Interfacial measurements in stratified types of flow. Part I: New optical measurement technique and dry angle measurements

The development of a new non-intrusive computerized image analysis and optical observation method for accurately detecting the vapor–liquid interface in stratified two-phase flows is presented. This technique is applied to a round horizontal sight glass tube using a monochromatic laser sheet for observing two-phase flow patterns and for measuring cross-sectional dry angles and void fractions in these types of flow. The cross-sectional image observed externally is distorted by refraction and is thus reconstructed by computer. From this image, the shape of the vapor–liquid interface is detected and the dry angle and void fraction are accurately determinable over a wide range of conditions for a glass tube of 13.6 mm diameter. Results for dry angle are reported here while the test facility and void fraction measurements are presented in Part II. R-22 and R-410A are the test fluids. Dry angles were quite close to values predicted for stratified flow and much larger than comparable values for air–water flows.     

Condensation heat transfer characteristics of R-22, R-134a and R-410A in small diameter tubes

The condensation heat transfer of pure refrigerants, R-22, R-134a and a binary refrigerant R-410A flowing in small diameter tubes was investigated experimentally. The condenser is a countflow heat exchanger which refrigerant flows in the inner tube and cooling water flows in the annulus. The heat exchanger is smooth, horizontal copper tube of 1.77, 3.36 and 5.35 mm inner diameter, respectively. The length of heat exchanger is 1220, 2660 and 3620 mm, respectively. The experiments were conducted at mass flux of 200–400 kg/m² s and saturation temperature of 40°C. The main results were summarized as follows: in case of single-phase flow, the single-phase Nusselt Number measured by experimental data was higher than that calculated by Gnielinski and Wu and Little correlation. The new single-phase correlation based on the experimental data was proposed in this study. In case of two-phase flow, the condensation heat transfer coefficient of R-410A for three tubes was slightly higher than that of R-22 and R-134a at the given mass flux. The condensation heat transfer coefficient of R-22 showed almost a similar value to that of R-134a. The condensation heat transfer coefficient for R-22, R-134a and R-410A increased with increasing mass flux and decreasing tube diameter. Most of the existing correlations which were proposed in the large diameter tube failed to predict condensation heat transfer. Therefore, the new condensation heat transfer correlation based on the experimental data was proposed in the present study.     

Condensation heat transfer characteristics of R-22, R-134a and R-410A in a single circular microtube

The condensation heat transfer coefficients of R-22, R-134a and R-410A in a single circular microtube were investigated experimentally. The experiments are conducted without oil in the refrigerant loop. The test section is a smooth, horizontal copper tube of 1.77 mm inner diameter. The experiments were conducted at mass flux of 450-1050 kg/m² s, saturation temperature of 40 °C. The test results showed that in case of single-phase flow, the single-phase Nusselt Number measured by experimental data was higher than that calculated by Gnielinski correlation. In case of two-phase flow, the condensation heat transfer coefficient of R-410A was higher than that of R-22 and R-134a at the given mass flux. The condensation heat transfer coefficient of R-22 showed almost a similar value to that of R-134a. Most of the existing correlations which were proposed in the large diameter tube failed to predict condensing heat transfer. And also, recently proposed correlation in the single circular microtube is considered not adequate for small diameter tube. Therefore, it is necessary to develop accurate and reliable correlation to predict heat transfer characteristics in the single circular microtube.     

Interfacial instability of turbulent two-phase stratified flow: Pressure-driven flow and non-Newtonian layers

We derive a flat-interface model to describe the flow of two horizontal, stably stratified fluids, where the bottom layer exhibits non-Newtonian rheology. The model takes into account the yield stress and power-law nature of the bottom fluid. In the light of the large viscosity contrast assumed to exist across the fluid interface, and for large pressure drops in the streamwise direction, the possibility for the upper Newtonian layer to display fully developed turbulence must be considered, and is described in our model. We develop a linear-stability analysis to predict the conditions under which the flat-interface state becomes unstable, and pay particular attention to characterizing the influence of the non-Newtonian rheology on the instability. Increasing the yield stress (up to the point where unyielded regions form in the bottom layer) is destabilizing; increasing the flow index, while bringing a broader spectrum of modes into play, is stabilizing. In addition, a second mode of instability is found, which depends on conditions in the bottom layer. For shear-thinning fluids, this second mode becomes more unstable, and yet more bottom-layer modes can become unstable for a suitable reduction in the flow index. One further difference between the Newtonian and non-Newtonian cases is the development of unyielded regions in the bottom layer, as the linear wave on the interface grows in time. These unyielded regions form in the trough of the wave, and can be observed in the linear analysis for a suitable parameter choice.     

