Pressure transient characteristics of non-uniform conductivity fractured wells in viscoelasticity polymer flooding based on oil-water two-phase flow

Polymer flooding in fractured wells has been extensively applied in oilfields to enhance oil recovery.In contrast to water,polymer solution exhibits non-Newtonian and nonlinear behavior such as effects of shear thinning and shear thickening,polymer convection,diffusion,adsorption retention,inaccessible pore volume and reduced effective permeability.Meanwhile,the flux density and fracture conductivity along the hydraulic fracture are generally non-uniform due to the effects of pressure distribution,formation damage,and proppant breakage.In this paper,we present an oil-water two-phase flow model that captures these complex non-Newtonian and nonlinear behavior,and non-uniform fracture characteristics in fractured polymer flooding.The hydraulic fracture is firstly divided into two parts:high-conductivity fracture near the wellbore and low-conductivity fracture in the far-wellbore section.A hybrid grid system,including perpendicular bisection(PEBI)and Cartesian grid,is applied to discrete the partial differential flow equations,and the local grid refinement method is applied in the near-wellbore region to accurately calculate the pressure distribution and shear rate of polymer solution.The combination of polymer behavior characterizations and numerical flow simulations are applied,resulting in the calculation for the distribution of water saturation,polymer concentration and reservoir pressure.Compared with the polymer flooding well with uniform fracture conductivity,this non-uniform fracture conductivity model exhibits the larger pressure difference,and the shorter bilinear flow period due to the decrease of fracture flow ability in the far-wellbore section.The field case of the fall-off test demonstrates that the proposed method characterizes fracture characteristics more accurately,and yields fracture half-lengths that better match engineering reality,enabling a quantitative segmented characterization of the near-wellbore section with high fracture conductivity and the far-wellbore section with low fracture conductivity.The novelty of this paper is the analysis of pressure performances caused by the fracture dynamics and polymer rheology,as well as an analysis method that derives formation and fracture parameters based on the pressure and its derivative curves.

A two-phase type-curve method with multiscale fluid transport mechanisms in hydraulically fractured shale reservoirs

The quantitative understanding of hydraulic fracture(HF)properties guides accurate production forecasts and reserve estimation.Type curve is a powerful technique to characterize HF and reservoir properties from flowback and long-term production data.However,two-phase flow of water and hydrocarbon after an HF stimulation together with the complex transport mechanisms in shale nanopores exacerbate the nonlinearity of the transport equation,causing errors in type-curve analysis.Accordingly,we propose a new two-phase type-curve method to estimate HF properties,such as HF volume and permeability of fracture,through the analysis of flowback data of multi-fractured shale wells.The proposed type curve is based on a semianalytical solution that couples the two-phase flow from the matrix with the flow in HF by incorporating matrix influx,slippage effect,stress dependence,and the spatial variation of fluid properties in inorganic and organic pores.For the first time,multiple fluid transport mechanisms are considered into two-phase type-curve analysis for shale reservoirs.We analyze the flowback data from a multi-fractured horizontal well in a shale gas reservoir to verify the field application of the proposed method.The results show that the fracture properties calculated by the type-curve method are in good agreement with the long-time production data.

Fluid−Structure Interaction of Two-Phase Flow Passing Through 90° Pipe Bend Under Slug Pattern Conditions

