Foam flow in vertical gas wells under liquid loading: Critical velocity and pressure drop prediction

Foam lift is one of the most cost effective methodologies for unloading gas wells. The surfactants are either injected intermittently or continuously to lift the liquid to the surface. By reducing the gravitational gradient and increasing the frictional gradient, the critical velocity at which liquid loading occurs is shifted to lower gas velocities. Currently, we do not have a methodology to predict the critical velocity (at the transition boundary of annular and intermittent flow) and the pressure drop under foam flow conditions.To address this, we measured several foam flow characteristics in both small scale and large scale facilities. Small scale facility involved measurement of foam carryover capacity as a function of time and surfactant concentration. Large scale facility involved measurement of liquid holdup, pressure drop, fraction of gas trapped in foam and foam holdup in 40-ft 2-in. and 4-in. tubing.We developed closure relationships for liquid hold up, foam holdup, fraction of gas trapped in the foam and interfacial friction factor by combining the small scale data with the data collected in the large scale experiments. These closure relationships are applicable to four different surfactants tested. A new transition criterion was developed and successfully used to predict onset of liquid loading under foam flow. Using a force balance over the gas core in annular flow, we developed a new procedure to calculate the pressure drop under foam flow conditions. We compared our model results with actual measurements in the large scale facility. Our model was reasonably able to predict the pressure drop within ±30%. The reason for such a large variance is that the small scale facility was not able to capture all the characteristics of the foam which were observed in the large scale facility. It is very difficult to reproduce the foam characteristics exactly in two different experiments. This is discussed further in this paper.The procedure developed is the only one currently available to calculate the pressure drop under the foam flow conditions using the small scale data. It is superior to conventional annular flow pressure drop prediction models which are currently available in the literature.     

Trapped Gas Fraction During Steady-State Foam Flow

Trapped or stationary gas contributes significantly to the extent of gas mobility reduction for aqueous foams. Simultaneous measurements of effluent bubble sizes and trapped gas saturation in sandstone are reported for the first time. Roughly 80% of the gas saturation in an aqueous foam is stationary at steady state in this permeable porous medium. The experiments show that as gas velocity increases, the trapped gas fraction decreases. Similarly, as injected gas–liquid ratio increases, the trapped gas fraction decreases. Hence, the absolute velocities of gas and aqueous surfactant solution are fundamental to foamed-gas mobility reduction for they help determine in situ foam texture. Effluent foam bubbles range in size from 60 to 120 μm in diameter. The smaller the effluent bubble, the smaller is the fraction of mobile gas. Scaling laws from network percolation theory are used to engender a mechanistic understanding of the various parameters identified as important in the experimental program. The closed form approimation predicts that the trapped gas fraction is a weak function of pressure gradient, foam-bubble size, and the permeability of the porous medium. Moreover, the theory reproduces well the newly obtained experimental data.     

Equivalent particle diameter and length scale for pressure drop in porous metals

The internal architecture of metal foam is significantly different from that of traditional porous media. This provides a set of challenges for understanding the fluid flow in this relatively new class of materials. This paper proposes that despite the geometrical differences between metal foam and traditional porous media, the Ergun correlation is a good fit for the linear pressure drop as a function of the Darcian velocity, provided that an appropriate equivalent particle diameter is used. The paper investigates an appropriate particle diameter considering the physics of energy dissipation, i.e. the viscous shear and the form drag. The above approach is supported by wind tunnel steady-state unidirectional pressure drop measurements for airflow through several isotropic open-cell aluminum foam samples having different porosities and pore densities. For each foam sample, the equivalent particle diameter correlated well with the surface area per unit volume of the foam. This was also very well valid for previous porous metal pressure drop data in the open literature.     

