Vortex flow adjacent to a stationary surface

The radius of transition from an inner core of solid body rotation to an outer free vortex motion was determined via the momentum integral method.     

Equilibrium of a pressurized plastic fluid in a wellbore crack

This paper investigates equilibrium of a pressurized plastic fluid invading a tensile wellbore crack in a linear elastic, permeable rock. The crack is initially filled by pore fluid at ambient pressure, that is immiscibly displaced by the plastic fluid invading from the wellbore. The plastic fluid comes to rest to form a “plug” within the elastically deformed crack when the limit equilibrium between the shear stresses generated at the fracture walls and the pressure drop between the wellbore wall and the crack tip is reached. The model assumes that the leak-off of the plastic fluid into the rock is negligible, while the displaced pore fluid in the crack tip region is freely exchanged with the surrounding permeable rock to maintain the ambient pressure level. When the crack length ? is small or large compared to the wellbore radius R, the problem reduces to that of a pressurized edge or Griffith’s crack, respectively, subjected to a uniform far-field confining stress. In these two end-member cases, the normalized solution for the net pressure distribution, the plug length, and the stress intensity factor at the crack tip is obtained as a function of two numbers – the normalized net fluid pressure at the crack inlet and at the crack tip (partial plugs only) – that embody the solution’s dependence on the wellbore and the far field loading, the fluid yield strength, and the rock modulus. In the general case of an intermediate crack length (? ～ R), the normalized solution is a function of two additional parameters, the length-to-radius ratio and a normalized measure of the far field stress anisotropy, respectively, which accurate approximation is devised from an end-member solution using a rescaling argument. The equilibrium plug solutions are used to evaluate the breakdown pressure, the critical wellbore pressure at which the crack propagation condition is first met, and to analyze the stability of the ensuing crack propagation.     

Newtonian and non-Newtonian liquids rotating adjacent to a stationary surface

The interaction of a rotating flow and a stationary surface is discussed for a second-order non-Newtonian liquid. Similarity solutions of the governing partial differential equations are obtained for the case of the outer flow in solid-body rotation. The results for the Newtonian case are compared with Bödewadt's series solution of this problem. The non-Newtonian solutions indicate that for certain values of the parameters characterizing the non-linear viscous response and normal stress effects a larger secondary flow is induced in the boundary layer than in the Newtonian case.Also at North Carolina State University Raleigh (N.C.), U.S.A.     

Motion of a cylinder adjacent to a free-surface: flow patterns and loading

The flow structure and loading due to combined translatory and sinusoidal motion of a cylinder adjacent to a free-surface are characterized using a cinema technique of high-image-density particle image velocimetry and simultaneous force measurements. The instantaneous patterns of vorticity and streamline topology are interpreted as a function of degree of submergence beneath the free-surface. The relative magnitudes of the peak vorticity and the circulation of vortices formed from the upper and lower surfaces of the cylinder, as well as vortex formation from the free-surface, are remarkably affected by the nominal submergence. The corresponding streamline topology, interpreted in terms of foci, saddle points, and multiple separation and reattachment points also exhibit substantial changes with submergence. All of these features affect the instantaneous loading of the cylinder. Calculation of instantaneous moments of vorticity and the incremental changes in these moments during the cylinder motion allow identification of those vortices that contribute most substantially to the instantaneous lift and drag. Furthermore, the calculated moments are in general accord with the time integrals of the measured lift and drag acting on the cylinder for sufficiently large submergence. Received: 18 May 1998/Accepted: 18 August 1999     

考虑水泥环完整性的油气井最大允许井口压力计算方法

在最大允许井口压力计算方法中考虑水泥环的完整性,以厚壁圆筒的弹塑性分析理论为基础,建立了套管-水泥环-地层组合体力学模型,将(内层)水泥环内壁发生破坏作为环空允许带压的限制条件。基于Drucker-Prager与拉伸破坏准则计算了某实例井各环空的最大允许井口压力,并将其结果与采用API RP 90标准推荐做法的计算结果进行了对比。计算结果表明:如果环空带压值增大,对于套管-水泥环-地层组合体而言,垂深最深与最浅处的水泥环内壁将最有可能率先发生破坏;采用本文方法的计算结果有可能会小于依据API RP 90标准的计算结果,依据API RP 90标准计算得到A、B环空的最大允许井口压力分别为17.04MPa和6.39MPa,而采用本文方法计算得到A、B环空的最大允许井口压力分别为15.1MPa和17.5MPa。     

水泥试样爆炸压裂实验及裂纹分形评价

通过室内爆炸压裂模拟实验,得到常压和20 MPa围压2种情况下的爆炸压裂效果,并对实验结果进行了对比分析,分析结果表明:围压对裂纹扩展路径、形状、弯曲度都有一定影响,与常压下的裂纹扩展情况相比,施加围压后得到的裂纹数量少、长度短、弯曲明显.在对爆炸压裂形成的裂纹进行定性描述的基础上,利用分形理论,对裂纹的分形特征进行了...     

