword(s):  
Exact Solution ◽  
Magnetic Dipole ◽  
Analytical Continuation ◽  
Dipole Source ◽  
Finite Conductivity ◽  
First Approximation ◽  
Sulfide Ore ◽  
Infinite Slab ◽  
Infinite Conductivity

A dyke of sulfide ore may be geophysically prospected by observing its response to a slowly oscillating magnetic dipole source. To a first approximation the field produced by a thin dyke is given by a dyke of infinite conductivity and vanishing thickness in a vacuum (Wesley, 1958, Geophysics, v. 23, p. 128). In order to identify the ore and to estimate the size of the deposit, it is necessary to consider further approximations involving the conductivity and thickness of the dyke. By a type of analytical continuation an approximation is found which agrees both with the exact solution for a dyke of infinite conductivity and vanishing thickness and with the exact solution (approximated only for ωσ large and also for σ small) for an infinite slab of finite conductivity and nonvanishing thickness, the dyke appearing as an infinite slab when both source and observer are near the dyke but far removed from the edge. The solution is very good provided the dyke is geometrically thin.

Effect of Non-Darcy Flow on the Constant-Pressure Production of Fractured Wells

1981 ◽  
Vol 21 (03) ◽  
pp. 390-400 ◽  
Author(s):  
K.H. Guppy ◽  
Heber Cinco-Ley ◽  
Henry J. Ramey
Keyword(s):  
Constant Pressure ◽  
Darcy Flow ◽  
Flow In Porous Media ◽  
Graphical Form ◽  
Finite Conductivity ◽  
Fracture Conductivity ◽  
Type Curves ◽  
Pressure Behavior ◽  
Apparent Conductivity ◽  
Constant Rate

Abstract In many low-permeability gas reservoirs, producing a well at constant rate is very difficult or, in many cases, impossible. Constant-pressure production is much easier to attain and more realistic in practice. This is seen when production occurs into a constant-pressure separator or during the reservoir depletion phase, when the rate-decline period occurs. Geothermal reservoirs, which produce fluids that drive backpressure turbines, and open-well production both incorporate the constant-pressure behavior. For finite-conductivity vertically fractured systems, solutions for the constant-pressure case have been presented in the literature. In many high-flow-rate wells, however, these solutions may not be useful since high velocities are attained in the fracture, which results in non-Darcy effects within the fracture. In this study, the effects of non-Darcy flow within the fracture are investigated. Unlike the constant-rate case, it was found that the fracture conductivity does not have a constant apparent conductivity but rather an apparent conductivity that varies with time. Semianalytical solutions as well as graphical solutions in the form of type curves are presented to illustrate this effect. An example is presented for analyzing rate data by using both solutions for Darcy and non-Darcy flow within the fracture. This example relies on good reservoir permeability from prefracture data to predict the non-Darcy effect accurately. Introduction To fully analyze the effects of constant-bottomhole-pressure production of hydraulically fractured wells, it is necessary that we understand the pressure behavior of finite-conductivity fracture systems producing at constant rate as well as the effects of non-Darcy flow on gas flow in porous media. Probably one of the most significant contributions in the transient pressure analysis theory for fractured wells was made by Gringarten et al.1,2 In the 1974 paper,2 general solutions were made for infinite-conductivity fractures. Cinco et al.3 found a more general solution for the case of finite-conductivity fractures and further extended this analysis in 1978 to present a graphical technique to estimate fracture conductivity.4 For the case of constant pressure at the wellbore, solutions were presented in graphical form by Agarwal et al.5 In his paper, a graph of log (1/qD) vs. log (tDxf) can be used to determine the conductivity of the fracture by using type-curve matching. Although such a contribution is of great interest, unique solutions are difficult to obtain. More recently, Guppy et al.6 showed that the Agarwal et al. solutions may be in error and presented new type curves for the solution to the constant-pressure case assuming Darcy flow in the fracture. That paper developed analytical solutions which can be applied directly to field data so as to calculate the fracture permeability-width (kfbf) product.

