On the Liquid-Flow Analog To Evaluate Gas Wells Producing in Shales

2013 ◽  
Vol 16 (02) ◽  
pp. 209-215 ◽  
Author(s):  
C.. Chen ◽  
R.. Raghavan
Keyword(s):  
Liquid Flow ◽  
Horizontal Well ◽  
Numerical Solutions ◽  
Transient Flow ◽  
Constant Pressure ◽  
Hydraulic Fractures ◽  
2D System ◽  
Infinite Conductivity ◽  
Linear Reservoir ◽  
Fractured Well

Summary Drawing on links to the analog considered by Al-Hussainy et al. (1966), we present a corresponding analog to correlate solutions for a fractured well producing at a constant pressure. A solution in terms of the similarity transformation for the pressure distribution in a linear reservoir filled with a real gas provides the basis. This solution is particularly suited to demonstrate that anomalous results will be obtained when long linear-flow trends typical of shales produced through a horizontal well consisting of multiple, infinite-conductivity fractures are evaluated in classical terms. The basis for the liquid-flow analog is re-examined by considering 2D numerical solutions for a fractured well producing a gas reservoir at a constant pressure. A method to correlate the nonlinear solutions with the corresponding liquid-flow solutions for fractured wells producing at a constant pressure during the infinite-acting period is provided. The phrase “analog” used here represents attempts to match values of both the well response and its derivative for a 2D system during transient flow. This correlation enables analysts to obtain estimates that are accurate in the manner of Al-Hussainy et al. (1966). An example illustrates the application of this recommendation for a horizontal well producing a shale reservoir through multiple hydraulic fractures.

Analysis of Pressure Data for Fractured Wells: The Constant-Pressure Outer Boundary

1978 ◽  
Vol 18 (02) ◽  
pp. 139-150 ◽  
Author(s):  
R. Raghavan ◽  
Nico Hadinoto
Keyword(s):  
Constant Pressure ◽  
Outer Boundary ◽  
Reservoir Pressure ◽  
Pressure Gradients ◽  
Pressure Data ◽  
Pressure Behavior ◽  
Fracture Length ◽  
Infinite Conductivity ◽  
Production Pattern ◽  
Fractured Well

Abstract Analysis of flowing and shut-in pressure behavior of a fractured well in a developed live-spot fluid injection-production pattern is presented. An idealization of this situation, a fractured well located at the center of a constant pressure square, is discussed. Both infinite-conductivity and uniform-flux fracture cases are considered. Application of log-log and semilog methods to determine formation permeability, fracture length, and average reservoir pressure A discussed. Introduction The analysis of pressure data in fractured wells has recovered considerable attention because of the large number of wells bat have been hydraulically fractured or that intersect natural fractures. All these studies, however were restricted to wells producing from infinite reservoirs or to cases producing from infinite reservoirs or to cases where the fractured well is located in a closed reservoir. In some cases, these results were not compatible with production performance and reservoir characteristics when applied to fractured injection wells. The literature did not consider a fractured well located in a drainage area with a constant-pressure outer boundary. The most common example of such a system would be a fractured well in a developed injection-production pattern. We studied pressure behavior (drawdown, buildup, injectivity, and falloff) for a fractured well located in a region where the outer boundaries are maintained at a constant pressure. The results apply to a fractured well in a five-slot injectionproduction pattern and also should be applicable to a fractured well in a water drive reservoir. We found important differences from other systems previously reported. previously reported. We first examined drawdown behavior for a fractured well located at the center of a constant-pressure square. Both infinite-conductivity and uniform-flux solutions were considered. The drawdown solutions then were used to examine buildup behavior by applying the superposition concept. Average reservoir pressure as a function of fracture penetration ratio (ratio of drainage length to fracture length) and dimensionless time also was tabulated. This represented important new information because, as shown by Kumar and Ramey, determination of average reservoir pressure for the constant-pressure outer boundary system was not as simple as that for the closed case since fluid crossed the outer boundary in an unknown quantity during both drawdown (injection) and buildup (falloff). MATHEMATICAL MODEL This study employed the usual assumptions of a homogeneous, isotropic reservoir in the form of a rectangular drainage region completely filled with a slightly compressible fluid of constant viscosity. Pressure gradients were small everywhere and Pressure gradients were small everywhere and gravity effects were neglected. The outer boundary of the system was at constant pressure and was equal to the initial pressure of the system. The plane of the fracture was located symmetrically plane of the fracture was located symmetrically within the reservoir, parallel to one of the sides of the boundary (Fig. 1). The fracture extended throughout the vertical extent of the formation and fluid was produced only through the fracture at a constant rate. Both the uniform-flux and the infinite-conductivity fracture solutions were considered. P. 139

scholarly journals Productivity Formulae of an Infinite-Conductivity Hydraulically Fractured Well Producing at Constant Wellbore Pressure Based on Numerical Solutions of a Weakly Singular Integral Equation of the First Kind

