scholarly journals Two-dimensional numerical investigations on the termination of bilinear flow in fractures

2013 ◽  
Vol 5 (1) ◽  
pp. 391-425
Author(s):  
◽  
R. Jung ◽  
J. Renner
Keyword(s):  
Flow Field ◽  
Master Curve ◽  
Radial Flow ◽  
Dimensionless Time ◽  
Linear Flow ◽  
Type Curves ◽  
Straight Line ◽  
The Matrix ◽  
Line Characteristic ◽  
Fourth Root

Abstract. Bilinear flow occurs when fluid is drained from a permeable matrix by producing it through an enclosed fracture of finite conductivity intersecting a well along its axis. The terminology reflects the combination of two approximately linear flow regimes, one in the matrix with flow essentially perpendicular to the fracture and one along the fracture itself associated with the non-negligible pressure drop in it. We investigated the characteristics, in particular the termination, of bilinear flow by numerical modeling allowing an examination of the entire flow field without prescribing the flow geometry in the matrix. Fracture storage capacity was neglected relying on previous findings that bilinear flow is associated with a quasi-steady flow in the fracture. Numerical results were generalized by dimensionless presentation. Definition of a dimensionless time that other than in previous approaches does not use geometrical parameters of the fracture permitted identifying the dimensionless well pressure for the infinitely long fracture as the master curve for type curves of all fractures with finite length from the beginning of bilinear flow up to fully developed radial flow. In log-log-scale the master curve's logarithmic derivative initially follows a 1/4-slope-straight line (characteristic for bilinear flow) and gradually bends into a horizontal line (characteristic for radial flow) for long times. During the bilinear flow period, isobars normalized to well pressure propagate with fourth and second root of time in fracture and matrix, respectively. The width-to-length ratio of the pressure field increases proportional to the fourth root of time during the bilinear period and starts to deviate from this relation close to the deviation of well pressure and its derivative from their fourth-root-of-time relations. At this time, isobars are already significantly inclined with respect to the fracture. The type curves of finite fractures all deviate counterclockwise from the master curve instead of clockwise or counterclockwise from the 1/4-slope-straight line as previously proposed. The counterclockwise deviation from the master curve was identified as the arrival of a normalized isobar reflected at the fracture tip sixteen times earlier. Nevertheless, two distinct regimes were found regarding pressure at the fracture tip when bilinear flow ends. For dimensionless fracture conductivities TD < 1, a significant pressure increase is not observed at the fracture tip until bilinear flow is succeeded by radial flow at a fixed dimensionless time. For TD > 10, the pressure at the fracture tip has reached substantial fractions of the associated change in well pressure when the flow field transforms towards intermittent formation linear flow at times that scale inversely with the fourth power of dimensionless fracture conductivity. Our results suggest that semi-log plots of normalized well pressure provide a means for the determination of hydraulic parameters of fracture and matrix after shorter test duration than for conventional analysis.

scholarly journals Two-dimensional numerical investigations on the termination of bilinear flow in fractures

Solid Earth ◽  
2013 ◽  
Vol 4 (2) ◽  
pp. 331-345 ◽  
Author(s):  
A. E. Ortiz R. ◽  
R. Jung ◽  
J. Renner
Keyword(s):  
Flow Field ◽  
Master Curve ◽  
Radial Flow ◽  
Dimensionless Time ◽  
Linear Flow ◽  
Type Curves ◽  
Straight Line ◽  
The Matrix ◽  
Line Characteristic ◽  
Fourth Root