Interfacial instabilities in a stratified flow of two superposed fluids

Here we shall present a linear stability analysis of a laminar, stratified flow of two superposed fluids which are a clear liquid and a suspension of solid particles. The investigation is based upon the assumption that the concentration remains constant within the suspension layer. Even for moderate flow-rates the base-state results for a shear induced resuspension flow justify the latter assumption. The numerical solutions display the existence of two different branches that contribute to convective instability: long and short waves which coexist in a certain range of parameters. Also, a range exists where the flow is absolutely unstable. That means a convectively unstable resuspension flow can be only observed for Reynolds numbers larger than a lower, critical Reynolds number but still smaller than a second critical Reynolds number. For flow rates which give rise to a Reynolds number larger than the second critical Reynolds number, the flow is absolutely unstable. In some cases, however, there exists a third bound beyond that the flow is convectively unstable again. Experiments show the same phenomena: for small flow-rates short waves were usually observed but occasionally also the coexistence of short and long waves. These findings are qualitatively in good agreement with the linear stability analysis. Larger flow-rates in the range of the second critical Reynolds number yield strong interfacial waves with wave breaking and detached particles. In this range, the measured flow-parameters, like the resuspension height and the pressure drop are far beyond the theoretical results. Evidently, a further increase of the Reynolds number indicates the transition to a less wavy interface. Finally, the linear stability analysis also predicts interfacial waves in the case of relatively small suspension heights. These results are in accordance with measurements for ripple-type instabilities as they occur under laminar and viscous conditions for a mono-layer of particles.     

Evaporation flow pattern and heat transfer of R-22 and R-134a in small diameter tubes

The flow patterns and heat transfer coefficients of R-22 and R-134a during evaporation in small diameter tubes were investigated experimentally. The evaporation flow patterns of R-22 and R-134a were observed in Pyrex sight glass tubes with 2 and 8 mm diameter tube, and heat transfer coefficients were measured in smooth and horizontal copper tubes with 1.77, 3.36 and 5.35 mm diameter tube, respectively. In the flow patterns during evaporation process, the annular flows in 2 mm glass tube occurred at a relatively lower vapor quality compared to 8 mm glass tube. The flow patterns in 2 mm glass tube did not agree with the Mandhane’s flow pattern maps. The evaporation heat transfer coefficients in the small diameter tubes (d _i  < 6 mm) were observed to be strongly affected by tube diameters, and to differ from those in the large diameter tubes. The heat transfer coefficients of 1.77 mm tube were higher than those of 3.36 mm and 5.35 mm tube. Most of the existing correlations failed to predict the evaporation heat transfer coefficient in small diameter tubes. Therefore, based on the experimental data, the new correlation is proposed to predict the evaporation heat transfer coefficients of R-22 and R-134a in small diameter tubes.     

Flow regime visualization and pressure drops of HFO-1234yf,R-134a and R-410A during downward two-phase flow in vertical return bends

This paper provides a qualitative visual observation of the two-phase flow patterns for HFO-1234yf and R-134a during downward flow in a vertical 6.7 mm inner diameter glass return bend. The different flow regimes observed are: slug, intermittent and annular flows. Bubble and vapor slug dynamical behaviors in downward slug flow are reported for HFO-1234yf. In addition, to determine the perturbation lengths up- and downstream of the return bend, the total pressure drop has been measured at different pressure tap location up- and downstream of the singularity. Furthermore, 285 pressure drop data points measured for two-phase flow of HFO-1234yf, R-134a and R-410A in vertical downward flow return bends are presented. The flow behavior in the return bend, which is subjected to the complex combined actions of gravity and centrifugal force was expressed in terms of the vapor Froude number. This experimental pressure drop database, which is included in the appendix, is compared to four well-known prediction methods available in the literature.     