Stabilized Productivity of a Hydraulically Fractured Well Producing at Constant Pressure

SPE Journal ◽  
2006 ◽  
Vol 11 (01) ◽  
pp. 120-131 ◽  
Author(s):  
Jacques Hagoort
Keyword(s):  
Flow Conditions ◽  
Productivity Improvement ◽  
Fracture Conductivity ◽  
Fracture Length ◽  
State Flow ◽  
Production Improvement ◽  
Constant Rate ◽  
Infinite Conductivity ◽  
State Production ◽  
Fractured Well

Summary This paper describes a simple and easy-to-construct numerical model for the calculation of the stabilized productivity of a hydraulically fractured well producing at a constant well pressure. The model takes into account both Darcy and non-Darcy pressure losses in the fracture. Dimensionless charts are presented that illustrate productivity improvement as a function of fracture length, fracture conductivity, and non-Darcy flow. For dimensionless fracture lengths in excess of 0.2, constant-pressure productivities are significantly lower than constant-rate productivities as predicted, for example, by the McGuire-Sikora productivity improvement chart. The maximum difference is 20% for an infinite-conductivity fracture with a length of unity. Both fracture conductivity and non-Darcy flow adversely affect well productivity; the reduction in productivity is larger for longer fractures. Introduction The productivity of a well is commonly expressed by a productivity index defined as the ratio of production rate and difference between average reservoir pressure and well pressure. Stabilized productivity refers to production from a well in the semisteady-state flow regime (i.e., the regime beyond the initial transient regime), during which flow in the reservoir is dominated by the reservoir boundaries. In the past, most studies on the stabilized productivity of hydraulically fractured wells were about steady-state production or semisteady-state production at a constant rate. As we shall demonstrate in this paper, the type of well boundary condition has a significant effect on productivity, especially for long fractures. For production by pressure depletion, characterized by declining production rates, constant well pressure is a more appropriate boundary condition. In the late 1950s, McGuire and Sikora (1960) presented a productivity improvement chart for fully penetrating fractured wells producing at a constant rate under semisteady-state flow conditions based on electrical analog model experiments. The chart shows production improvement vs. fracture conductivity for various fracture lengths. The McGuire-Sikora chart is a classic in the fracturing literature and is being used to this day. In the early 1960s, Prats (1961) presented a theoretical study on the productivity of a fully penetrating fractured well under steady-state flow conditions. He showed that the effect of a fracture can be represented by an apparent or effective wellbore radius, which depends on fracture length and fracture conductivity. For fractures that are relatively small and have an infinite conductivity, the effective wellbore radius is equal to half the fracture half-length. In a follow-up study, Prats et al. (1962) demonstrated that this result also holds for stabilized flow of a slightly compressible liquid. In the mid-1970s, Holditch presented a production improvement chart (included in Lee 1989) based on experiments with a numerical reservoir simulator, which essentially confirmed the earlier results of McGuire and Sikora. Although based on production at constant rate, the McGuire-Sikora and Holditch charts are also being used for production at declining production rates (Lee 1989).

Transient Flow in Elliptical Systems

1979 ◽  
Vol 19 (06) ◽  
pp. 401-410 ◽  
Author(s):  
Fikri Kucuk ◽  
William E. Brigham
Keyword(s):  
Porous Medium ◽  
Line Source ◽  
Vertical Fracture ◽  
Constant Rate ◽  
Flow Geometry ◽  
Elliptical Flow ◽  
Long Time ◽  
Infinite Conductivity ◽  
Fractured Well ◽  
Anisotropic Reservoirs

