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

Analysis of Pressure Data From Vertically Fractured Injection Wells

1981 ◽  
Vol 21 (01) ◽  
pp. 5-20
Author(s):  
Curtis O. Bennett ◽  
Albert C. Reynolds ◽  
Rajagopal Raghavan
Keyword(s):  
Enhanced Recovery ◽  
Outer Boundary ◽  
Fracture Plane ◽  
Pressure Data ◽  
Sweep Efficiency ◽  
Injection Wells ◽  
Type Curve ◽  
Pressure Behavior ◽  
Fracture Length ◽  
Fracture Orientation

Abstract This study investigates the flowing and shut-inpressure behavior of a fractured well located in asquare drainage region with the outer boundary at aconstant pressure. The fracture plane lies on one ofthe diagonals of the square. The report shows how toanalyze pressure data for a five-spot pattern when thefracture orientation is most favorable (from theviewpoint of sweep efficiency). Comparisons aremade with studies in the literature that assume anunfavorable fracture orientation. Fractureorientation must be considered in the analysis ofpressure data for the following conditions:smallfracture-penetration ratios,large flowing timesprior to shut-in, andlarge values of fractureflow capacity. Insights into the application of type-curve analysisto estimate drainage volumes are presented. Claimsin the literature regarding the determination of thedrainage volume by type-curve matching appear tobe unrealistic. Introduction No quantitative data are available on the effect of thecompass orientation of a vertical fracture on pressuretransient data (injection or falloff). This is surprisingsince pressure falloff tests are the principal means ofdetermining the efficacy of a fluid-injectionprogram - e.g., the effective formation flowcapacity, injectivity, skin factor, average reservoirpressure, and position of the flood front. Perhaps the dearth of information on this topic isdue to the fracture lengths being small comparedwith interwell distances in most waterflood orgas-injection projects. If the fracture length is smallcompared with the interwell distance, the orientationof the fracture should have a negligible effect on theshape of the pressure vs. time curve. However, withnew enhanced recovery projects that require closerwell spacing, interwell distances are of the same orderof magnitude as the created fracture length. In such instances, compass orientation of a vertical fracturecan have a significant effect on pressure data. All studies of the transient pressure behavior offractured wells in a bounded drainage region haveassumed that the fracture plane is parallel to theboundaries, which were considered to be eitherclosed or at constant pressure. Raghavan andHadinoto showed that the constant pressureouter-boundary solutions can be applied to a fractured wellin a five-spot injection-production pattern. However, the assumption that the fracture plane is parallel tothe boundaries of the square drainage region impliesthat the fracture is aligned directly with two of theadjacent producers. Clearly, this is only one of themany compass orientations that may exist in thefield; if consideration is given to the sweep efficiency of the flood, then this orientation would be the leastdesirable since the sweep efficiency at breakthroughwill be minimal. The most favorable fractureorientation would be the one in which the fractureplane lies along the diagonal of the square drainageregion (Fig. 1). As already mentioned, the fractureorientation may not have a significant effect ontransient data if the fracture lengths are small - butfor long fracture lengths, the effect of the orientationon the pressure behavior of injection wells can besignificant. Thus, it appears necessary to determinethe effect of fracture orientation on the pressurebehavior of fractured injection wells. SPEJ P. 5^

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).

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.

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.

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

Total Pressure Correction of a Sub-Miniature Five-Hole Probe in Areas of Pressure Gradients

2012 ◽  
Author(s):  
J. Town ◽  
A. Akturk ◽  
C. Camcı
Keyword(s):  
Pressure Measurement ◽  
Measurement Errors ◽  
Total Pressure ◽  
Pressure Gradients ◽  
Pressure Data ◽  
Flow Conditions ◽  
Complex Flow ◽  
Ducted Fan ◽  
The Difference ◽  
Best Fit

Five-hole probes, being a dependable and accurate aerodynamic tools, are excellent choices for measuring complex flow fields. However, total pressure gradients can induce measurement errors. The combined effect of the different flow conditions on the ports causes the measured total pressure to be prone to a greater error. This paper proposes a way to correct the total pressure measurement. The correction is based on the difference between the measured total pressure data of a Kiel probe and a sub-miniature prism-type five-hole probe. By comparing them in a ducted fan related flow field, a line of best fit was constructed. The line of best fit is dependent on the slope of the line in a total pressure versus span and difference in total pressure between the probes at the same location. A computer program, performs the comparison and creates the correction equation. The equation is subsequently applied to the five-hole probe total pressure measurement, and the other dependent values are adjusted. The validity of the correction is then tested by placing the Kiel probe and the five-hole probe in ducted fans with a variety of different tip clearances.