Numerical simulations of evolution characteristics of slug flow across a 90°pipe bend have been carried out to study the fluid−structure interaction response induced by internal slug flow.The two-phase flow patterns and turbulence were modelled by using the volume of fluid(VOF)model and the Realizable k−εturbulence model respectively.Firstly,validation of the CFD model was carried out and the desirable results were obtained.The different flow patterns and the time-average mean void fraction was coincident with the reported experimental data.Simulations of different cases of slug flow have been carried out to show the effects of superficial gas and liquid velocity on the evolution characteristics of slug flow.Then,a one-way coupled fluid-structure interaction framework was established to investigate the slug flow interaction with a 90°pipe bend under various superficial liquid and gas velocities.It was found that the maximum total deformation and equivalent stress increased with the increasing superficial gas velocity,while decreased with the increasing superficial liquid velocity.In addition,the total deformation and equivalent stress has obvious periodic fluctuation.Furthermore,the distribution position of maximum deformation and stress was related to the evolution of slug flow.With the increasing superficial gas velocity,the maximum total deformation was mainly located at the 90°pipe bend.But as the superficial liquid velocity increases,the maximum total deformation was mainly located in the horizontal pipe section.Consequently,the slug flow with higher superficial gas velocity will induce more serious cyclical impact on the 90°pipe bend.

SECOND-ORDER MOMENT MODEL FOR DENSE TWO-PHASE TURBULENT FLOW OF BINGHAM FLUID WITH PARTICLES

The USM-θ model of Bingham fluid for dense two-phase turbulent flow was developed, which combines the second-order moment model for two-phase turbulence with the particle kinetic theory for the inter-particle collision. In this model, phases interaction and the extra term of Bingham fluid yield stress are taken into account. An algorithm for USM-θ model in dense two-phase flow was proposed, in which the influence of particle volume fraction is accounted for. This model was used to simulate turbulent flow of Bingham fluid single-phase and dense liquid-particle two-phase in pipe. It is shown USM-θ model has better prediction result than the five-equation model, in which the particle-particle collision is modeled by the particle kinetic theory, while the turbulence of both phase is simulated by the two-equation turbulence model. The USM-θ model was then used to simulate the dense two-phase turbulent up flow of Bingham fluid with particles. With the increasing of the yield stress, the velocities of Bingham and particle decrease near the pipe centre. Comparing the two-phase flow of Bingham-particle with that of liquid-particle, it is found the source term of yield stress has significant effect on flow.

MATHEMATICAL MODEL OF TWO-PHASE FLUID NONLINEAR FLOW IN LOW-PERMEABILITY POROUS MEDIA WITH APPLICATIONS

A mathematical model of two-phase fluid nonlinear flow in the direction of normal of ellipse through low-permeability porous media was established according to a nonlinear flow law expressed in a continuous function with three parameters, a mass conservation law and a concept of turbulent ellipses. A solution to the model was obtained by using a finite difference method and an extrapolation method. Formulas of calculating development index not only before but also after water breaks through an oil well in the condition of two-phase fluid nonlinear flow in the media were derived. An example was discussed. Water saturation distribution was presented. The moving law of drainage front was found. Laws of change of pressure difference with time were recognized. Results show that there is much difference of water saturation distribution between nonlinear flow and linear flow; that drainage front by water moves faster, water breaks through sooner and the index gets worse because of the nonlinear flow; and that dimensionless pressure difference gets larger at the same dimensionless time and difficulty of oil development becomes bigger by the nonlinear flow. Thus, it is necessary that influence of nonlinear flow on development indexes of the oil fields be taken into account. The results provide water-flooding development of the oilfields with scientific basis.

Numerical Investigation on Dynamic Response Characteristics of Fluid-Structure Interaction of Gas-Liquid Two-Phase Flow in Horizontal Pipe

Fluid-structure interaction(FSI)of gas-liquid two-phase fow in the horizontal pipe is investigated numerically in the present study.The volume of fluid model and standard k-e turbulence model are integrated to simulate the typical gas-liquid two-phase fow patterns.First,validation of the numerical model is conducted and the typical fow patterns are consistent with the Baker chart.Then,the FSI framework is established to investigate the dynamic responses of the interaction between the horizontal pipe and gas-liquid two-phase fow.The results show that the dynamic response under stratified fow condition is relatively flat and the maximum pipe deformation and equivalent stress are 1.8 mm and 7.5 MPa respectively.Meanwhile,the dynamic responses induced by slug fow,wave fow and annular fow show obvious periodic fuctuations.Furthermore,the dynamic response characteristics under slug flow condition are maximum;the maximum pipe deformation and equivalent stress can reach 4mm and 17.5 MPa,respectively.The principal direction of total deformation is different under various flow patterns.Therefore,the periodic equivalent stress will form the cyclic impact on the pipe wall and affect the fatigue life of the horizontal pipe.The present study may serve as a reference for FSI simulation under gas-liquid two-phase transport conditions.

