Capillary Bundle Research Articles

Non-Newtonian Rheology in a Capillary Tube with Varying Radius

Fluid blobs in an immiscible Newtonian fluid flowing in a capillary tube with varying radius show highly nonlinear behavior. We consider here a generalization of previously obtained results to blobs of non-Newtonian fluids. We compute here the yield pressure drop and the mean flow rate in two cases: (i) When a single blob is injected, (ii) When many blobs are randomly injected into the tube. We find that the capillary effects emerge from the non-uniformity of the tube radius and contribute to the threshold pressure for flow to occur. Furthermore, in the presence of many blobs the threshold value depends on the number of blobs and their relative distances which are randomly distributed. For a capillary fiber bundle of identical parallel tubes, we calculate the probability distribution of the threshold pressure and the mean flow rate. We consider two geometries: tubes of sinusoidal shape, for which we derive explicit expressions, and triangular-shaped tubes, for which we find that essential singularities are developed. We perform numerical simulations confirming our analytical results.

Transient Taylor-Aris dispersion in N-capillary systems: Convergence properties of the band broadening in polydisperse multi-capillary columns with diffusional bridging

We report on a generic mathematical study of the transient plate height regime in bundles of polydisperse capillaries with diffusional bridging. I.e., we have investigated how the plate height in such systems can be expected to vary with the residence time, or equivalently, with the column length, before reaching a long-time limit wherein the plate height becomes constant. This is important because, in case of systems with a large number of capillaries (N), the transient regime solution is practically much more relevant than the long-time limit solution. The problem has been investigated using a simplified, yet representative geometry that is amenable to a semi-analytical solution, only requiring a numerical integration of the velocity field across a line (2D geometries) or a square or rectangle (3D geometries). The availability of this semi-analytical approach allows to consider the required large number of different random capillary diameter drawings (order of 10,000) needed to establish a sufficiently averaged result (deviation from true mean on the order of 1%). The general solution could be simplified into a set of two-parameter expressions describing either the long-time limit (τ→∞) as well as that of an infinite number of capillaries (N→∞). The combinations of N and τ where these expressions are valid to within an accuracy of 5% could be delimited with simple analytical expressions as well. It has also been observed that the long-time limit solution for the additional plate height (Δhτ→∞) remains dependant on the number of capillaries N, even when N tends to infinity. It is found that Δhτ→∞ can be expected to increase in proportion with N-1 (2D-case) or with ln(N-1) (3D-case). This dependency of Δhτ→∞ on N is a result that is not obtained when modelling the capillary bundle using a representative binary capillary system approach as has been done up-till-now in literature.

A Dynamic Pore Network Model for Imbibition Simulation Considering Corner Film Flow

AbstractWetting films can develop in the corners of angular pores under strong wetting conditions. Modeling the dynamics of corner film remains elusive using direct numerical simulations because of the significant scale difference between main meniscus and corner film flow. In this paper, the modified interacting capillary bundle model (ICB), developed in our previous work to describe accurately corner film dynamics in a single square tube, is incorporated into a single‐pressure dynamic pore network model (DPNM) to simulate imbibition in strongly wetting porous media with corner film flow. The traditional pore network is decomposed into several layers of interacting subpore networks where the 0th layer of subpore network simulates the main meniscus flow and higher layers the corner film flow. The fluid flow between different layers is captured by interlayer throats. In addition, the snap‐off mechanism caused by the thickening of wetting corner film is considered. The accuracy of the developed model is validated for four cases: spontaneous imbibition in a single square tube, wetting fluid redistribution through corner films under a capillary pressure difference, snap off in a narrow throat connecting two large pores, and imbibition dynamics in a real microfluidic porous geometry. The validated model is then used to simulate both spontaneous and controlled imbibition in a pore network with random pore size distribution. The interaction between corner film and main meniscus flow in porous media is analyzed from a pore‐scale perspective.

