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.

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 Numerical Simulation of Unsteady-State Flow in Dual Porous Coalbed Methane Horizontal Wells with Complex Boundary Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Cheng-yong Li ◽  
Jun Zhou ◽  
Xiang-yi Yi ◽  
Yi Luo ◽  
Ping-zhi Gong
Keyword(s):  
Boundary Conditions ◽  
Coalbed Methane ◽  
Gas Flow ◽  
Flow Characteristics ◽  
Horizontal Wells ◽  
Basic Solution ◽  
Pressure Derivative ◽  
Type Curves ◽  
Complex Boundary ◽  
Lord Kelvin

The bottom-hole pressure response which can reflect the gas flow characteristics is important to study. A mathematical model for description of gas from porous coalbed methane (CBM) reservoirs with complex boundary conditions flowing into horizontal wells has been developed. Meanwhile, basic solution of boundary elements has been acquired by combination of Lord Kelvin point source solution, the integral of Bessel function, and Poisson superimpose formula for CBM horizontal wells with complex boundary conditions. Using this model, type curves of dimensionless pressure and pressure derivative are obtained, and flow characteristics of horizontal wells in complex boundary reservoirs and relevant factors are accordingly analyzed.

Study on the Flow Characteristics of Shengli Oilfield Super Heavy Oil

2017 ◽  
Vol 10 (1) ◽  
pp. 69-78 ◽  
Author(s):  
Wang Shou-long ◽  
Li Ai-fen ◽  
Peng Rui-gang ◽  
Yu Miao ◽  
Fu Shuai-shi
Keyword(s):  
Pressure Drop ◽  
Pressure Gradient ◽  
Heavy Oil ◽  
Flow Characteristics ◽  
Fluid Viscosity ◽  
Linear Flow ◽  
Start Up ◽  
Non Linear ◽  
Core Permeability ◽  
Velocity Pressure

Objective:The rheological properties of oil severely affect the determination of percolation theory, development program, production technology and oil-gathering and transferring process, especially for super heavy oil reservoirs. This paper illustrated the basic seepage morphology of super heavy oil in micro pores based on its rheological characteristics.Methods:The non-linear flow law and start-up pressure gradient of super heavy oil under irreducible water saturation at different temperatures were performed with different permeable sand packs. Meanwhile, the empirical formulas between start-up pressure gradient, the parameters describing the velocity-pressure drop curve and the ratio of gas permeability of a core to fluid viscosity were established.Results:The results demonstrate that temperature and core permeability have significant effect on the non-linear flow characteristics of super heavy oil. The relationship between start-up pressure gradient of oil, the parameters representing the velocity-pressure drop curve and the ratio of core permeability to fluid viscosity could be described as a power function.Conclusion:Above all, the quantitative description of the seepage law of super heavy oil reservoir was proposed in this paper, and finally the empirical diagram for determining the minimum and maximum start-up pressure of heavy oil with different viscosity in different permeable formations was obtained.

Integration of Partial Differential Equation for Transient Linear Flow of Gas-Condensate Fluids in Porous Structures

1964 ◽  
Vol 4 (04) ◽  
pp. 291-306 ◽  
Author(s):  
C. Kenneth Eilerts
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Effective Permeability ◽  
Flow Characteristics ◽  
Compressibility Factor ◽  
Variable Coefficients ◽  
Back Pressure ◽  
Absolute Permeability ◽  
Partial Differential ◽  
Linear Flow