MICROGRAVITY EXPERIMENTS OF TWO-PHASE FLOW PATTERNS ABOARD MIR SPACE STATION

A first experimental study on two-phase how patterns at a long-term, steady microgravity condition was conducted on board the Russian Space Station 'MIR' in August 1999. Carbogal and air are used as the liquid and the gas phase, respectively. Bubble, slug, slug-annular transitional, and annular hows are observed. A new region of annular how with lower liquid superficial velocity is discovered, and the region of the slug-annular transitional flow is wider than that observed by experiments on board the parabolic aircraft. The main patterns are bubble, slug-annular transitional and annular flows based on the experiments on board MIR space station. Some influences on the two-phase how patterns in the present experiments are discussed.

Prediction of Flowing Bottomhole Pressures for Two-Phase Coalbed Methane Wells

A method is proposed to predict the flowing bottomhole pressures （FBHPs） for two-phase coalbed methane （CBM） wells. The mathematical models for both gas column pressure and two-phase fluid column pressure were developed based on the well liquid flow equation. FBHPs during the production were predicted by considering the effect of entrained liquid on gravitational gradients. Comparison of calculated BHPs by Cullender-Smith and proposed method was also studied. The results show that the proposed algorithm gives the desired accuracy of calculating BHPs in the low- productivity and low-pressure CBM wells. FBHP is resulted from the combined action of wellhead pressure, gas column pressure and fluid column pressure. Variation of kinetic energy term, compressibility and friction factors with depth increments and liquid holdup with velocity should be considered to simulate the real BHPs adequately. BHP is a function of depth of each column segment. The small errors of less than 1.5% between the calculated and measured values are obtained with each segment within 25 m. Adjusting BHPs can effectively increase production pressure drop, which is beneficial to CBM desorption and enhances reservoir productivity. The increment of pressure drop from 5.37 MPa2 to 8.66 MPa2 leads to an increase of CBM production from 3270 m3/d to 6700 m3/d and is attributed to a decrease in BHP from 2.25 MPa to 1.33 MPa.

CFD studies of scraper built in SPA clarifying tank based on mixture solid-liquid two-phase flow model

In the industrial process of producing the strong phosphoric acid(SPA),clarification of the solution is essential to the ultimate product.However,the large viscosity of sediment and the induced interface interaction result in difficulties when the SPA is clarified.CFD numerical methodology was applied to simulate internal flow field and performance of the low speed scraper based on Mixture solidliquid two-phase flow model.Sediment deposition was generated by loading solid particles at the bottom of clarifying vessel.The moving mesh and RNG k-εmodel were used to simulate the rotational turbulent flow in clarifying tank.Variables studied,amongst others,were the scraper rotation speed and the mounting height,which could affect the solid suspension height.Features of flow field and solid volume fraction distribution in computational domain were presented and analyzed.The numerical reports of the scraper torque and velocities of inlet and outlet filed were obtained.It seems the torque value of rotatio-nal axis and particle suspending height augment with an increasing rotating speed.Meanwhile,a high revolving speed is good for the deposition discharge.The particle fraction distribution in meridional surface and horizontal surface at fixed rotation speed were analyzed to determine the corresponding optimal installation height.The simulating results reflect the flow field is marginally stirred by the scraper and proper working parameters are obtained,in which case the comprehensive properties of the scraper and the clarifying tank are superior.