Effects of Bubbles on the Hydraulic Conductivity of Porous Materials – Theoretical Results

In a porous material, both the pressure drop across a bubble and its speed are nonlinear functions of the fluid velocity. Nonlinear dynamics of bubbles in turn affect the macroscopic hydraulic conductivity, and thus the fluid velocity. We treat a porous medium as a network of tubes and combine critical path analysis with pore-scale results to predict the effects of bubble dynamics on the macroscopic hydraulic conductivity and bubble density. Critical path analysis uses percolation theory to find the dominant (approximately) one-dimensional flow paths. We find that in steady state, along percolating pathways, bubble density decreases with increasing fluid velocity, and bubble density is thus smallest in the smallest (critical) tubes. We find that the hydraulic conductivity increases monotonically with increasing capillary number up to Ca 10^–2, but may decrease for larger capillary numbers due to the relative decrease of bubble density in the critical pores. We also identify processes that can provide a positive feedback between bubble density and fluid flow along the critical paths. The feedback amplifies statistical fluctuations in the density of bubbles, producing fluctuations in the hydraulic conductivity.     

Simulation of Entry-Region Flow in Open-Cell Metal Foam and Experimental Validation

Open-cell metal foam is distinguished from traditional porous media by its very high porosities (often greater than 90 %), and its web-like open structure and good permeability. As such, the foam is a very attractive core for many engineered systems, e.g., heat exchangers, filtration devices, catalysts, and reactors. The flow field inside the foam is rather complex due to flow reversal and vigorous mixing. This complexity is increased by the possible presence of an entry region. The entrance region in metal foam is usually underestimated and ignored, just like its counterpart in traditional porous media. In this paper, the actual entry length is determined by simulation and direct experiment on commercial open-cell aluminum foam. It is shown to be dependent on flow velocity and to reach a constant value for higher velocities. The complex and intrinsically random architecture of the foam is idealized using a unit geometrical model, in order to numerically investigate the flow field and pressure drop inside the foam. The Navier–Stokes equations are solved directly, and velocity and pressure fields are obtained for various approach velocities using a commercial numerical package. The entry length is ascertained from the behavior of the velocity field close to the entrance. Comparisons to experimental data were also carried out. The commercial foam that was used in the experiment had 10 ppi and porosity of 91.2 %. Air was forced to flow inside the foam using an open-loop wind tunnel. Good qualitative agreement between the modeling and experimental results are obtained. The agreement lends confidence to the modeling approach and the determined entry length.     

Simulations of flow through open cell metal foams using an idealized periodic cell structure

A new approach to modeling the flow through a porous medium with a well defined structure is presented. This approach entailed modeling an idealized open cell metal foam based on a fundamental periodic unit of eight cells and solving the flow through the three-dimensional cellular unit. To model an infinitely large matrix, periodic boundary conditions were set on the walls parallel to the flow direction, while a pseudo-periodic boundary condition with a prescribed volumetric flow rate was set over the inlet–outlet pair of the unit cell. The pressure drop data of the flow through the cellular unit were then compared on a length-normalized basis against experimental data. The pressure drop values predicted by the simulations were consistently 25% lower than the values obtained in the experiments on a similar foam and under identical flow conditions. One explanation for the discrepancy between the two sets of data is the lack of pressure drop increasing wall effects in the simulations. The increase in the pressure drop from wall effects in the simulation was quantified.     

Unsteady non-darcian fluid flow in parallel-plates channels partially filled with porous materials

Using Green's function method, analytical solutions are obtained for the problem of transient fluid flow in parallel-plate channels partially filled with porous materials. The unsteadiness in the fluid flow is caused either by a sudden change in the imposed pressure gradient or (and) by a sudden change in the velocity of the channel boundaries. The Brinkman-extended Darcy model is used to model the flow inside the porous domain. Received on 9 April 1997     

Particle Flow Code Method-Based Meso-scale Identification for Seepage Failure of Soil Levee

The development and validation of a grid-based pore-scale numerical modelling methodology applied to five different commercial metal foam samples is described. The 3-D digital representation of the foam geometry was obtained by the use of X-ray microcomputer tomography scans, and macroscopic properties such as porosity, specific surface and pore size distribution are directly calculated from tomographic data. Pressure drop measurements were performed on all the samples under a wide range of flow velocities, with focus on the turbulent flow regime. Airflow pore-scale simulations were carried out solving the continuity and Navier–Stokes equations using a commercial finite volume code. The feasibility of using Reynolds-averaged Navier–Stokes models to account for the turbulence within the pore space was evaluated. Macroscopic transport quantities are calculated from the pore-scale simulations by averaging. Permeability and Forchheimer coefficient values are obtained from the pressure gradient data for both experiments and simulations and used for validation. Results have shown that viscous losses are practically negligible under the conditions investigated and pressure losses are dominated by inertial effects. Simulations performed on samples with varying thickness in the flow direction showed the pressure gradient to be affected by the sample thickness. However, as the thickness increased, the pressure gradient tended towards an asymptotic value.     