A novel model for oil-water two-phase relative permeability of shale formation during flowback based on fractal method

As an abundant type of unconventional hydrocarbon resources on the earth, shale oil has been widely concerned by scholars across the world. During flowback period, flow-back resistance of fracturing fluids increases because of the existence of shale oil. Therefore, it is of practical significance to study the flow-back capability of fracturing fluid for high well performance. In this paper, fractal method of porous media has been used for calculation of relative permeability of oil-water two phase flow in shale pores. First, we built up flow equation of single nanopore with four layers: boundary layer, bulk wetting phase layer, oil-water two phase layer, and nonwetting phase layer. Furthermore, boundary slip, boundary, viscosity change in boundary layer and threshold pressure gradient is considered for correction flow rate equation. The pores of shale matrix can be simplified into capillary bundle model, and fractal theory is suitable for this model. Aperture fractal dimension and tortuosity fractal dimension is used to expand its scale to porosity media with fractal method. Then the model is verified by experimential results and classic models. Furthermore, serval main factors affecting the relative permeability of water are primarily analyzed, and the more essential factors out of the aforementioned ones are extracted and further studied. The results show that with the increase in the thickness of oil-water two phase layer, the relative permeability of water is affected more seriously; The relative permeability of water is positively correlated with slip length and thickness of boundary layer. This work provides a more convenient approach to calculate relative permeability with no experiments, and provide permeability for numerical simulation in flow back period.

A comprehensive review on the flow behaviour in shale gas reservoirs: Multi‐scale, multi‐phase, and multi‐physics

AbstractAn accurate description of fluid flow is critical for the prediction of the productivity of shale gas reservoirs. In this paper, we present the current advances and a systematic summary on the fluid flow in shale gas reservoirs. First, shale pore structures, consisting of organic pores and inorganic pores ranging from nanoscale to micro‐scale, and reservoir fluids, including free gas, absorbed gas, and water, are systematically presented. Thereafter, multi‐physics flow phenomena motivated by scale effects, such as continuum flow, slip flow, transition flow, free molecular flow, and surface diffusion for gas, as well as the effective viscosity and slip boundary condition for water, are carefully summarized. Meanwhile, these flow mechanisms are discussed with molecular dynamics (MD) simulations and theoretical analysis. Subsequently, on the basis of upscaling approaches, including capillary bundle models, the lattice Boltzmann method (LBM), and pore network models (PNMs), fluid flow through heterogeneous shale matrix is reviewed and the influences of scale effects and pore structures are clarified. Additionally, shale gas well performance is discussed by combining the multiple transport mechanisms and a fracturing‐shut‐in‐flowback‐production process. Our review concluded that the fluid flow behaviour in shale gas reservoirs is a complex multi‐scale process accompanied by multi‐physical phenomena and multi‐fluid distributions. Keeping this in mind is helpful for predicting the shale gas production and recoverable gas resources. We expect this study can not only help by providing a better understanding of the fluid flow in shale reservoirs but also provide significant implications to address other multiphase flow processes.

Dynamic track model of miscible CO2 geological utilizations with complex microscopic pore-throat structures

Monitoring field-scale CO2 geological utilizations is of paramount importance but challenging due to the complexities in microscopic heterogeneity. In this paper, complex geological characterizations and fluid properties are specifically analysed and a prediction model is developed, which is capable to track the dynamic behaviour of miscible CO2 multiphase flow in microscopically-heterogeneous porous media. More specifically, first, pore-throat sizes and distributions were characterised from the constant-rate mercury injection and the CO2-displacement seepage resistance was evaluated from a capillary bundle model. The differences of seepage resistance caused from the throat changing and coupling diffusion-dissolution effects and viscosity-resistance reductions were specifically studied in the process of continuous miscible CO2 displacement. Accordingly, a comprehensive mathematical model was developed with the time-node analysis method and validated through comparison with the experimental results. The leading edge of CO2 displacement as well as the timing of gas breakthrough and displacement completion are determined to be different with varying throat sizes. It is further found that the gas breakthrough time and the time required to complete the displacement are reduced with increasing throat size and their differences also decrease. Moreover, the interval area of injection and production wells could be characterised as pure CO2, diffusion and pure oil zones, whose positions could be dynamically tracked from the recovery performance. The calculated oil recovery and gas-oil ratio from the developed model share good agreement with experimental results, with deviations of 2.2 and 0.7%. Such the validated mathematical model is then applied to successfully predict the dynamic performance of an actual reservoir (H3).