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Chaolang Hu ◽  
Jing Lu ◽  
Xiaoming He
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Constant Pressure ◽  
Numerical Results ◽  
Production Performance ◽  
Weakly Singular ◽  
Wellbore Pressure ◽  
Infinite Conductivity ◽  
Fractured Well

In order to increase productivity, it is important to study the performance of a hydraulically fractured well producing at constant wellbore pressure. This paper constructs a new productivity formula, which is obtained by solving a weakly singular integral equation of the first kind, for an infinite-conductivity hydraulically fractured well producing at constant pressure. And the two key components of this paper are a weakly singular integral equation of the first kind and a steady-state productivity formula. A new midrectangle algorithm and a Galerkin method are presented in order to solve the weakly singular integral equation. The numerical results of these two methods are in accordance with each other. And then the solutions of the weakly singular integral equation are utilized for the productivity formula of hydraulic fractured wells producing at constant pressure, which provide fast analytical tools to evaluate production performance of infinite-conductivity fractured wells. The paper also shows equipotential threads, which are generated from the numerical results, with different fluid potential values. These threads can be approximately taken as a family of ellipses whose focuses are the two endpoints of the fracture, which is in accordance with the regular assumption in Kuchuk and Brigham, 1979.

scholarly journals Production Decline Analysis for Two-Phase Flow in Multifractured Horizontal Well in Shale Gas Reservoirs

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Wei-Yang Xie ◽  
Xiao-Ping Li ◽  
Lie-Hui Zhang ◽  
Xiao-Hua Tan ◽  
Jun-Chao Wang ◽  
...  
Keyword(s):  
Shale Gas ◽  
Horizontal Well ◽  
Flow Model ◽  
Transient Flow ◽  
Two Phase Flow ◽  
Hydraulic Fractures ◽  
Phase Flow ◽  
Gas Reservoirs ◽  
Two Phase ◽  
Shale Gas Reservoirs

After multistage fracturing, the flowback of fracturing fluid will cause two-phase flow through hydraulic fractures in shale gas reservoirs. With the consideration of two-phase flow and desorbed gas transient diffusion in shale gas reservoirs, a two-phase transient flow model of multistage fractured horizontal well in shale gas reservoirs was created. Accurate solution to this flow model is obtained by the use of source function theory, Laplace transform, three-dimensional eigenvalue method, and orthogonal transformation. According to the model’s solution, the bilogarithmic type curves of the two-phase model are illustrated, and the production decline performance under the effects of hydraulic fractures and shale gas reservoir properties are discussed. The result obtained in this paper has important significance to understand pressure response characteristics and production decline law of two-phase flow in shale gas reservoirs. Moreover, it provides the theoretical basis for exploiting this reservoir efficiently.

Modeling Infinite-Conductivity Vertical Fractures With Source and Sink Terms

1983 ◽  
Vol 23 (04) ◽  
pp. 633-644 ◽  
Author(s):  
Long X. Nghiem
Keyword(s):  
Single Phase ◽  
Constant Pressure ◽  
Outer Boundary ◽  
Fracture Plane ◽  
Hydraulic Fractures ◽  
Low Permeability ◽  
Infinite Conductivity ◽  
Source Sink ◽  
Source And Sink ◽  
Vertical Fractures

