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

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

The Productivity of a Well Completed With a Vertical Penny-Shaped Fracture

SPE Journal ◽  
2011 ◽  
Vol 16 (02) ◽  
pp. 401-410 ◽  
Author(s):  
Jacques Hagoort
Keyword(s):  
Simple Formula ◽  
Single Phase ◽  
Constant Pressure ◽  
Productivity Index ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Fracture Conductivity ◽  
Pressure Surface ◽  
Flow Capacity ◽  
Infinite Conductivity

Summary In this work, we present a simple polynomial relationship for the effective well radius of a well completed with a vertical, infinite-conductivity, penny-shaped hydraulic fracture as a function of fracture diameter. It is based on an analytical solution for steady-state, single-phase flow to a circular, constant-pressure surface in an infinite porous medium. This solution is extended to a vertical, penny-shaped fracture in the center of a plane circular reservoir. The effective well radius increases with increasing fracture diameter to approximately 0.2 times the fracture diameter for a fracture diameter equal to the reservoir thickness. As a rule of thumb, the productivity of a well with an infinite-conductivity, penny-shaped fracture exceeds the openhole productivity for fracture diameters larger than one-third of the reservoir thickness. The productivity of a well with a fracture diameter equal to the reservoir thickness is approximately twice the openhole productivity. The adverse effect of fracture conductivity can be estimated by a simple formula that relates fracture efficiency to the productivity index of a well with an infinite-conductivity fracture and to the ratio of the flow capacity of the fracture to that of the reservoir.

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.

Experience in Optimizing the Location and Parameters of Multistage Hydraulic Fractures for a Multilateral Well Based on Reservoir Simulation

2021 ◽  
Author(s):  
Vil Syrtlanov ◽  
Yury Golovatskiy ◽  
Konstantin Chistikov ◽  
Dmitriy Bormashov
Keyword(s):  
Hydraulic Fracturing ◽  
Reservoir Simulation ◽  
Optimal Placement ◽  
Hydraulic Fractures ◽  
Low Permeability ◽  
Advantages And Disadvantages ◽  
Modeling Methods ◽  
Multistage Hydraulic Fracturing ◽  
Determination Of Parameters

Abstract This work presents the approaches used for the optimal placement and determination of parameters of hydraulic fractures in horizontal and multilateral wells in a low-permeability reservoir using various methods, including 3D modeling. The results of the production rate of a multilateral dualwellbore well are analyzed after the actual hydraulic fracturing performed on the basis of calculations. The advantages and disadvantages of modeling methods are evaluated, recommendations are given to improve the reliability of calculations for models with hydraulic fracturing (HF)/ multistage hydraulic fracturing (MHF).

Can We Engineer Better Multistage Horizontal Completions? Evidence of the Importance of Near-wellbore Fracture Geometry from Theory, Lab and Field Experiments

2015 ◽  
Author(s):  
B.. Lecampion ◽  
J.. Desroches ◽  
X.. Weng ◽  
J.. Burghardt ◽  
J.E.. E. Brown
Keyword(s):  
Pressure Drop ◽  
Field Experiments ◽  
Horizontal Wells ◽  
Fracture Plane ◽  
Hydraulic Fractures ◽  
Multiple Fractures ◽  
Fracture Geometry ◽  
Complex Fracture ◽  
Limited Entry ◽  
The Impact

Abstract There is accepted evidence that multistage fracturing of horizontal wells in shale reservoirs results in significant production variation from perforation cluster to perforation cluster. Typically, between 30 and 40% of the clusters do not significantly contribute to production while the majority of the production comes from only 20 to 30% of the clusters. Based on numerical modeling, laboratory and field experiments, we investigate the process of simultaneously initiating and propagating several hydraulic fractures. In particular, we clarify the interplay between the impact of perforation friction and stress shadow on the stability of the propagation of multiple fractures. We show that a sufficiently large perforation pressure drop (limited entry) can counteract the stress interference between different growing fractures. We also discuss the robustness of the current design practices (cluster location, limited entry) in the presence of characterized stress heterogeneities. Laboratory experiments highlight the complexity of the fracture geometry in the near-wellbore region. Such complex fracture path results from local stress perturbations around the well and the perforations, as well as the rock fabric. The fracture complexity (i.e., the merging of multiple fractures and the reorientation towards the preferred far-field fracture plane) induces a strong nonlinear pressure drop on a scale of a few meters. Single entry field experiments in horizontal wells show that this near-wellbore effect is larger in magnitude than perforation friction and is highly variable between clusters, without being predictable. Through a combination of field measurements and modeling, we show that such variability results in a very heterogeneous slurry rate distribution; and therefore, proppant intake between clusters during a stage, even in the presence of limited entry techniques. We also note that the estimated distribution of proppant intake between clusters appears similar to published production log data. We conclude that understanding and accounting for the complex fracture geometry in the near-wellbore is an important missing link to better engineer horizontal well multistage completions.

scholarly journals Estimating Organ Contribution to Grain Filling and Potential for Source Upregulation in Wheat Cultivars with a Contrasting Source–Sink Balance