Direct Observation of Trapped Gas Bubbles by Capillarity in Sandy Porous Media

We investigated the mechanism of residual gas trapping at a microscopic level. We imaged trapped air bubbles in a Berea sandstone chip after spontaneous imbibition at atmospheric pressure. The pore structure and trapped bubbles were observed by microfocused X-ray computed tomography. Distributions of trapped bubbles in Berea and Tako sandstone were imaged in coreflooding at a capillary number of 1.0 × 10⁻⁶. Trapped bubbles are of two types, those occupying the center of the pore with a pore-scale size and others having a pore-network scale size. In low-porosity media such as sandstone, connected bubbles contribute greatly to trapped gas saturation. Effects of capillary number and injected water volume were investigated using a packed bed of glass beads 600μm in diameter, which had high porosity (38%). The trapped N₂ bubbles are stable against the water flow rate corresponding to a capillary number of 1.0 × 10⁻⁴.     

Analytical considerations of flow/thermal coupling of nanofluids in foam metals with local thermal non-equilibrium (LTNE) phenomena and inhomogeneous nanoparticle distribution

Foam metals with micro pores own excellent thermal performance, however, poor heat conductive ability of most heat-transfer fluids restricts further heat transfer improvement. Combination of foam metal and nanofluid with highly conductive nanoparticles is a promising solution. Convective thermal characteristics of nanofluids in porous foams are theoretically investigated in this work. Effects of Brownian motion and thermophoretic diffusion of nanoparticles in the base fluid on thermal performance are considered. The nanoparticle and the base-fluid are considered to be in thermal equilibrium and the temperature difference between the nanofluid and foam ligaments is especially considered. Compared with the base-fluid flow in a duct, the velocity distribution for the nanofluid flow in a porous foam is more uniform with a decreased dimensionless temperature. The pressure drop of the nanofluid increases with an increase in the concentration of the nanoparticles. By employing foam metals and nanofluid, the cross-sectional temperature becomes closer to the wall temperature. Simultaneously, notable difference between solid and fluid temperatures can be observed, revealing the LTNE effect of the nanofluid on the porous foam. It is found that the Nusselt number first increases and then decreases with an increase in nanoparticle concentration. Furthermore, the Nusselt number decreases with an increase in the foam porosity. It is found that the thermal performance of a nanofluid in a plain tube is different from that in the foam metals.     

Numerical Simulation of Yield Stress Fluid Flow in Capillary Bundles: Influence of the Form and the Axial Variation in the Cross Section

In this paper, we investigate possible improvements that can be made to the bundle of capillaries model in order to better represent the flow of yield stress fluids through porous media. This was examined by performing extensive and progressive numerical simulations and by introducing the non-circularity of channels’ cross section and/or its variability along the channels’ axis. It is shown that if only the non-circularity of channels’ cross section is taken into account, a moderate influence is observed on both critical pressure gradient for the flow onset and the flow rate/pressure gradient \({Q}({\nabla P})\) relationship. However, the axial variation in capillaries’ cross section has proved to be more impacting the computed flow rate/pressure gradient data. We show hence that when available pore throat and pore body size distributions are used to construct the bundle of axially varying capillaries, the obtained \({Q}({\nabla P})\) data do fit well experimental results corresponding to the flow of a Bingham-like fluid through a bed of randomly packed mono-sized spheres.     