Nghiem, Long X., SPE, Computer Modeling Group Abstract This paper describes a method for handling infinite-conductivity vertical fractures in reservoir simulation by using source and sink terms. It begins with a review of the concept of source/sink in reservoir simulation, and uses that concept to develop a method for computing the flow into or out of the fracture by assuming elliptical tow and by using the pressures of the blocks surrounding those containing the fracture. The assumption that the flow into the fracture is everywhere perpendicular to the fracture plane (i. e., linear flow) and the effect of the skin factor are also investigated. Test runs showed excellent agreement between computed results and those obtained by analytical and variational methods for single-phase systems. The formulation was extended to multiphase systems. and simulation of a waterflood yielded physically reasonable results. Introduction The simulation of fractured wells is of considerable interest because of the large number of wells that have been hydraulically fractured to increase the injectivity/ productivity in low-permeability formations. Hydraulic productivity in low-permeability formations. Hydraulic fracturing usually yields a vertical fracture plane that intersects the wellbore. Agarwal et al. showed that hydraulic fractures for which the dimensionless fracture flow capacity, FCD, defined as . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) is greater than 500 can be represented by infinite-conductivity vertical fractures. In Eq. 1 kf is the fracture permeability, vi, is the fracture width, k is the formation permeability, and x., is the fracture half-length. permeability, and x., is the fracture half-length. Conventional hydraulic fractures as opposed to massive hydraulic fractures usually fall into this category. Conceptually, an infinite-conductivity vertical fracture (ICVF) has no thickness and no pressure drop along the fracture plane and is represented by a finite line source or sink in an areal representation of the reservoir. Russell and Truitt simulated single-phase flow into an ICVF parallel to a boundary of a square reservoir by using finite differences. The fracture was located symmetrically within a no-flow boundary drainage area and was treated by these authors as a boundary condition. A more general problem was later solved by Gringarten et al., who used the analytical method of source/sink and Green's function proposed by Gringarten and Ramey. Gringarten et al. considered a fracture within a no-flow boundary rectangular reservoir. Using the same analytical method, Raghavan and Hadinoto obtained solutions for a fracture within a constant-pressure outer boundary. In both cases, the fracture was parallel to a boundary of the reservoir. Using Galerkin's method, Bennett et al. investigated the case of a fracture lying along one of the diagonals of a constant-pressure outer boundary square reservoir. The solutions for all these cases are reported in the form of tables and plots of dimensionless pressure drop vs. dimensionless time for different fracture-penetration ratios. This paper presents a method for modeling ICVF's with source/sink terms. The fracture is treated as a singularity and it is assumed that elliptical flow applies in the neighborhood of the fracture. The flow into or out of the fracture is computed from the fracture pressure and the pressures of the blocks surrounding those containing the fracture. SPEJ p. 633

Efficient Localized Nonlinear Solution Strategies for Unconventional-Reservoir Simulation with Complex Fractures

2021 ◽  
Author(s):  
Jiamin Jiang
Keyword(s):  
Transient Flow ◽  
A Priori ◽  
Time Discretization ◽  
Unconventional Reservoirs ◽  
Hydraulic Fractures ◽  
Nonlinear Solution ◽  
Compositional Model ◽  
Simulation Speed ◽  
Flow Mechanisms ◽  
Solution Accuracy

Abstract It is very challenging to simulate unconventional reservoirs efficiently and accurately. Transient flow can last for a long time and sharp solution (pressure, saturation, compositions) gradients are induced because of the severe permeability contrast between fracture and matrix. Although high-resolution models for well and fracture are required to achieve adequate resolution, they are computationally too demanding for practical field models with many stages of hydraulic fracture. The paper aims to innovate localization strategies that take advantage of locality on timestep and Newton iteration levels. The strategies readily accommodate to complicated flow mechanisms and multiscale fracture networks in unconventional reservoirs. Large simulation speed-up can be obtained if performing localized computations only for the solution regions that will change. We develop an a-priori method to exploit the locality, based on the diffusive character of the Newton updates of pressure. The method makes adequate estimate of the active computational gridblock for the next iterate. The active gridblock set marks the ones need to be solved, and then the solution to local linear system is accordingly computed. Fully Implicit Scheme is used for time discretization. We study several challenging multi-phase and compositional model cases with explicit fractures. The test results demonstrate that significant solution locality of variables exist on timestep and iteration levels. A nonlinear solution update usually has sparsity, and the nonlinear convergence is restricted by a limited fraction of the simulation model. Through aggressive localization, the proposed methods can prevent overly conservative estimate, and thus achieve significant computational speedup. In comparison to a standard Newton method, the novel solver techniques achieve greatly improved solving efficiency. Furthermore, the Newton convergence exhibits no degradation, and there is no impact on the solution accuracy. Previous works in the literature largely relate to the meshing aspect that accommodates to horizontal wells and hydraulic fractures. We instead develop new nonlinear strategies to perform localization. In particular, the adaptive DD method produces proper domain partitions according to the fluid flow and nonlinear updates. This results in an effective strategy that maintains solution accuracy and convergence behavior.

scholarly journals Analysis of Flow Behavior for Acid Fracturing Wells in Fractured-Vuggy Carbonate Reservoirs

2018 ◽  
Vol 2018 ◽  
pp. 1-20 ◽  
Author(s):  
Mingxian Wang ◽  
Zifei Fan ◽  
Xuyang Dong ◽  
Heng Song ◽  
Wenqi Zhao ◽  
...  
Keyword(s):  
Composite System ◽  
Numerical Solutions ◽  
Transient Flow ◽  
Flow Behavior ◽  
Outer Region ◽  
Carbonate Reservoirs ◽  
Single Fracture ◽  
Specific Flow ◽  
Type Curves ◽  
Acid Fracturing