Deformation of a tensile membrane adjacent to a moving wall caused by viscous flow

Theoretical analysis is done for deformation of a tensile membrane caused by viscous flow between the membrane and a solid boundary (wafer) placed adjacent to the membrane. The membrane and its support are assumed to have circular shapes. The boundary is assumed to move perpendicularly to the membrane by a small fraction of the gap width. The Hele–Shaw flow theory is applied to the flow. Variation of the central gap width is obtained, and it is shown that the maximum membrane deformation depends on the membrane tension and the final gap width. The dependence agrees with experimental results.     

Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet

An analysis is made for the steady two-dimensional magneto-hydrodynamic flow of an incompressible viscous and electrically conducting fluid over a stretching vertical sheet in its own plane. The stretching velocity, the surface temperature and the transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The transformed boundary layer equations are solved numerically for some values of the involved parameters, namely the magnetic parameter M, the velocity exponent parameter m, the temperature exponent parameter n and the buoyancy parameter λ, while the Prandtl number Pr is fixed, namely Pr = 1, using a finite difference scheme known as the Keller-box method. Similarity solutions are obtained in the presence of the buoyancy force if n = 2m−1. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. It is found that both the skin friction coefficient and the local Nusselt number decrease as the magnetic parameter M increases for fixed λ and m. For m = 0.2 (i.e. n = −0.6), although the sheet and the fluid are at different temperatures, there is no local heat transfer at the surface of the sheet except at the singular point of the origin (fixed point).     

A semi-analytic solution of a wellbore in a non-isothermal low-permeability porous medium under non-hydrostatic stresses

The transient stress, displacement, pore pressure and temperature fields around a wellbore in a thermo-poro-elastic (THM) medium subject to non-hydrostatic remote stresses are analyzed under non-isothermal plane-strain conditions. The linear THM model proposed by Coussy (1989) is adopted in the analysis with a focus on thermal effects in low-permeability saturated rocks, characterized by a latent heat associated with local changes of fluid mass content. Non-dimensionalized parameters are identified by reformulating the fully-coupled governing equations and boundary conditions. The wellbore problem is simplified by decomposing it into axisymmetric and deviatoric loading cases. The corresponding analytical solutions are obtained in Laplace space. The inverse Laplace transforms are performed numerically to find the time-dependent distributions of field variables in the rock mass around the wellbore. These numerical results show that although the pore pressure diffusion has little influence on temperature and stress, temperature changes can strongly affect the pore pressure and stress around the wellbore. The temperature change can lead to changes in near-well stresses and the resulting significant change in wellbore breakdown pressure illustrates the importance of considering the THM coupling.     

Fracture of the hanging wall of an excavation adjacent to a decline fault

Summary This paper presents the mathermatical modelling of a real rock mechanics problem concerning mining in the neighbourhood of the fault. The excavation starts at the fault surface and is carried out in the direction to the interior of the region.

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Bruch der überhängenden Wand beim Abbau in der Nähe von schrägen Verwerfungen

Übersicht Diese Arbeit stellt das mathematische Modell eines realen Problems der Felsmechanik vor, bei dem der Abbau in der Nähe von Verwerfungen erfolgt. Der Abbau beginnt an der Oberfläche der Verwerfung und wird in das Innere der Verwerfungszone weitergeführt.

     

Free convection in a saturated porous medium adjacent to a non-isothermal vertical impermeable wall

Boundary layer analysis is performed for free convection in a saturated porous medium adjacent to non-isothermal vertical impermeable surfaces. The impermeable surface temperature is assumed to be an arbitrary function of the distance along the surface. The solutions are obtained in the form of perturbations to the isothermal case. Using the differentials of the wall temperature, which are functions of distance along the surface, as perturbation elements, universal functions are derived. These universal functions can be used to estimate the heat transfer to any type of wall temperature variation. Solutions for some specialized wall temperature variations are derived using these universal functions and are compared with the solutions that are available in the literature. The agreement is found to be good. The case of uniform wall heat flux turns out to be a special case of non-isothermal wall solution.Es wurde eine Untersuchung der Grenzschicht bei freier Konvektion in einem gesättigten porösen Medium, das an eine nicht isotherme undurchlässige Oberfläche angrenzt, durchgeführt. Die Temperatur der undurchlässigen Oberfläche wurde als beliebige Ortsfunktion der Oberfläche angenommen. Lösungen werden in Form von Störungen des isothermen Falles erhalten. Unter Benutzung der Differentiale der Wandtemperatur, welche Ortsfunktionen der Oberfläche sind, werden als Störfunktion universelle Funktionen erhalten. Diese universellen Funktionen können benutzt werden, um die Wärmeübertragung für beliebige Variationen der Wandtemperatur zu bestimmen. Für einige spezielle Wandtemperaturen werden Lösungen unter Benutzung dieser universellen Funktionen abgeleitet und mit Lösungen aus der Literatur verglichen; gute Übereinstimmung kann festgestellt werden. Der Fall der einheitlichen Wandwärmestromdichte stellt sich als Spezialfall der nicht isothermen Lösung heraus.     