Abstract. Bilinear flow occurs when fluid is drained from a permeable matrix by producing it through an enclosed fracture of finite conductivity intersecting a well along its axis. The terminology reflects the combination of two approximately linear flow regimes: one in the matrix with flow essentially perpendicular to the fracture, and one along the fracture itself associated with the non-negligible pressure drop in it. We investigated the characteristics, in particular the termination, of bilinear flow by numerical modeling allowing for an examination of the entire flow field without prescribing the flow geometry in the matrix. Fracture storage capacity was neglected relying on previous findings that bilinear flow is associated with a quasi-steady flow in the fracture. Numerical results were generalized by dimensionless presentation. Definition of a dimensionless time that, other than in previous approaches, does not use geometrical parameters of the fracture permitted identifying the dimensionless well pressure for the infinitely long fracture as the master curve for type curves of all fractures with finite length from the beginning of bilinear flow up to fully developed radial flow. In log–log scale the master curve's logarithmic derivative initially follows a 1/4-slope straight line (characteristic for bilinear flow) and gradually bends into a horizontal line (characteristic for radial flow) for long times. During the bilinear flow period, isobars normalized to well pressure propagate with the fourth and second root of time in fracture and matrix, respectively. The width-to-length ratio of the pressure field increases proportional to the fourth root of time during the bilinear period, and starts to deviate from this relation close to the deviation of well pressure and its derivative from their fourth-root-of-time relations. At this time, isobars are already significantly inclined with respect to the fracture. The type curves of finite fractures all deviate counterclockwise from the master curve instead of clockwise or counterclockwise from the 1/4-slope straight line as previously proposed. The counterclockwise deviation from the master curve was identified as the arrival of a normalized isobar reflected at the fracture tip 16 times earlier. Nevertheless, two distinct regimes were found in regard to pressure at the fracture tip when bilinear flow ends. For dimensionless fracture conductivities TD < 1, a significant pressure increase is not observed at the fracture tip until bilinear flow is succeeded by radial flow at a fixed dimensionless time. For TD > 10, the pressure at the fracture tip has reached substantial fractions of the associated change in well pressure when the flow field transforms towards intermittent formation linear flow at times that scale inversely with the fourth power of dimensionless fracture conductivity. Our results suggest that semi-log plots of normalized well pressure provide a means for the determination of hydraulic parameters of fracture and matrix after shorter test duration than for conventional analysis.

scholarly journals A Semianalytical Solution for Multifractured Horizontal Wells in Box-Shaped Reservoirs

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Lei Wang ◽  
Xiaodong Wang ◽  
He Zhang ◽  
Yunpeng Hu ◽  
Chen Li
Keyword(s):  
Radial Flow ◽  
Flow Characteristics ◽  
Horizontal Wells ◽  
Flow Region ◽  
Linear Flow ◽  
Type Curves ◽  
Production Ratio ◽  
Dirac Function ◽  
Sensitive Parameters ◽  
First Time

This paper presented a 3D point sink model through using Dirac function. Then, 3D point sink solution in boxed reservoirs was obtained through using Laplace transform and Fourier transform methods. Based on the flux and pressure equivalent conditions in Laplace space, a semianalytical solution for multifractured horizontal wells was also proposed for the first time. The production rate distribution was discussed in detail for multifractured horizontal wells. The calculative results show the outermost fractures had higher production ratio due to larger drainage area and the inner fractures were lower due to the strong interface between fractures. Type curves were established to analyze the flow characteristics, which would be divided into six stages, for example, bilinear flow region, the first linear flow region, the first radial flow region, the second linear flow region, the second radial flow region, and the boundary dominated flow region, respectively. Finally, effects of some sensitive parameters on type curves were also analyzed in detail.

scholarly journals A Novel Radial-Composite Model of Pressure Transient Analysis for Multistage Fracturing Horizontal Wells with Stimulated Reservoir Volume

Geofluids ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Yunpeng Hu ◽  
Xiaoling Zhang ◽  
Ziyun Cheng ◽  
Wei Ding ◽  
Liangchao Qu ◽  
...  
Keyword(s):  
Horizontal Well ◽  
Storage Capacity ◽  
Radial Flow ◽  
Horizontal Wells ◽  
Outer Zone ◽  
Middle Zone ◽  
Linear Flow ◽  
Type Curves ◽  
Stimulated Reservoir Volume ◽  
Reservoir Volume