Flow regimes and two-phase pressure gradient in horizontal straight tubes: Experimental results for HFO-1234yf, R-134a and R-410A

Two-phase flow regime visualizations of HFO-1234yf and R-134a in a 6.70 mm inner diameter glass straight tube have been simultaneous investigated by top and side views with a high speed high resolution camera. No major difference was observed between both refrigerants. HFO-1234yf flow regimes were satisfactorily predicted by the Wojtan et al. [1] flow pattern map. In addition, 819 pressure drop data points measured during two-phase flow of refrigerants HFO-1234yf, R-134a and R-410A in horizontal straight tubes are presented. The tube diameter (D) varies from 7.90 to 10.85 mm. The mass velocity ranges from 187 to 1702 kg m⁻² s⁻¹ and the saturation temperatures from 4.8 °C to 20.7 °C. The results are compared against 10 well-known two-phase frictional pressure drop prediction methods. For the entire database, the best accuracy is given by the method of Müller-Steinhagen and Heck [2] with around 90% of the data predicted within a ±30% error band. An analysis was carried out on the maximum pressure gradient and on the corresponding vapor quality. A statistical analysis for each flow regime was also carried out.     

A segregated method for compressible flow computation. Part II: general divariant compressible flows

Typically, segregated methods have been used for the computation of incompressible flows whereas coupled solvers, for compressible flows. Compared to coupled solvers, segregated methods present the advantage of computational savings in RAM memory and CPU time, although at the cost of an inferior robustness. However, previously published segregated algorithms for general compressible flows are known to present pitfalls, like convergence to wrong solutions, lack of robustness in the presence of strong discontinuities, such as normal and oblique shocks, and complicated boundary condition imposition. Therefore, in this paper a segregated method for non‐isothermal compressible flows is proposed that preserves the thermodynamic coupling and overcomes the criticisms of existing methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Interfacial energy and dissipation in martensitic phase transformations. Part II: Size effects in pseudoelasticity

This paper is a continuation of the Part I (H. Petryk, S. Stupkiewicz, Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory. J. Mech. Phys. Solids, 2010, doi:10.1016/j.jmps.2009.11.003). A fully three-dimensional model of an evolving martensitic microstructure is examined, taking into account size effects due to the interfacial energy and also dissipation related to annihilation of interfaces. The elastic micro-strain energy at microstructured interfaces is determined with the help of finite element computations and is approximated analytically. Three interface levels are examined: of grain boundaries attained by parallel martensite plates, of interfaces between austenite and twinned martensite, and of twin interfaces within the martensite phase. Minimization of the incremental energy supply, being the sum of the increments in the free energy and dissipation of the bulk and interfacial type at all levels, is used as the evolution rule, based on the theory presented in Part I. An example of the formation and evolution of a rank-three laminated microstructure of finite characteristic dimensions in a pseudoelastic CuAlNi shape memory alloy is examined quantitatively.     

Visual observation of two-phase flow pattern of R-22, R-134a, and R-407C in a 6.5-mm smooth tube

Two-phase flow pattern and friction characteristics for R-22, R-134a, and R-407C inside a 6.5 mm smooth tube are reported in this study. The range of mass flux is between 50 and 700 kg/(m² s). The experimental data show that the two-phase friction multipliers are strongly related to the flow pattern. For a stratified, wavy flow pattern a mass-flux dependence of the multipliers is seen. The flow pattern transition for the mixture refrigerant shows a considerable delay, compared with that of pure refrigerant.     

Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). Herep _c ¦_y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p _c }¦_x represents the large-scale capillary pressure evaluated at the centroid.In addition to{p _c } ^c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as , , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.Roman Letters A scalar that maps {}*/t onto - A scalar that maps {}*/t onto - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - A interfacial area between the -region and the -region contained within, m² - a vector that maps ({}*/t) onto , m - a vector that maps ({}*/t) onto , m - b vector that maps ({p}– g) onto , m - b vector that maps ({p}– g) onto , m - B second order tensor that maps ({p}– g) onto , m² - B second order tensor that maps ({p}– g) onto , m² - c vector that maps ({}*/t) onto , m - c vector that maps ({}*/t) onto , m - C second order tensor that maps ({}*/t) onto , m² - C second order tensor that maps ({}*/t) onto . m² - D third order tensor that maps ( ) onto , m - D third order tensor that maps ( ) onto , m - D second order tensor that maps ( ) onto , m² - D second order tensor that maps ( ) onto , m² - E third order tensor that maps () onto , m - E third order tensor that maps () onto , m - E second order tensor that maps () onto - E second order tensor that maps () onto - p _c =(), capillary pressure relationship in the-region - p _c =(), capillary pressure relationship in the-region - g gravitational vector, m/s² - largest of either or - - - i unit base vector in thex-direction - I unit tensor - K local volume-averaged-phase permeability, m² - K local volume-averaged-phase permeability in the-region, m² - K local volume-averaged-phase permeability in the-region, m² - {K } large-scale intrinsic phase average permeability for the-phase, m² - K –{K }, large-scale spatial deviation for the-phase permeability, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K –{K }, large-scale spatial deviation for the-phase permeability in the-region, m² - K ^* large-scale permeability for the-phase, m² - L characteristic length associated with local volume-averaged quantities, m - characteristic length associated with large-scale averaged quantities, m - I _i i = 1, 2, 3, lattice vectors for a unit cell, m - l characteristic length associated with the-region, m - ; characteristic length associated with the-region, m - l _H characteristic length associated with a local heterogeneity, m - - n unit normal vector pointing from the-region toward the-region (n =–n ) - n unit normal vector pointing from the-region toward the-region (n =–n ) - p pressure in the-phase, N/m² - p local volume-averaged intrinsic phase average pressure in the-phase, N/m² - {p } large-scale intrinsic phase average pressure in the capillary region of the-phase, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p local volume-averaged intrinsic phase average pressure for the-phase in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - p –{p }, large scale spatial deviation for the-phase pressure in the-region, N/m² - P _c p –{p }, capillary pressure, N/m² - {p_c}^c large-scale capillary pressure, N/m² - r ₀ radius of the local averaging volume, m - R ₀ radius of the large-scale averaging volume, m - r position vector, m - , m - S /, local volume-averaged saturation for the-phase - S ^* {}*{}*, large-scale average saturation for the-phaset time, s - t time, s - u , m - U , m² - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - v local volume-averaged phase average velocity for the-phase in the-region, m/s - {v } large-scale intrinsic phase average velocity for the-phase in the capillary region of the-phase, m/s - {v } large-scale phase average velocity for the-phase in the capillary region of the-phase, m/s - v –{v }, large-scale spatial deviation for the-phase velocity, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - v –{v }, large-scale spatial deviation for the-phase velocity in the-region, m/s - V local averaging volume, m³ - V volume of the-phase in, m³ - V large-scale averaging volume, m³ - V capillary region for the-phase within, m³ - V capillary region for the-phase within, m³ - V _c intersection of m³ - V volume of the-region within, m³ - V volume of the-region within, m³ - V () capillary region for the-phase within the-region, m³ - V () capillary region for the-phase within the-region, m³ - V () , region in which the-phase is trapped at the irreducible saturation, m³ - y position vector relative to the centroid of the large-scale averaging volume, m Greek Letters local volume-averaged porosity - local volume-averaged volume fraction for the-phase - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region - local volume-averaged volume fraction for the-phase in the-region (This is directly related to the irreducible saturation.) - {} large-scale intrinsic phase average volume fraction for the-phase - {} large-scale phase average volume fraction for the-phase - {}* large-scale spatial average volume fraction for the-phase - –{}, large-scale spatial deviation for the-phase volume fraction - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - –{}, large-scale spatial deviation for the-phase volume fraction in the-region - a generic local volume-averaged quantity associated with the-phase - mass density of the-phase, kg/m³ - mass density of the-phase, kg/m³ - viscosity of the-phase, N s/m² - viscosity of the-phase, N s/m² - interfacial tension of the - phase system, N/m - , N/m - , volume fraction of the-phase capillary (active) region - , volume fraction of the-phase capillary (active) region - , volume fraction of the-region ( + =1) - , volume fraction of the-region ( + =1) - {p }– g, N/m³ - {p }– g, N/m³     