Abstract Finite difference equations were programmed and used to integrate the second-order, second-degree, partial differential equation with variable coefficients that represents the transient linear flow of gas-condensate fluids. Effect was given to the change with pressure of the compressibility factor, the viscosity, and the effective permeability and to change of the absolute permeability with distance. Integrations used as illustrations include recovery of fluid from a reservoir at a constant production rate followed by recovery at a declining rate calculated to maintain a constant pressure at the producing boundary. The time required to attain such a limiting pressure and the fraction of the reserve recovered in that time vary markedly with properties of the fluid represented by the coefficients. Fluid also is returned to the reservoir at a constant rate, until initial formation pressure is attained at the input boundary, and then at a calculated rate that will maintain but not exceed the limiting pressure. The computing programs were used to calculate the results that would be obtained in a series of back-pressure tests made at selected intervals of reservoir depletion. If effect is given to the variations in properties of the fluid that actually occur, then a series of back-pressure curves one for each stage of reserve depletion -- is required to indicate open-flow capacity and related flow characteristics dependably. The isochronal performance method for determining flow characteristics of a well was simulated by computation. Introduction The back-pressure test procedure is based on a derivation of the equation for steady-state radial flow of a gas, the properties of which are of necessity assumed to remain unchanged in applying the test results. The properties of most natural gases being recovered from reservoirs change as the reserve is depleted and pressures decline, and the results of an early back-pressure test may not be dependable for estimating the future delivery capacity of a well. The compressibility factor of a fluid under an initial pressure of 10,000 psia can change 45 per cent and the viscosity can change 70 per cent during the productive life of the reservoir. There are indications that the effective permeability to the flowing fluid can become 50 per cent of the original absolute permeability before enough liquid collects in the structure about a well as pressure declines to permit flow of liquid into the well. The advent of programmed electronic computing made practicable the integration of nonlinear, second-order, partial differential equations pertaining to flow in reservoirs. Aronofsky and Porter represented the compressibility factor and the viscosity by a linear relationship, and integrated the equation for radial flow of gas for pressures up to 1,200 psi. Bruce, Peaceman, Rachford and Rice integrated the partial differential equations for both linear and radial unsteady-state flow of ideal gas in porous media. Their published results were a substantial guide in this study of integration of the partial differential equation of linear flow with coefficients of the equation variable. The computing program was developed to treat effective permeability as being both distance-dependent and pressure-dependent. In this study all examples of effective permeability are assumptions designed primarily to aid in developing programs for giving effect to this and other variable coefficients. The accumulation of data for expressing the pressure dependency of the effective permeability is the objective of a concurrent investigation. SPEJ P. 291^

scholarly journals Bayesian Optimization Design of Inlet Volute for Supercritical Carbon Dioxide Radial-Flow Turbine

Machines ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 218
Author(s):  
Chao Bian ◽  
Shaojie Zhang ◽  
Jinguang Yang ◽  
Haitao Liu ◽  
Feng Zhao ◽  
...  
Keyword(s):  
Optimization Design ◽  
Radial Flow ◽  
Flow Characteristics ◽  
Bayesian Optimization ◽  
Working Fluid ◽  
Cross Sectional ◽  
Pressure Loss Coefficient ◽  
Flow Capacity ◽  
Cycle Efficiency ◽  
Total Pressure Loss Coefficient

The radial-flow turbine, a key component of the supercritical CO2 (S-CO2) Brayton cycle, has a significant impact on the cycle efficiency. The inlet volute is an important flow component that introduces working fluid into the centripetal turbine. In-depth research on it will help improve the performance of the turbine and the entire cycle. This article aims to improve the volute flow capacity by optimizing the cross-sectional geometry of the volute, thereby improving the volute performance, both at design and non-design points. The Gaussian process surrogate model based parameter sensitivity analysis is first conducted, and then the optimization process is implemented by Bayesian optimization (BO) wherein the acquisition function is used to query optimal design. The results show that the optimized volute has better and more uniform flow characteristics at design and non-design points. It has a smoother off-design conditions performance curve. The total pressure loss coefficient at the design point of the optimized volute is reduced by 33.26%, and the flow deformation is reduced by 54.55%.

scholarly journals Heat transfer and flow region characteristics study in a non-annular channel between rotor and stator

2012 ◽  
Vol 16 (2) ◽  
pp. 593-603 ◽  
Author(s):  
M. Nili-Ahmadabadi ◽  
H. Karrabi
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Flow Characteristics ◽  
Annular Channel ◽  
Flow Region ◽  
Air Gap ◽  
Heat Transfer Characteristics ◽  
Transfer Characteristics ◽  
Flow And Heat Transfer ◽  
Slip Ring

This paper will present the results of the experimental investigation of heat transfer in a non-annular channel between rotor and stator similar to a real generator. Numerous experiments and numerical studies have examined flow and heat transfer characteristics of a fluid in an annulus with a rotating inner cylinder. In the current study, turbulent flow region and heat transfer characteristics have been studied in the air gap between the rotor and stator of a generator. The test rig has been built in a way which shows a very good agreement with the geometry of a real generator. The boundary condition supplies a non-homogenous heat flux through the passing air channel. The experimental devices and data acquisition method are carefully described in the paper. Surface-mounted thermocouples are located on the both stator and rotor surfaces and one slip ring transfers the collected temperature from rotor to the instrument display. The rotational speed of rotor is fixed at three under: 300rpm, 900 rpm and 1500 rpm. Based on these speeds and hydraulic diameter of the air gap, the Reynolds number has been considered in the range: 4000<Rez<30000. Heat transfer and pressure drop coefficients are deduced from the obtained data based on a theoretical investigation and are expressed as a formula containing effective Reynolds number. To confirm the results, a comparison is presented with Gazley?s (1985) data report. The presented method and established correlations can be applied to other electric machines having similar heat flow characteristics.

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