In the process of stimulated reservoir volume of tight reservoir, horizontal well can form three zones, the inner zone is multistage fracturing zone, the middle zone is skin damage zone, and the outer zone is undamaged zone. In this paper, a transient well test analysis model of multistage fracturing horizontal well in three area composite reservoir is proposed. Based on Laplace transformation, point source integration, and superposition principle, the infinite conductivity multifracture model of three area composite reservoir is obtained. The linear equations of finite conductivity multifracture in Laplace space are established by using the equal conditions of flow and pressure at the fracture wall. Gauss-Newton iteration method and Stehfest number are used to obtain the solution of wellbore pressure. The accuracy of the results is verified by numerical simulation. Then, the flow characteristics of multistage fracturing horizontal wells in three area composite reservoirs are analyzed by type curves. The flow is divided into ten stages, which are the bilinear flow, the linear flow, the first radial flow stage, the inner zone linear flow, the inner zone radial flow, the middle zone linear flow, the middle zone radial flow, the outer zone linear flow, the outer zone radial flow, and the boundary dominated flow. The pressure derivative curves show different characteristics in different flow stages. The influences of fracture conductivity, fracture spacing, radius ratio of the middle zone to inner zone, radius ratio of the outer zone to the middle zone, permeability ratio of inner zone to the middle zone, permeability ratio of inner zone to outer zone, storage capacity ratio of inner zone to the middle zone, and storage capacity ratio of inner zone to outer zone on type curves are analyzed. Finally, the application and reliability of the proposed model are verified by a case example.

Radial Flow Velocity Field for Predicting Upper-Bound Solutions for Plane Strain Extrusion

1968 ◽  
Vol 90 (1) ◽  
pp. 45-50
Author(s):  
R. G. Fenton
Keyword(s):  
Velocity Field ◽  
Flow Field ◽  
Flow Velocity ◽  
Plane Strain ◽  
Upper Bound ◽  
Radial Flow ◽  
Flow Velocity Field ◽  
Wide Range ◽  
Ram Pressure ◽  
Considerable Range

The upper bound of the average ram pressure, based on an assumed radial flow velocity field, is derived for plane strain extrusion. Ram pressures are calculated for a complete range of reduction ratios and die angles, considering a wide range of frictional conditions. Results are compared with upper-bound ram pressures obtained by considering velocity fields other than the radial flow field, and it is shown that for a considerable range of reduction ratios and die angles, the radial flow field yields better upper bounds for the average ram pressure.

Modelling Motion Patterns with Circular-Linear Flow Field Maps

2020 ◽  
pp. 65-113
Author(s):  
Tomasz Piotr Kucner ◽  
Achim J. Lilienthal ◽  
Martin Magnusson ◽  
Luigi Palmieri ◽  
Chittaranjan Srinivas Swaminathan
Keyword(s):  
Flow Field ◽  
Linear Flow ◽  
Motion Patterns

A New Analytical Method for Analyzing Linear Flow in Tight/Shale Gas Reservoirs: Constant-Flowing-Pressure Boundary Condition

2012 ◽  
Vol 15 (03) ◽  
pp. 370-384 ◽  
Author(s):  
Morteza Nobakht ◽  
C.R.. R. Clarkson
Keyword(s):  
Shale Gas ◽  
Analytical Method ◽  
Flow Analysis ◽  
Initial Pressure ◽  
Reliable Estimate ◽  
Square Root ◽  
Linear Flow ◽  
Straight Line ◽  
Half Length ◽  
Time Plot

Summary Many tight/shale gas wells exhibit linear flow, which can last for several years. Linear flow can be analyzed using a square-root-of-time plot, a plot of rate-normalized pressure vs. the square root of time. Linear flow appears as a straight line on this plot, and the slope of this line can be used to calculate the product of fracture half-length and the square root of permeability. In this paper, linear flow from a fractured well in a tight/shale gas reservoir under a constant-flowing-pressure constraint is studied. It is shown that the slope of the square-root-of-time plot results in an overestimation of fracture half-length, if permeability is known. The degree of this overestimation is influenced by initial pressure, flowing pressure, and formation compressibility. An analytical method is presented to correct the slope of the square-root-of-time plot to improve the overestimation of fracture halflength. The method is validated using a number of numerically simulated cases. As expected, the square-root-of-time plots for these simulated cases appear as a straight line during linear flow for constant flowing pressure. It is found that the newly developed analytical method results in a more reliable estimate of fracture half-length, if permeability is known. Our approach, which is fully analytical, results in an improvement in linear-flow analysis over previously presented methods. Finally, the application of this method to multifractured horizontal wells is discussed and the method is applied to three field examples.