Improved averaging method for turbulent flow simulation. Part II: Calculations and verification

This is the second of two articles intended to develop, apply and verify a new method for averaging the momentum and mass transport equations for turbulence. Part I presented the theoretical development of a new space-time filter (STF) averaging procedure. The new method, as well as all existing averaging procedures, are applied to the one-dimensional transient equations of momentum and scalar transport in a Burgers' flow field. Dense-grid ‘exact’ results from the unaveraged equations are presented to depict the dynamic behaviour of the flow field and serve as a basis for verifying the coarse-grid STF predictions. In this paper, a finite difference procedure is used to numerically solve the new STF averaged equations, as well as the other forms of the averaged equations derived in Part I. All averaged equations are solved on the same coarse grid. The velocity and scalar fields, predicted from each equation form, are intercompared according to a verification procedure based on the statistical and spectral properties of the results. It is found that the new STF procedure improves coarse-grid dynamic predictions over the existing methods of averaging.     

Laminar boundary layers of co-current gas-liquid stratified flows—II. Velocity measurements

An experimental investigation of the velocity distribution in the laminar boundary layer at a moving interface in co-current gas-liquid flow has been made. The velocity profiles were measured by hot wire technique at different distances from the initial phase interaction location. The experimental results compare well with theoretical predictions which are solutions of the Prandtl's equations.     

Molecular Mixing and Flowfield Measurements in a Recirculating Shear Flow. Part II: Supersonic Flow

Fundamental aspects of mixing between two gaseous streams in a complex geometry are studied and discussed. In the present paper, a supersonic top-stream is expanded over a 30° ramp, through which a secondary lower-stream is injected. The mass flux through the secondary stream is purposely insufficient to provide the entrainment requirements of the resulting shear layer, causing it to attach to the lower guidewall. Part of the shear layer fluid is directed upstream forming a recirculation zone, with enhanced mixing characteristics. The pressure coefficient of the device is quantified as a function of velocity ratio. The effect of heat release on the pressure coefficient is also reported. Molecular mixing was measured employing “flip” experiments based on the hypergolic hydrogen-fluorine chemical reaction. The amount of mixing for the expansion-ramp geometry is found to be higher than in classical free shear layers. However, as in free shear layers, the level of mixing decreases with increasing top-stream velocity. Results for a similar configuration with subsonic/transonic flow in the top stream are reported in Part I of this two-part series.     

Experimental study of turbulence in isothermal and stably stratified homogeneous shear flows: Mean flow measurements

The evolution of freestream turbulence under the combined action of linear shear and stable linear temperature profile is investigated. The experiment is carried out in a small, open circuit, low-speed test cell that uses air as working fluid. The temperature gradient formed at the entrance to the test section by means of an array of 24 horizontal, differentially heated elements is varied to get a maximum Brunt-Vaisala frequency N_o[=({g/T_m}{∂T/∂y})^1/2] of 3.1⁻¹. Linear velocity profiles are produced using screens of variable mesh size. The Reynolds number Re_M based on centre-line velocity and mesh size is varied from 80 to 175. Isothermal studies are carried out in four different experiments with varying velocity gradients. The effect of inlet turbulence level on growth of turbulence is studied in these flows by keeping the shear parameter Sh (=(x/u)(∂u/∂y)) constant. The range of shear parameters considered is 2.5–7.0. Shear and stratification combined produce a maximum gradient Richardson number Ri_g (= N_o²/(∂u/∂y)²) of 0.0145. Results have been presented in terms of evolution of variance of velocity fluctuations, Reynolds shear stress and temperature fluctuations. Measurements show the following: In isothermal flows the growth rate of turbulence quantities depends on both shear parameter and inlet turbulence level. There are distinct stages in the evolution of the flow and that can be identified by the power-law exponent of growth of turbulence. Shear is seen to promote the growth of turbulence and accelerate it towards a fully developed equilibrium state. Stratification initially suppresses the growth of turbulence and, hence, enhances the degree of underdevelopment. Under these conditions shear becomes active and subsequently enhances the growth rate of turbulence quantities.     