Multiscale, Multiphysics Network Modeling of Shale Matrix Gas Flows

We present a pore network model to determine the permeability of shale gas matrix. Contrary to the conventional reservoirs, where permeability is only a function of topology and morphology of the pores, the permeability in shale depends on pressure as well. In addition to traditional viscous flow of Hagen–Poiseuille or Darcy type, we included slip flow and Knudsen diffusion in our network model to simulate gas flow in shale systems that contain pores on both micrometer and nanometer scales. This is the first network model in 3D that combines pores with nanometer and micrometer sizes with different flow physics mechanisms on both scales. Our results showed that estimated apparent permeability is significantly higher when the additional physical phenomena are considered, especially at lower pressures and in networks where nanopores dominate. We performed sensitivity analyses on three different network models with equal porosity; constant cross-section model (CCM), enlarged cross-section model (ECM) and shrunk length model (SLM). For the porous systems with variable pore sizes, the apparent permeability is highly dependent on the fraction of nanopores and the pores’ connectivity. The overall permeability in each model decreased as the fraction of nanopores increased.     

Interface behavior and void formation during infiltration of liquids into porous structures

A numerical approach was proposed to simulate time-dependent evolution of the liquid front during the pore-level infiltration of liquids into porous structures. It considers the multiphase problem of liquid penetration into the pore(s) initially occupied with air and the simultaneous escaping of air. The Volume-of-Fluid (VOF) method was employed using a two-dimensional model of the graphite pore structure. The proposed method is capable of tracking the evolution of liquid front and yields the infiltration criteria for wetting and non-wetting liquids. Contribution of various driving forces (resulting from pressure gradient, gravity and interfacial effects) to infiltration and interface behavior including the liquid front shape, position and velocity was investigated. Interface pinning (temporary and permanent) and wicking flow through the pore(s) were investigated during infiltration of wetting liquids, whereas pore-level fingering and void formation (entrapment of air within the pore) were observed for non-wetting liquids. The results were verified against the results of coupled VOF level-set method, known to be more accurate for interface tracking. Moreover, the results of liquid penetration length during the wicking flow through a network of pores in series were validated with good agreement against the experimental results of unidirectional horizontal infiltration of graphite foam, and a modified Washburn equation.     

Characterization of Single-Phase Flow Through Carbonate Rocks: Quantitative Comparison of NMR Flow Propagator Measurements with a Realistic Pore Network Model

We report here the quantitative comparisons between the measured NMR flow propagator of a carbonate rock and the flow propagator calculated with a porous network extracted from the micro-CT image of the twin plug. We developed a numerical model based on a particle tracking algorithm in pore space. The particle tracking in throats is described using the first arrival time distribution. As pores have an important volume fraction in the sample considered, we implemented a time-delay mechanism for particle transport in the pores. We consider that the nodes have volume and there is a transport of the tracking particles inside the nodes, which leads to an “apparent” time-delay. Simulations of flow propagator show good agreement with low field NMR experiments performed on the twin plug of the sample used for pore network extraction with a single adjustable parameter (that describes the dynamics in the pores). These results lead us to a better understanding of the connection between pore structure and the behavior of NMR flow propagator in fluid-saturated rocks and are essential in interpreting the experimental data and correlating NMR parameters to petrophysical properties.     

The Effect of Microstructure on Models for the Flow of a Bingham Fluid in Porous Media: One-Dimensional Flows

The Buckingham–Reiner models for the one-dimensional flow of a Bingham fluid along a uniform pipe or channel are well-known, but are modified here to cover much more general one-dimensional configurations. These include selections of channels with different widths, and five different probability density functions describing distributions of channel widths. It is found that the manner in which breakthrough occurs at the threshold pressure gradient depends very strongly on the type of distribution of pores and that a pseudo-threshold pressure gradient, which might be inferred from measurements of flow at relatively high pressure gradients, may be more than twice the magnitude of the true threshold gradient.     

Numerical investigation of static flow instability in a low-pressure subcooled boiling channel

A three-dimensional two-fluid model to predict subcooled boiling flow at low pressure is presented. The model is adopted to investigate the two-phase flow and heat transfer characteristics in a heated channel. The presence of bubbles as a consequence of heating flow through a vertical rectangular channel has a significant effect on the overall pressure drop along the channel. Numerical results were compared against a series experimental data performed at various conditions – mass flux, heat flux, inlet temperature and exit pressure. Good agreement on the overall pressure drop was achieved. The onset of flow instability velocity was also accurately determined when compared against measurements. Predicted results of void fraction provided useful information towards a more fundamental understanding of the occurrence of onset of nucleate boiling, onset of significant voiding and onset of flow instability. The phenomenon of boiling onset oscillations was also predicted through the use of the two-fluid model.     