Unsteady-State Spherical Flow With Storage and Skin

1985 ◽  
Vol 25 (06) ◽  
pp. 804-822 ◽  
Author(s):  
Jeffrey A. Joseph ◽  
Leonard F. Koederitz
Keyword(s):  
Boundary Condition ◽  
Radial Flow ◽  
Linear Partial Differential Equation ◽  
Flow Profile ◽  
Unsteady State ◽  
Well Test ◽  
Linear Flow ◽  
Dimensionless Groups ◽  
Short Time ◽  
Suitable Expression

Abstract This paper presents short-time interpretation methods for radial-spherical (or radial-hemispherical) flow in homogeneous and isotropic reservoirs inclusive of wellbore storage, wellbore phase redistribution, and damage skin effects. New dimensionless groups are introduced to facilitate the classic transformation from radial flow in the sphere to linear flow in the rod. Analytical expressions, type curves (in log-log and semilog format), and tabulated solutions are presented, both in terms of pressure and rate, for all flow problems considered. A new empirical equation to estimate the duration of wellbore and near-wellbore effects under spherical flow is also proposed. Introduction The majority of the reported research on unsteady-state flow theory applicable to well testing usually assumes a cylindrical (typically a radial-cylindrical) flow profile because this condition is valid for many test situations. Certain well tests, however, are better modeled by assuming a spherical flow symmetry (e.g., wireline formation testing, vertical interference testing, and perhaps even some tests conducted in wellbores that do not fully penetrate the productive horizon or are selectively penetrate the productive horizon or are selectively completed). Plugged perforations or blockage of a large part of an openhole interval may also promote spherical flow. Numerous solutions are available in the literature for almost every conceivable cylindrical flow problem; unfortunately, the companion spherical problem has not received as much attention, and comparatively few papers have been published on this topic. papers have been published on this topic. The most common inner boundary condition in well test analysis is that of a constant production rate. But with the advent of downhole tools capable of the simultaneous measurement of pressures and flow rates, this idealized inner boundary condition has been refined and more sophisticated models have been proposed. Therefore, similar methods must be developed for spherical flow analysis, especially for short-time interpretations. This general problem has recently been addressed elsewhere. Theory The fundamental linear partial differential equation (PDE) describing fluid flow in an infinite medium characterized by a radial-spherical symmetry is (1) The assumptions incorporated into this diffusion equation are similar to those imposed on the radial-cylindrical diffusivity equation and are discussed at length in Ref. 9. In solving Eq. 1, the classic approach is illustrated by Carslaw and Jaeger (later used by Chatas, and Brigham et al.). According to Carslaw and Jaeger, mapping b=pr will always reduce the problem of radial flow in the sphere (Eq. 1) to an equivalent problem of linear flow in the rod for which general solutions are usually known. (For example, see Ref. 17 for particular solutions in petroleum applications.) Note that in this study, we assumed that the medium is spherically isotropic; hence k in Eq. 1 is the constant spherical permeability. This assumption, however, does not preclude analysis in systems possessing simple anisotropy (i.e., uniform but unequal horizontal and vertical permeability components). In this case, k as used in this paper should be replaced by k, an equivalent or average (but constant) spherical permeability. Chatas presented a suitable expression (his Eq. 10) obtained presented a suitable expression (his Eq. 10) obtained from a volume integral. It is desirable to transform Eq. 1 to a nondimensional form, thereby rendering its applicability universal. The following new, dimensionless groups accomplish this and have the added feature that solutions are obtained directly in terms of the dimensionless pressure drop, PD, not the usual b (or bD) groups. ......................(2) .......................(3) .........................(4) The quantity rsw is an equivalent or pseudospherical wellbore radius used to represent the actual cylindrical sink (or source) of radius rw. SPEJ p. 804