Abstract This study presents analytical solutions to elliptical flow problems that are applicable to infinite-conductivity vertically fractured wells, elliptically shaped reservoirs, and anisotropic reservoirs producing at a constant rate or pressure. Type curves and tables are presented for the dimensionless flow rate and the dimensionless wellbore pressure for various inner boundary conditions ranging from K = 1 1, which corresponds to a circle, to K =, which corresponds to a vertical fracture. For elliptical reservoirs, K is the ratio of the major to minor axes of the inner boundary ellipse; for anisotropic reservoirs, it is the square root of the ratio of maximum to minimum permeabilities. Introduction Flow in a homogeneous and isotropic porous medium usually will be radial or linear, depending on the shape of the boundary. But in the area surrounding a vertical fracture, an anisotropic formation, or an aquifer with an elliptical inner boundary, flow will be elliptical.The study of elliptical flow in porous media is more recent than the usual radial and linear flow studies, but even elliptical flow studies date back at least several decades. The earliest discussion of steady-state elliptical flow usually is attributed to Muskat. He presented a steady-state analytical solution for the now from a finite-length line source into an infinitely large reservoir.One of the classic papers on elliptic flow by Prats et al. considered flow of compressible fluids from a vertically fractured well in a closed elliptical reservoir producing at a constant pressure. Prats et al. also producing at a constant pressure. Prats et al. also presented a solution for long times for the presented a solution for long times for the constant-rate case.Gringarten et al. found that older studies by Russell and Truitt (where flow is to a vertically fractured well) are unsuitable for short-time analysis. Gringarten et al. presented analytical solutions for fractures with infinite conductivity and with uniform flux. These solutions were for both closed squares and infinite reservoirs produced at a constant rate.In the last few years considerable work has been done on fracture systems, including numerical solutions and a semianalytical solutions for both finite and infinite fracture conductivities. Most of these studies, however, have not used the concept that the fracture is an elliptical flow system. Nevertheless, the results they obtain are important for well testing.Another problem related to elliptical flow is flow through an anisotropic porous medium. For this problem, a line source solution and a long-time problem, a line source solution and a long-time approximation presented by Earlougher are available for the constant-rate case.The purpose of this paper is to study elliptical flow in a broad sense with regard to reservoir engineering problems and to see whether these problems can be problems and to see whether these problems can be solved and whether elliptical problems can be handled in a unified, consistent manner. Development of Elliptical Flow Models The flow from an isotropic and homogeneous medium to a map usually will be radial, but lack of homogeneity will distort the radial flow geometry. In particular, flow will be elliptical through a porous particular, flow will be elliptical through a porous medium with directional permeability distribution (simple anisotropy). The inner geometry of a well also can distort radial flow geometry. For example, the flow will be elliptical if the well has an infinite-conductivity vertical fracture. Elliptical flow also will be encountered in flow from an aquifer to a reservoir that has an elliptical boundary at the oil/water contact. SPEJ P. 401

scholarly journals CONFORMAL MODELS OF MAGNETOHYDRODYNAMIC TURBULENCE

1996 ◽  
Vol 11 (29) ◽  
pp. 5261-5277 ◽  
Author(s):  
OMDUTH COCEAL ◽  
STEVEN THOMAS
Keyword(s):  
Three Dimensional ◽  
Approximate Solutions ◽  
Velocity Fields ◽  
Point Of View ◽  
Three Dimensions ◽  
Finite Conductivity ◽  
Magnetohydrodynamic Turbulence ◽  
Infinite Conductivity ◽  
Conformal Models

Following the previous work of Ferretti and Yang on the role of magnetic fields in the theory of conformal turbulence, we show that nonunitary minimal model solutions to two-dimensional magnetohydrodynamics (MHD) obtained by dimensional reduction from three dimensions exist under different (and more restrictive) conditions. From a three-dimensional point of view, these conditions are equivalent to perpendicular flow, in which the magnetic and velocity fields are orthogonal. We extend the analysis to the finite conductivity case and present some approximate solutions, whose connection with the exact ones of the infinite conductivity case is also discussed.

The Rayleigh—Jeffreys problem with boundary slab of finite conductivity

1968 ◽  
Vol 32 (2) ◽  
pp. 393-398 ◽  
Author(s):  
D. A. Nield
Keyword(s):  
Finite Thickness ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Wave Number ◽  
Linear Perturbation ◽  
Finite Conductivity ◽  
Solid Layer ◽  
Onset Of Convection ◽  
Infinite Conductivity ◽  
Bounded Below

Linear perturbation analysis is applied to the problem of the onset of convection in a horizontal layer of fluid heated uniformly from below, when the fluid is bounded below by a rigid plate of inlinite conductivity and above by a solid layer of finite conductivity and finite thickness. The critical Rayleigh number and wave-number are found for various thickness ratios and thermal conductivity ratios. Both numbers are reduced by the presence of a boundary of finite (rather than infinite) conductivity in qualitative agreement with the observation of Koschmieder (1966).