Agronomy ◽  
2020 ◽  
Vol 10 (10) ◽  
pp. 1527
Author(s):  
Carolina Rivera-Amado ◽  
Gemma Molero ◽  
Eliseo Trujillo-Negrellos ◽  
Matthew Reynolds ◽  
John Foulkes
Keyword(s):  
Grain Growth ◽  
Photosynthetic Rate ◽  
Grain Filling ◽  
Leaf Sheath ◽  
Grain Weight ◽  
Yield Potential ◽  
Wheat Cultivars ◽  
Sink Limitation ◽  
Source Sink ◽  
Source And Sink

Grain filling may be limited by the joint source and sink capacity in modern wheat cultivars, indicating a need to research the co-limitation of yield by both photosynthesis and the number and potential size of grains. The extent to which the post-anthesis source may be limiting final grain size can be estimated by partial degraining of spikes, while defoliation and shading treatments can be useful to estimate if any excess photosynthetic capacity exists. In the current study, degraining was applied to a set of 26 elite spring wheat cultivars from the International Maize and Wheat Improvement Center (CIMMYT)’s core germplasm (CIMCOG) panel, while lamina defoliation and shading through stem-and-leaf-sheath covering treatments were applied to a subset of the same cultivars. Responses to source treatments in grain weight, pre-anthesis reserve contribution to grain weight, dry-matter translocation efficiency, and flag-leaf and spike photosynthetic rate were measured and compared to an unmanipulated control treatment. Grain weight responses to degraining among cultivars ranged from no response to increases of 28%, suggesting a range of responses from sink limitation, to probable source and sink co-limitation of grain growth. Grain weight’s response to degraining increased linearly with the years of cultivar release from 1966 to 2009, indicating that the current highest yield potential CIMMYT spring wheats have a co-limitation of grain growth by source and sink. This may have been due to an increase in grain sink strength with years of cultivar release with no commensurate increase in post-anthesis source capacity. The relatively low decreases in grain weight with defoliation compared to decreases in light interception by defoliation indicated that sink limitation was still likely predominating in the cultivars with co-limitation. The stem-and-leaf-sheath covering treatment decreased grain weight by nearly 10%, indicating that stem-and-leafsheath photosynthesis plays a key role in grain growth during grain filling. In addition, pre-anthesis reserve contribution to grain weight was increased by ca. 50% in response to lamina defoliation. Our results showed that increasing the post-anthesis source capacity, through increases in stem-and-leaf-sheath photosynthetic rate during grain filling and pre-anthesis reserve contribution to grain weight, is an important objective in enhancing yield potential in wheat through maintaining a source–sink balance.

Evaluation of Induced Thermal Pressurization in Clearwater Shale Caprock in Electromagnetic Steam-Assisted Gravity-Drainage Projects

SPE Journal ◽  
2013 ◽  
Vol 19 (03) ◽  
pp. 443-462 ◽  
Author(s):  
Sahar Ghannadi ◽  
Mazda Irani ◽  
Rick Chalaturnyk
Keyword(s):  
Pore Pressure ◽  
Analytical Solutions ◽  
Oil Sands ◽  
Shear Failure ◽  
Gravity Drainage ◽  
Hydraulic Fractures ◽  
Low Permeability ◽  
Steam Assisted Gravity Drainage ◽  
Thermal Pressurization ◽  
Shale Formation

Summary Inductive methods, such as electromagnetic steam-assisted gravity drainage (EM-SAGD), have been identified as technically and economically feasible recovery methods for shallow oil-sands reservoirs with overburdens of more than 30 m (Koolman et al. 2008). However, in EM-SAGD projects, the caprock overlying oil-sands reservoirs is also electromagnetically heated along with the bitumen reservoir. Because permeability is low in Alberta thermal-project caprock formations (i.e., the Clearwater shale formation in the Athabasca deposit and the Colorado shale formation in the Cold Lake deposit), the pore pressure resulting from the thermal expansion of pore fluids may not be balanced with the fluid loss caused by flow and the fluid-volume changes resulting from pore dilation. In extreme cases, the water boils, and the pore pressure increases dramatically as a result of the phase change in the water, which causes profound effective-stress reduction. After this condition is established, pore pressure increases can lead to shear failure of the caprock, the creation of microcracks and hydraulic fractures, and subsequent caprock integrity failure. It is typically believed that low-permeability caprocks impede the transmission of pore pressure from the reservoir, making them more resistant to shear failure (Collins 2005, 2007). In cases of induced thermal pressurization, low-permeability caprocks are not always more resistant. In this study, analytical solutions are obtained for temperature and pore-pressure rises caused by the constant EM heating rate of the caprock. These analytical solutions show that pore-pressure increases from EM heating depend on the permeability and compressibility of the caprock formation. For stiff or low-compressibility media, thermal pressurization can cause fluid pressures to approach hydrostatic pressure, and shear strength to approach zero for low-cohesive-strength units of the caprock (units of the caprock with high silt and sand percentage) and sections of the caprock with pre-existing fractures with no cohesion.

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