Current density and polarization curves for radial flow field patterns applied to PEMFCs (Proton Exchange Membrane Fuel Cells)

Energy ◽  
2010 ◽  
Vol 35 (2) ◽  
pp. 920-927 ◽  
Author(s):  
S. Cano-Andrade ◽  
A. Hernandez-Guerrero ◽  
M.R. von Spakovsky ◽  
C.E. Damian-Ascencio ◽  
J.C. Rubio-Arana
Keyword(s):  
Fuel Cells ◽  
Flow Field ◽  
Current Density ◽  
Proton Exchange Membrane ◽  
Radial Flow ◽  
Proton Exchange ◽  
Polarization Curves ◽  
Field Patterns ◽  
Exchange Membrane

Study of Flow Characteristics of Oil and Water in the Process of Stirring

2013 ◽  
Vol 353-356 ◽  
pp. 3190-3193
Author(s):  
Zong Rui Hao ◽  
Juan Xu ◽  
Hai Yan Bie ◽  
Zhong Hai Zhou
Keyword(s):  
Flow Field ◽  
Flow Pattern ◽  
Cross Section ◽  
Radial Flow ◽  
Flow Characteristics ◽  
Stirred Tank ◽  
Turbulent Model ◽  
Vof Model ◽  
Circular Flows ◽  
Oil Water

To study the flow pattern in the process of oil-water stirring in three paddle stirring tank, RNG k-ε turbulent model and VOF model are adopted to simulate the flow field at different time in the stirred tank with the baffle. The results showed that, in the stirring process, inverted cone manifold was formed in the center of the stirring shaft. The stratified area was formed in the baffle and gradually transported to the bottom of the tank. The two circular flows were formed among three groups of blades. And the axially acting of the fluid was strong, which made homogeneous stirring in the stirred tank. At the same time the radial flow of the cross-section inside the tank increased because of the baffle.

Welltest analysis of hydraulically fractured tight gas reservoirs: a field example from Perth Basin, Western Australia

2012 ◽  
Vol 52 (1) ◽  
pp. 587 ◽  
Author(s):  
Hassan Bahrami ◽  
Vineeth Jayan ◽  
Reza Rezaee ◽  
Dr Mofazzal Hossain
Keyword(s):  
Radial Flow ◽  
Second Derivative ◽  
Tight Gas ◽  
Transient Pressure ◽  
Gas Reservoirs ◽  
Type Curves ◽  
Tight Gas Reservoirs ◽  
Diagnostic Plots ◽  
Reservoir Permeability ◽  
Reservoir Flow

Welltest interpretation requires the diagnosis of reservoir flow regimes to determine basic reservoir characteristics. In hydraulically fractured tight gas reservoirs, the reservoir flow regimes may not clearly be revealed on diagnostic plots of transient pressure and its derivative due to extensive wellbore storage effect, fracture characteristics, heterogeneity, and complexity of reservoir. Thus, the use of conventional welltest analysis in interpreting the limited acquired data may fail to provide reliable results, causing erroneous outcomes. To overcome such issues, the second derivative of transient pressure may help eliminate a number of uncertainties associated with welltest analysis and provide a better estimate of the reservoir dynamic parameters. This paper describes a new approach regarding welltest interpretation for hydraulically fractured tight gas reservoirs—using the second derivative of transient pressure. Reservoir simulations are run for several cases of non-fractured and hydraulically fractured wells to generate different type curves of pressure second derivative, and for use in welltest analysis. A field example from a Western Australian hydraulically fractured tight gas welltest analysis is shown, in which the radial flow regime could not be identified using standard pressure build-up diagnostic plots; therefore, it was not possible to have a reliable estimate of reservoir permeability. The proposed second derivative of pressure approach was used to predict the radial flow regime trend based on the generated type curves by reservoir simulation, to estimate the reservoir permeability and skin factor. Using this analysis approach, the permeability derived from the welltest was in good agreement with the average core permeability in the well, thus confirming the methodology’s reliability.

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