Planar laser-induced fluorescence (PLIF) measurements of liquid film thickness in annular flow. Part II: Analysis and comparison to models

Film thickness distributions in upward vertical air–water annular flow have been determined using planar laser-induced fluorescence (PLIF). Film thickness data are frequently used to estimate interfacial shear and pressure loss. This film roughness concept has been used in a number of models for annular flow of varying complexity. The PLIF data are presently applied to the single-zone interfacial shear correlation of Wallis; the more detailed model of Owen and Hewitt; and the two-zone (base film and waves) model of Hurlburt, Fore, and Bauer. For the present data, these models all under-predict the importance of increasing liquid flow on pressure loss and interfacial shear. Since high liquid flow rates in annular flow induce disturbance wave and entrainment activity, further modeling in these areas is advised.     

Macro-mechanical modeling of blast-wave mitigation in foams. Part II: reliability of pressure measurements

A phenomenological study of the process occurring when a plane shock wave reflected off an aqueous foam column filling the test section of a vertical shock tube has been undertaken. The experiments were conducted with initial shock wave Mach numbers in the range $1.25\le {M}_\mathrm{s} \le 1.7$ and foam column heights in the range 100–450 mm. Miniature piezotrone circuit electronic pressure transducers were used to record the pressure histories upstream and alongside the foam column. The aim of these experiments was to find a simple way to eliminate a spatial averaging as an artifact of the pressure history recorded by the side-on transducer. For this purpose, we discuss first the common behaviors of the pressure traces in extended time scales. These observations evidently quantify the low frequency variations of the pressure field within the different flow domains of the shock tube. Thereafter, we focus on the fronts of the pressure signals, which, in turn, characterize the high-frequency response of the foam column to the shock wave impact. Since the front shape and the amplitude of the pressure signal most likely play a significant role in the foam destruction, phase changes and/or other physical factors, such as high capacity, viscosity, etc., the common practice of the data processing is revised and discussed in detail. Generally, side-on pressure measurements must be used with great caution when performed in wet aqueous foams, because the low sound speed is especially prone to this effect. Since the spatial averaged recorded pressure signals do not reproduce well the real behaviors of the pressure rise, the recorded shape of the shock wave front in the foam appears much thicker. It is also found that when a thin liquid film wet the sensing membrane, the transducer sensitivity was changed. As a result, the pressure recorded in the foam could exceed the real amplitude of the post-shock wave flow. A simple procedure, which allows correcting this imperfection, is discussed in detail.     

An investigation of hydraulic fracturing. Part II: two-phase flow model

In the previous work presented in Part I (Theoret. Appl. Fracture Mech. 18, 89–102 (1993)), hydraulic fracture in an infinitely large saturated porous medium is analyzed under an assumption of one-phase flow in the medium. The investigation is extended in this paper to the case of a two phase saturated immiscible flow of oil and water in the porous medium. The medium is initially saturated with oil. Flow in the medium is induced by diffusion of water injected into the fracture. The quasi-static growth of the fracture for a prescribed injection rate is analyzed based on the assumptions that the pressure in the fracture is uniform and that the permeating flow in the medium is unidirectional. The constant fracture toughness criterion for plane strain deformation is employed and the effect of capillary pressure is neglected. Empirical formulas are used for the permeabilities of the oil and water phases. It is seen that the distributions of water saturation and pore pressure in the medium are governed by two nonlinear partial differential equations. Numerical solutions are obtained by a finite difference scheme with iterations. It is found that the injected water is restricted within a layer near the surface of the fracture whose thickness is small compared with the length of the fracture. Thus the flow in the medium is governed essentially by the oil phase. To compare our problem with the corresponding problem of one-phase flow, we find that the difference in crack growth in these two problems is small for the ration of kinematic viscosities of the oil and water phases within the practical range. Hence our study confirms the validity of the one phase flow assumption used in the previous work for prediction of hydraulic fracture growth.