Effects of nonlinear velocity slip and temperature jump onpseudo-plastic power-law fluid over moving permeable surface in presence of magnetic field

Flow and heat transfer of a pseudo-plastic power-law fluid over a stretching permeable surface with the magnetic effect is investigated. In the boundary conditions,the nonlinear temperature jump and the velocity slip are considered. Semi-similarity equations are obtained and solved by bvp4c with MATLAB. The problem can be considered as an extension of the previous work done by Mahmoud(Mahmoud, M. A. A. Slip velocity effect on a non-Newtonian power-law fluid over a moving permeable surface with heat generation. Mathematical and Computer Modelling, 54, 1228–1237(2011)). Efforts are made to discuss the effects of the power-law number, slip velocity, and temperature jump on the dimensionless velocity and temperature distribution.

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.

The axisymmetric long-wave interfacial stability of core-annular flow of power-law fluid with surfactant

The long wave stability of core-annular flow of power-law fluids with an axial pressure gradient is investigated at low Reynolds number. The interface between the two fluids is populated with an insoluble surfactant. The analytic solution for the growth rate of perturbation is obtained with long wave approximation. We are mainly concerned with the effects of shear-thinning/thickening property and interfacial surfactant on the flow stability. The results show that the influence of shear-thinning/thickening property accounts to the change of the capillary number. For a clean interface, the shear-thinning property enhances the capillary instability when the interface is close to the pipe wall. The converse is true when the interface is close to the pipe centerline. For shear-thickening fluids, the situation is reversed. When the interface is close to the pipe centerline, the capillary instability can be restrained due to the influence of surfactant. A parameter set can be found under which the flow is linearly stable.

Mixture flow of particles and power-law fluid in round peristaltic tube

The erythrocyte and blood flowing in the blood vessel can be treated as the two-phase flow of the mixture of particles and a power-law fluid in a peristaltic tube.In the present work, the peristaltic transport of a power-law fluid and the suspension of particles in a tube is investigated by a perturbation method using the long wavelength approximation. The influence of different parameters on the velocity profile and streamlines is explored. Results show that there is a deflection of the flow field when the power-law index n = 0.5 or 1.5 compared with the Newtonian fluid where the trapping zone is symmetric to a certain cross section. The flux rate and reflux of the material are identified,and the conditions under which the reflux appears are determined. Moreover, a reflux phenomenon occurs near the wall. The trapping zone is related to not only the tube geometry and the flow flux but also the fluid properties. Both the length and width of the trapping zone increase with an increase in θ or φ. The trapping zone is more difficult to produce in the shear-thinning fluid than the shear-thickening fluid.

Effect of thermal conductivity on heat transfer for a power-law non-Newtonian fluid over a continuous stretched surface with various injection parameters

The two-dimensional non-Newtonian steady flow on a power-law stretched surface with suction or injection is studied. Thermal conductivity is assumed to vary as a linear function of temperature. The transformed governing equations in the present study are solved numerically using the Runge-Kutta method. Through a comparison, results for a special case of the problem show excellent agreement with those in a previous work. Two cases are considered, one corresponding to a cooled surface temperature and the other to a uniform surface temperature. Numerical results show that the thermal conductivity variation parameter, the injection parameter, and the power-law index have significant influences on the temperature profiles and the Nusselt number.

MHD flow of power-law fluid on moving surface with power-law velocity and special injection/blowing

The problem of magnetohydrodynamic （MHD） flow on a moving surface with the power-law velocity and special injection/blowing is investigated. A scaling group transformation is used to reduce the governing equations to a system of ordinary differen- tial equations. The skin friction coefficients of the MHD boundary layer flow are derived, and the approximate solutions of the flow characteristics are obtained with the homotopy analysis method （HAM）. The approximate solutions are easily computed by use of a high order iterative procedure, and the effects of the power-law index, the magnetic parameter, and the special suction/blowing parameter on the dynamics are analyzed. The obtained results are compared with the numerical results published in the literature, verifying the reliability of the approximate solutions.