Unsteady mixed convection flow in stagnation region adjacent to a vertical surface

The unsteady two-dimensional laminar mixed convection flow in the stagnation region of a vertical surface has been studied where the buoyancy forces are due to both the temperature and concentration gradients. The unsteadiness in the flow and temperature fields is caused by the time-dependent free stream velocity. Both arbitrary wall temperature and concentration, and arbitrary surface heat and mass flux variations have been considered. The Navier-Stokes equations, the energy equation and the concentration equation, which are coupled nonlinear partial differential equations with three independent variables, have been reduced to a set of nonlinear ordinary differential equations. The analysis has also been done using boundary layer approximations and the difference between the solutions has been discussed. The governing ordinary differential equations for buoyancy assisting and buoyancy opposing regions have been solved numerically using a shooting method. The skin friction, heat transfer and mass transfer coefficients increase with the buoyancy parameter. However, the skin friction coefficient increases with the parameter , which represents the unsteadiness in the free stream velocity, but the heat and mass transfer coefficients decrease. In the case of buoyancy opposed flow, the solution does not exist beyond a certain critical value of the buoyancy parameter. Also, for a certain range of the buoyancy parameter dual solutions exist.Die zweidimensionale laminare Mischkonvektionsströmung im Staubereich einer vertikalen Oberfläche, in der Temperatur- und Konzentrationsgradienten die Auftriebskräfte erzeugen, wurde untersucht. Die Unstetigkeiten im Strömungs- und im Temperaturfeld liegen in der zeitabhängigen freien Strömungsge-schwindigkeit begründet. Die willkürliche Wandtempereratur und Konzentration sowie die willkürliche Oberflächenwärme- und die Massenstromschwankungen wurden in Betracht gezogen. Die Navier-Stokes-Gleichung, die Energiegleichung und die Konzentrationsgleichung, die drei nicht lineare, partielle Differentialgleichungen mit drei unabhängigen Variablen darstellen, sind auf eine Gruppe von nicht linearen, gewöhnlichen Differentialgleichungen reduziert worden. Die Berechnung ist auch mit der Grenzflächenap-proximation gemacht worden und die Unterschiede der Ergebnisse wurden diskutiert.Die bestehenden gewöhnlichen Differentialgleichungen für auftriebsuntertützte und auftriebshemmende Bereiche sind numerisch mit dem Shooting-Verfahren gelöst worden. Die Oberflächenreibung, die Wärme- und Stoffübertragungskoeffizienten steigen mit dem Auftriebsparameter. Der Oberflächenreibungskoeffizient steigt mit dem Parameter, der auch für die Unstetigkeit der freien Strömungsgeschwindigkeit verantwortlich ist. Die Wärme- und Stoffübertragungskoeffizienten sinken dann. Im Fall der auftriebsgehemmten Strömung, existiert nach einem bestimmten kritischen Wert des Auftriebsparameters keine Lösung mehr. Für eine bestimmte Reihe von Auftriebsparametern gibt es zwei Lösungen.     

Mixed convection adjacent to inclined flat surfaces embedded in a porous medium

An analysis is performed to study the flow and heat transfer characteristics of laminar mixed convection boundary layer flows from inclined (including horizontal and vertical) surfaces embedded in a saturated porous medium with constant aiding external flows and uniform surface temperature. Both the streamwise and normal components of the buoyancy forces are retained in the momentum equations. Nondimensionalization of the boundary layer equations results in the following three governing parameter: (1)Gr/Re, the ratio of the Grashof number to the Reynolds number; (2)Pe _x =Re _x Pr, the Peclet number; (3) φ, the angle of inclination from the horizontal. The resulting nonsimilar equations are solved by an efficient implicit finite-difference scheme. Numerical results are presented for flows with different values ofGr/Re in the range of 0 to 50, over a wide range of the Peclet numbersPe _x, and various values of φ ranging from 0 to 90 degrees. It is found that the local surface heat transfer rate increases with increasing the local Peclet number. In addition, as the plate is tilted from the horizontal to the vertical orientation, the local Nusselt number increases for a given Peclet number and the effect of the buoyancy force on the surface heat transfer rate increases.     