多孔介质中稳定泡沫的流动计算及实验研究

将多孔介质简化为一簇变截面毛管束,根据多孔介质的颗粒直径、颗粒排列方式、孔喉尺度比及束缚水饱和度,计算出变截面毛细管的喉道半径和孔隙半径. 在考虑多孔介质喉道和孔隙中单个气泡的受力和变形基础上,利用动量守恒定理,推导出单个孔隙单元内液相的压力分布和孔隙单元两端的压差计算公式,最终得到多孔介质的压力分布计算公式. 利用长U型填砂管对稳定泡沫的流动特性进行了实验研究. 研究结果表明:稳定泡沫流动时多孔介质中的压力分布呈线性下降,影响泡沫在多孔介质中流动特性的因素包括:多孔介质的孔喉结构、泡沫流体的流量和干度、气液界面张力、气泡尺寸,其中孔喉结构和泡沫干度是影响泡沫封堵能力的主要因素.关键词: 稳定泡沫;多孔介质;变截面毛管;流动;表观粘度;压力分布;实验研究      

Flux in Porous Media with Memory: Models and Experiments

The classic constitutive equation relating fluid flux to a gradient in potential (pressure head plus gravitational energy) through a porous medium was discovered by Darcy in the mid 1800s. This law states that the flux is proportional to the pressure gradient. However, the passage of the fluid through the porous matrix may cause a local variation of the permeability. For example, the flow may perturb the porous formation by causing particle migration resulting in pore clogging or chemically reacting with the medium to enlarge the pores or diminish the size of the pores. In order to adequately represent these phenomena, we modify the constitutive equations by introducing a memory formalism operating on both the pressure gradient–flux and the pressure–density variations. The memory formalism is then represented with fractional order derivatives. We perform a number of laboratory experiments in uniformly packed columns where a constant pressure is applied on the lower boundary. Both homogeneous and heterogeneous media of different characteristic particle size dimension were employed. The low value assumed by the memory parameters, and in particular by the fractional order, demonstrates that memory is largely influencing the experiments. The data and theory show how mechanical compaction can decrease permeability, and consequently flux.     

Towards a theory of nonequilibrium multiphase filtration flow

A new theoretical model of joint filtration flow of immiscible incompressible fluids is presented. This model takes into account relaxation processes due to the exchange of the fluids between pores of different sizes, and these relaxation processes are driven by capillary forces. The fluids occupy connected regions in the four-dimensional space formed by three coordinates and the pore length scale. When fluid exchange between pores of given sizes is effected by way of successive flow through pores of all intermediate sizes, the fluid pressure within each region is governed by a hyperbolic equation, the role of time being played by the pore linear scale. Pressure jumps across hypersurfaces separating these regions are equal to corresponding values of capillary pressure. A supplementary condition at any such hypersurface requires the speed of its displacement in the four-dimensional space to coincide with the normal velocity components of both the adjoining fluids. As a result, a new formulation of multiphase filtration flow problems is gained with allowance made for capillary relaxation in the porous space.     

Experiments on transport of hydrophobic particles and gas bubbles in porous media

Adhesion of hydrophobic colloids (clay minerals) on the surface of bubbles of air and the transport of the composite units formed by bubbles and mineral particles were observed in a glass micro model.When a clay mineral suspension flowed in a porous medium that contained bubbles of air trapped in small pores, particles accumulated preferentially on the upstream portion of the bubbles, and quasi-stable bubble-mineral particle units were formed. With an increase in the flow velocity, the particles moved along the interface between the bubble and the liquid and accumulated on the downstream portion of the bubbles. A large stress could mobilize the units which, occasionally, accumulated in larger voids.The mechanism suggested is adhesion of the particles on the surface of the bubble due to compression of their diffuse electrical double layer. The adsorbed particles can be moved by shear stresses which act in the region of water molecules between the well-organized layers of water on the surfaces of the bubble and the clay particles. A large enough shear stress causes the bubbles to become more streamlined, allowing them to move in the channel system. If in contact, the common lamina of the bubbles can withdraw and rupture.Bubbles transport from 20 to 50 times more particles than can be transported by average suspension.