Exact solution to peristaltic transport of power-law fluid in asymmetric channel with compliant walls

Effects of compliant wall properties on the peristaltic flow of a non-Newtonian fluid in an asymmetric channel are investigated.The rheological characteristics are characterized by the constitutive equations of a power-law fluid.Long wavelength and low Reynolds number approximations are adopted in the presentation of mathematical developments.Exact solutions are established for the stream function and velocity.The streamlines pattern and trapping are given due attention.Salient features of the key parameters entering into the present flow are displayed and important conclusions are pointed out.

THE TRANSIENT ELLIPTIC FLOW OF POWER-LAW FLUID IN FRACTAL POROUS MEDIA

The steady oil production and pressure distribution formulae of vertically fractured well for power-law non-Newtonian fluid were derived on the basis of the elliptic flow model in fractal reservoirs. The corresponding transient flow in fractal reservoirs was studied by numerical differentiation method: the influence of fractal index to transient pressure of vertically fractured well was analyzed. Finally the approximate analytical solution of transient flow was given by average mass conservation law. The study shows that using elliptic flow method to analyze the flow of vertically fractured well is a simple method.

Multi-scale complexity entropy causality plane: An intrinsic measure for indicating two-phase flow structures

We extend the complexity entropy causality plane（CECP） to propose a multi-scale complexity entropy causality plane（MS-CECP） and further use the proposed method to discriminate the deterministic characteristics of different oil-in-water flows. We first take several typical time series for example to investigate the characteristic of the MS-CECP and find that the MS-CECP not only describes the continuous loss of dynamical structure with the increase of scale, but also reflects the determinacy of the system. Then we calculate the MS-CECP for the conductance fluctuating signals measured from oil–water two-phase flow loop test facility. The results indicate that the MS-CECP could be an intrinsic measure for indicating oil-in-water two-phase flow structures.

Heat transfer model of two-phase flow across tube bundle in submerged combustion vaporizer

In order to optimize the design of the submerged combustion vaporizer(SCV), an experimental apparatus was set up to investigate the heat transfer character outside the tube bundle in SCV. Several experiments were conducted using water and CO_2 as the heat transfer media in the tubes, respectively. The results indicated that hot air flux, the initial liquid level height and the tube pitch ratio had great influence on the heat transfer coefficient outside the tube bundle(ho). Finally, the air flux associated factor β and height associated factor γ were introduced to propose a new hocorrelation. After verified by experiments using cold water, high pressure CO_2 and liquid N_2 as heat transfer media, respectively, it was found that the biggest deviation between the predicted and the experimental values was less than 25%.

High-strength aluminum alloys hollow billet prepared by two-phase zone continuous casting

The two-phase zone continuous casting(TZCC)technique was used to continuously cast high-strength aluminum alloy hollow billets,and a verified 3D model of TZCC was used to simulate the flow and temperature fields at casting speeds of 2-6 mm·min^(-1).Hollow billets under the same conditions were prepared,and their macro/microstructures were analyzed by an optical microscope and a scanning electron microscope.During the TZCC process,a circular fluid flow appears in front of the mushy zone,and the induction heated stepped mold and convective heat transfer result in a curved solidification front with depressed region near the inner wall and a vertical temperature gradient.The deflection of the solidification front decreases and the average cooling rate in the mushy zone increases with increasing casting speed.Experimental results for a 2D12 alloy show that hot tearing periodically appears in the hollow billet accompanied by macrosegregation near the inner wall at casting speeds of 2 and 4 mm·min^(-1),while macroscopic defects of hot tearing and macrosegregation weaken and the average size of columnar crystals in the hollow billets decreases with further increasing casting speed.2D12 aluminum alloy hollow billets with no macroscopic defects,the finest columnar crystals,and excellent mechanical properties were prepared by TZCC at a casting speed of 6 mm·min^(-1),which is beneficial for the further plastic forming process.