Simultaneous initiation and growth of multiple radial hydraulic fractures from a horizontal wellbore

Multi-stage fracturing is the current preferred method of completion of horizontal wells in unconventional hydrocarbon reservoirs. Its core component consists in simultaneously initiating and propagating an array of hydraulic fractures. We develop a numerical model for the initiation and growth of an array of parallel radial hydraulic fractures. The solution accounts for fracture growth, coupling between elastic deformation and fluid flow in the fractures, elastic stress interactions between fractures and fluid flow in the wellbore. We also take into account the presence of a local pressure drop (function of the entering flow rate) at the connection between the well and the fracture, i.e., a choke-like effect due to current well completion practices, also referred to as entry friction. The partitioning of the fluid into the different fractures at any given time is part of the solution and is a critical indicator of simultaneous (balanced fluid partitioning) versus preferential growth. We validate our numerical model against reference solutions and a laboratory experiment for the initiation and growth of a single radial hydraulic fracture. We then investigate the impact of stress interaction on preferential growth of a subset of fractures in the array. Our results show that a sufficiently large local entry friction provides a strong feedback in the system and thus can counteract elastic stress interaction between fractures, thereby ensuring simultaneous growth. We propose a dimensionless number capturing the competition between stress interaction and local entry friction. This dimensionless number is a function of rock properties, fracture spacing and injection parameters. We verify that it captures the transition from the case of simultaneous growth (entry friction larger than interaction stress) to the case of preferential growth of some fractures (interaction stress larger than entry friction). We also discuss the implication of these results for multi-stage fracturing engineering practices.     

Free convection in a saturated porous medium adjacent to a vertical impermeable wall subjected to a non-uniform heat flux

Boundary layer analysis is performed for free convection in a fluid-saturated porous medium adjacent to a vertical impermeable wall subjected to a non-uniform heat flux. The wall heat flux is assumed to be an arbitrary function of the distance along the surface. The solutions are obtained in the form of perturbations to the uniform heat flux case. Using the differentials of the wall heat flux, which are functions of the distance along the surface, as perturbation elements, universal functions are obtained. These universal functions can be used to estimate the heat transfer for any type of wall heat flux variation. For the wall heat flux variation as a power function of the distance from the origin, the solutions obtained by using these universal functions have been compared with those obtained by similarity analysis and the agreement is found to be good. Further, solutions are presented for wall heat flux varying exponentially and sinusoidally but comparison could not be drawn due to non-availability of solutions in the literature.     

Steady film condensation and boiling adjacent to a body of arbitrary shape in a porous medium

The problems of steady film condensation and boiling over a body of arbitrary shape embedded in a porous medium have been attacked by means of the similarity transformation. A new similarity variable is suggested to solve the problems once for all possible two-dimensional and axisymmetric bodies of arbitrary geometrical configuration. Upon transforming the governing equations and boundary conditions using the similarity variables, the resulting set of equations transforms into that obtained for the case of a vertical flat plate. Thus, the numerical values furnished by Parmentier and Cheng for a flat surface are readily applicable to two-phase flows over two-dimensional and axisymmetric bodies of arbitrary shape.     

Sound propagation through a discretely inhomogeneous thermoelastic plane layer adjacent to heat-conducting liquids

An analytical solution of the problem of the propagation of a plane sound wave through a discretely inhomogeneous thermoelastic layer adjacent to inviscid heat-conducting liquids is obtained. Results of calculations of the dependences of the transmission coefficient on the wave incidence angle and frequency for discretely inhomogeneous and continuously inhomogeneous thermoelastic layers are given. It is shown that a thermoelastic layer with continuously inhomogeneous thickness can be simulated using a system of homogeneous thermoelastic layers.     

Analysis of wellbore crossflows for nonstationary operation of a well in an inhomogeneous multilayer bed

This paper considers the nonstationary filtration of a homogeneous elastic fluid in a rigid, inhomogeneous, multilayer bed with no hydrodynamic communication between interbeds. The effect of wellbore crossflow between interbeds after stopping of well operation is studied numerically. It is shown that the volume and time of fluid crossflow between the interbeds are comparable to the volume of injection and the time of chemical treatment of wells. It is concluded that crossflows can be used for selective injection of chemicals. A procedure is proposed to control the volume of selective fluid injection into a well by changing the injection rate.