This study develops a mathematical model for transient flow analysis of acid fracturing wells in fractured-vuggy carbonate reservoirs. This model considers a composite system with the inner region containing finite number of artificial fractures and wormholes and the outer region showing a triple-porosity medium. Both analytical and numerical solutions are derived in this work, and the comparison between two solutions verifies the model accurately. Flow behavior is analyzed thoroughly by examining the standard log-log type curves. Flow in this composite system can be divided into six or eight main flow regimes comprehensively. Three or two characteristic V-shaped segments can be observed on pressure derivative curves. Each V-shaped segment corresponds to a specific flow regime. One or two of the V-shaped segments may be absent in particular cases. Effects of interregional diffusivity ratio and interregional conductivity ratio on transient responses are strong in the early-flow period. The shape and position of type curves are also influenced by interporosity coefficients, storativity ratios, and reservoir radius significantly. Finally, we show the differences between our model and the similar model with single fracture or without acid fracturing and further investigate the pseudo-skin factor caused by acid fracturing.

Unsteady natural convection in a cavity with internal heating and cooling

1984 ◽  
Vol 140 ◽  
pp. 135-151 ◽  
Author(s):  
John C. Patterson
Keyword(s):  
Steady State ◽  
Natural Convection ◽  
Numerical Solutions ◽  
Transient Flow ◽  
Scaling Analysis ◽  
Steady State Flow ◽  
Internal Heating ◽  
Sources And Sinks ◽  
Heating And Cooling ◽  
Unsteady Natural Convection

The problem of transient natural convection in a cavity of aspect ratio A < 1 driven by internal buoyancy sources and sinks distributed linearly in the horizontal and uniformly in the vertical is considered. Scaling analysis is used to show that a number of possible transient flow regions are possible, collapsing ultimately onto one of conductive, transitional, or convective steady-state flow regimes. A number of numerical solutions are obtained, and their relationships to the scaling analysis are discussed.

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).

Investigation of the Effects of Completion Geometry on Single-Phase Liquid Flow Behavior in Horizontal Wells

2000 ◽  
Vol 123 (2) ◽  
pp. 119-126 ◽  
Author(s):  
Weipeng Jiang ◽  
Cem Sarica ◽  
Erdal Ozkan ◽  
Mohan Kelkar
Keyword(s):  
Liquid Flow ◽  
Horizontal Well ◽  
Single Phase ◽  
Flow Behavior ◽  
Horizontal Wells ◽  
Main Flow ◽  
Test Facility ◽  
Small Scale ◽  
Phase Liquid ◽  
Single Phase Liquid Flow

The fluids in horizontal wells can exhibit complicated flow behaviors, in part due to interaction between the main flow and the influxes along the wellbore, and due to completion geometries. An existing small-scale test facility at Tulsa University Fluid Flow Projects (TUFFP) was used to simulate the flow in a horizontal well completed with either circular perforations or slotted liners. Single phase liquid flow experiments were conducted with Reynolds numbers ranging approximately from 5000 to 65,000 and influx to main flow rate ratios ranging from 1/50 to 1/1000. For both the perforation and slot cases, three different completion densities and three different completion phasings are considered. Based on the experimental data, new friction factor correlations for horizontal well with multiple perforation completion or multiple slots completion were developed using the principles of conservation of mass and momentum.

Simplified Transient Numerical Model of a Supersonic Jet Impacting a Substrate

2017 ◽  
Author(s):  
Sichang Xu ◽  
Patrick Pomerleau-Perron ◽  
Gary W. Rankin
Keyword(s):  
Flow Field ◽  
Numerical Solutions ◽  
Transient Flow ◽  
Fluid Dynamic ◽  
Supersonic Jet ◽  
Axially Symmetric ◽  
Rotary Valve ◽  
Time Curve ◽  
Jet Impact ◽  
Impact Region

The transient flow field near the surface of a substrate impacted by a pulsating supersonic jet emerging from a long tube is investigated using a simplified axially symmetric numerical approach. In the system being modeled, the pulses are created using a rotary valve located at the tube entrance. This flow situation approximates the conditions existing in the Shock-Induced Cold Spray process for coating surfaces with metallic particles. Previous numerical studies of transient supersonic jets either focused on jets emerging from orifices or did not give details of the complex supersonic flow field in the jet impact region. The current approximate numerical method considers the flow within the long tube and in the jet impact region. The procedure involves two stages. The upstream pressure variation with time is first determined using a one-dimensional compressible flow approximation of the entire tube and rotary valve arrangement. The resulting pressure versus time curve serves as the transient inlet boundary condition for an axially symmetric computational fluid dynamic solution of the flow through the tube and region of jet impact on the substrate. The numerical solutions of substrate pressure on the jet centerline versus time are compared with available experimental results and predict certain general features of the substrate pressure traces. Although the simplified model is only in fair agreement with some aspects of the experimental curves, it is shown to be useful in explaining certain peculiar flow features. With the aid of the numerical solution, an explanation for the movement and instability of the bow shock wave which forms ahead of the substrate is described.

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