ON THE KELVIN–HELMHOLTZ PROBLEM IN MAGNETOHYDRODYNAMICS WITH FINITE CONDUCTIVITY

1965 ◽  
Vol 43 (7) ◽  
pp. 1342-1346
Author(s):  
R. A. Wentzell
Keyword(s):  
Magnetic Field ◽  
Finite Conductivity ◽  
Type Problem ◽  
Horizontal Magnetic Field ◽  
Helmholtz Problem ◽  
Infinite Conductivity ◽  
The Magnetic Field

A study has been made of the effect of large but finite conductivity upon Kelvin–Helmholtz-type instabilities in the presence of a horizontal magnetic field. For finite conductivity, the selected Kelvin–Helmholtz-type problem which is stable at infinite conductivity is stable no matter how large the magnetic field. These instabilities grow aperiodically at a rate proportional to (resistivity)[Formula: see text].

scholarly journals On the Cylindrical Probe Method of Measuring Thermal Conductivity with Special Reference to Soils. I. Extension of Theory and Discussion of Probe Characteristics

1958 ◽  
Vol 11 (2) ◽  
pp. 255 ◽  
Author(s):  
DA de Vries ◽  
AJ Peck
Keyword(s):  
Thermal Conductivity ◽  
Contact Resistance ◽  
Special Reference ◽  
Line Source ◽  
Thermal Contact Resistance ◽  
Thermal Contact ◽  
Complete Agreement ◽  
Finite Conductivity ◽  
Cylindrical Probe ◽  
Infinite Conductivity

The theory of cylindrical probes for measuring thermal conductivity is extended to the case of a probe of finite conductivity containing a line source at its centre. This provides a more realistic approximation to most actual probes than the theory for a probe of infinite conductivity developed by other authors. New experimental results are presented which are in complete agreement with theory It is shown how an estimate can be obtained of the magnitude of a possible thermal contact resistance between the probe and the medium and how its influence on the measured conductivity can be assessed.

scholarly journals Productivity Analysis of Volume Fractured Wells under Different Working Systems

Geofluids ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Hongfei Ma ◽  
Wenqi Zhao ◽  
Meng Sun ◽  
Xiaodong Wang ◽  
Lun Zhao ◽  
...  
Keyword(s):  
Prediction Models ◽  
Finite Conductivity ◽  
Productivity Analysis ◽  
Well Performance ◽  
Fracture Conductivity ◽  
Initial Flow ◽  
Stimulated Reservoir Volume ◽  
Constant Rate ◽  
Wellbore Pressure

The volume fracturing technique has been widely used to improve the productivity of ultralow-permeability reservoirs. This paper presents a new semianalytical model to simulate the pressure transient and production behaviour of finite conductivity vertical fractured wells with stimulated reservoir volume (SRV) in heterogeneous reservoirs. The model is based on the five-linear flow model, the Warren-Root model, and fracture conductivity influence function. The model is validated by comparing its results with a numerical model. One novelty of this model is its consideration of three different kinds of production prediction models. Constant rate, constant pressure, and compound working systems are taken into account. This paper illustrates the effects of the SRV size and shape, mobility ratio, initial flow rate, limiting wellbore pressure, and hydraulic fracture parameters under different working systems. Results show that the SRV and parameters of fractures have a significant influence on long-term well performance. Moreover, the initial rate can extend the constant rate period by 418%, and limiting wellbore pressure can effectively improve the cumulative recovery rate by 23%. Therefore, this model can predict long-term wells’ behaviour and provide practical guiding significance for hydraulic fracturing design.

THE EFFECT OF FINITE CONDUCTIVITY ON GRAVITY WAVES IN A HORIZONTAL MAGNETIC FIELD

1965 ◽  
Vol 43 (4) ◽  
pp. 645-652 ◽  
Author(s):  
R. A. Wentzell ◽  
J. H. Blackwell
Keyword(s):  
Magnetic Field ◽  
Electrical Conductivity ◽  
Gravity Waves ◽  
Gravitational Force ◽  
Finite Conductivity ◽  
Horizontal Magnetic Field ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Infinite Conductivity ◽  
Mode Solution

A study has been made of the behavior of the plane interface between a vacuum and an electrically conducting fluid subject to a normal gravitational force and a magnetic field parallel to the interface. The system is examined for perturbations which bend the lines of force, without restriction to the extensively used idealization of infinite electrical conductivity. The eigenvalue spectra obtained, which are surprisingly different from the simpler ones corresponding to infinite conductivity, are examined by approximate and numerical techniques over the complete range of electrical conductivity from infinity to zero. The disappearance of a normal mode solution above a critical value of conductivity is an interesting feature of the effect of finite conductivity on magnetohydrodynamic stability.

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