Low-Cost Measurement of Hydraulic Fracture Dimensions Using Pressure Data in Treatment and Observation Wells – A Workflow for Every Well

2021 ◽  
Author(s):  
A. Kirby Nicholson ◽  
Robert C. Bachman ◽  
R. Yvonne Scherz ◽  
Robert V. Hawkes
Keyword(s):  
Hydraulic Fracturing ◽  
Real Time ◽  
Hydraulic Fracture ◽  
Full Scale ◽  
Pressure Data ◽  
Observation Well ◽  
High Quality ◽  
Fracture Length ◽  
Geometry Model ◽  
Observation Wells

Abstract Pressure and stage volume are the least expensive and most readily available data for diagnostic analysis of hydraulic fracturing operations. Case history data from the Midland Basin is used to demonstrate how high-quality, time-synchronized pressure measurements at a treatment and an offsetting shut-in producing well can provide the necessary input to calculate fracture geometries at both wells and estimate perforation cluster efficiency at the treatment well. No special wellbore monitoring equipment is required. In summary, the methods outlined in this paper quantifies fracture geometries as compared to the more general observations of Daneshy (2020) and Haustveit et al. (2020). Pressures collected in Diagnostic Fracture Injection Tests (DFITs), select toe-stage full-scale fracture treatments, and offset observation wells are used to demonstrate a simple workflow. The pressure data combined with Volume to First Response (Vfr) at the observation well is used to create a geometry model of fracture length, width, and height estimates at the treatment well as illustrated in Figure 1. The producing fracture length of the observation well is also determined. Pressure Transient Analysis (PTA) techniques, a Perkins-Kern-Nordgren (PKN) fracture propagation model and offset well Fracture Driven Interaction (FDI) pressures are used to quantify hydraulic fracture dimensions. The PTA-derived Farfield Fracture Extension Pressure, FFEP, concept was introduced in Nicholson et al. (2019) and is summarized in Appendix B of this paper. FFEP replaces Instantaneous Shut-In Pressure, ISIP, for use in net pressure calculations. FFEP is determined and utilized in both DFITs and full-scale fracture inter-stage fall-off data. The use of the Primary Pressure Derivative (PPD) to accurately identify FFEP simplifies and speeds up the analysis, allowing for real time treatment decisions. This new technique is called Rapid-PTA. Additionally, the plotted shape and gradient of the observation-well pressure response can identify whether FDI's are hydraulic or poroelastic before a fracture stage is completed and may be used to change stage volume on the fly. Figure 1Fracture Geometry Model with FDI Pressure Matching Case studies are presented showing the full workflow required to generate the fracture geometry model. The component inputs for the model are presented including a toe-stage DFIT, inter-stage pressure fall-off, and the FDI pressure build-up. We discuss how to optimize these hydraulic fractures in hindsight (look-back) and what might have been done in real time during the completion operations given this workflow and field-ready advanced data-handling capability. Hydraulic fracturing operations can be optimized in real time using new Rapid-PTA techniques for high quality pressure data collected on treating and observation wells. This process opens the door for more advanced geometry modeling and for rapid design changes to save costs and improve well productivity and ultimate recovery.

Optimization of Spacing and Penetration Ratio for Infinite-Conductivity Fractures in Unconventional Reservoirs: A Section-Based Approach

SPE Journal ◽  
2017 ◽  
Vol 22 (06) ◽  
pp. 1877-1892 ◽  
Author(s):  
S.. Liu ◽  
P. P. Valkó
Keyword(s):  
Shale Gas ◽  
Optimization Method ◽  
Limited Range ◽  
Productivity Index ◽  
Optimal Decision ◽  
Fracture Length ◽  
Penetration Ratio ◽  
Feasible Range ◽  
Infinite Conductivity ◽  
Total Fracture

Summary In this paper, we consider the development plan of shale gas or tight oil with multiple multistage fractured laterals in a large square drainage area that we call a “section” (usually 640 acres in the US). We propose a convenient section-based optimization of the fracture array with two integer variables, the number of columns (horizontal laterals) and rows (fractures created in a lateral), to provide some general statements regarding spacing of wells and fractures. The approach is dependent on a reliable and efficient productivity-index (PI) calculation for the boundary-dominated state (BDS). The dimensionless PI is obtained by solving a time-independent eigenvalue problem by use of the finite-element method (FEM) combined with the Richardson extrapolation. The results of the case study demonstrate two decisive factors: the dimensionless total fracture length, related to the total amount of proppant and fracturing fluid available for the section, and the feasible range of actual fracture half-lengths, related to current fracturing-technology limitations. Under the constraint of dimensionless total fracture length, increasing the number of columns (horizontal laterals) increases the total PI but with only diminishing returns, whereas the optimal fracture-penetration ratio decreases somewhat, but is still near unity. When adding the technological constraint of a limited range of fracture half-lengths that can be routinely and reliably created, only a few choices remain admissible, and the optimal decision can be easily made. These general statements for the ideal homogeneous and isotropic formation can serve as a reference in the more-detailed optimization works. In other words, we offer a first-pass method for decision making in early stages when detailed inputs are not yet available. The information derived from the section-based optimization method and the efficient and reliable algorithm for PI calculation should help the design of multistage fracturing in shale-gas or ultralow-permeability oil formations.

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