Thermokinetics of capillary absorption and its application in cement-based materials

Capillary water can transmit in capillary tube induced by hydrostatic pressure, which has an important impact on the durability and shrinkage of concrete. In this study, the ultimate height of capillary rise under various conditions was analyzed based on thermodynamic equilibrium of capillary water where capillary water is subjected to six types of force. Subsequently, a logarithmic model of capillary rising with three parameters (Jurins's height hc, permeability coefficient k, characteristic time τ) was proposed according to the analytical solution of kinetics function of capillary absorption. To better understand capillary behavior in capillary bundle with different radius, the spatio-temporal distribution of saturation, and the variation of water content with time and saturation are predicted according to the kinetics of capillary rise in single capillary. Additionally, the kinetics of capillary absorption in porous materials was discussed. Finally, the developed model was verified by testing the true rising height of water in a mortar specimen using X ray computed tomography. The equivalent height of adsorbed water in mortar with varied saturation under different time was obtained by gravimetric method. The results show that the logarithmic model gives a good description of the kinetics data of water absorption in mortar.

Pore structure analysis and permeability prediction of shale oil reservoirs with HPMI and NMR: A case study of the Permian Lucaogou Formation in the Jimsar Sag, Junggar Basin, NW China

The pore structure characteristics and formation permeability are important reservoir information for the development of shale oil. The pore structures of the shale oil formation in the Lucaogou Formation in the Jimsar Sag are characterized with high-pressure mercury injection (HPMI), nuclear magnetic resonance (NMR), and fractal theory. The capillary bundle model, geometric model, and wetting phase model are used to obtain the fractal dimension of the capillary pressure curve. Among them, Df1 from the capillary bundle model is between 2.1937 and 3.4131, Df2 from the geometric model is within a range of 1.0414–3.575, and Df3 from the wetting phase model is from 0.5691 to 2.7047. For NMR spectra, the pore space is divided into micropores, mesopores, and macropores based on the peak value of the NMR T2 spectra, and the fractal dimension of different pore types is calculated, where Dfmi is from 1.564 to 2.301, Dfme is in a range from 2.082 to 2.941, and Dfma is between 2.831 and 2.991. The relationships between the fractal dimension and petrophysical parameters are analyzed, and the results indicate that among the three types of models, the fractal dimension of the capillary bundle model has the highest correlation with the core petrophysical parameters. Therefore, it is recommended to use the capillary bundle model to evaluate the fractal characteristics of pore structures. In particular, the fractal dimension is positively correlated with the throat coefficient and negatively correlated with the size of the pore throat. In contrast, the fractal dimension of NMR has a poor correlation with core petrophysical parameters. There is no obvious correlation between the fractal dimension of micropores and petrophysical parameters, while the fractal dimension of mesopores and macropores is only correlated with the throat-sorting coefficient, and the correlation coefficient is less than 0.15. Thus, the fractal dimension from NMR data is not suitable for studying the fractal characteristics of shale oil cores in the Lucaogou Formation. In addition, we predicted the permeability of shale cores using the Swanson model, Pittman model, and Winland model. The permeability prediction results show that the Swanson model has the greatest accuracy, followed by the Winland model. Among them, the accuracy index (ACI) of the Swanson model is 0.901, and the ACI of the Winland model is between 0.483 and 0.897. The permeability prediction by the Pittman model is poorest among these three models.

Edge wetting: Steady state of rivulets in wedges

The geometry of rough, textured, fractured, and porous media is topologically complicated. Such media are commonly modeled as bundles of capillary tubes. However, angle-containing geometries can serve as a more realistic portrayal of their internal structure. A basic element inherent to all of these media is an open wedge-like channel. The classical theory of capillarity ignoring intermolecular interactions implies that liquid entering a wedge must propagate indefinitely along its spine when the liquid–gas interface is concave. This is well known as the Concus–Finn condition. In the present paper, we show that steady-state rivulets can be formed in such channels when surface forces are taken into account. We present a simple model based on the disjoining pressure approach and analyze the shape of rivulets in wedges. We also consider the case when the walls of the wedge are soft and can be deformed by the liquid.

A FRACTAL MODEL OF POWER-LAW FLUID THROUGH CHARGED FIBROUS POROUS MEDIA BY USING THE FRACTIONAL-DERIVATIVE THEORY

The study characterizes power-law fluid through charged fibrous porous media with spatial fractional-derivative and fractal geometry. Seepage flow of power-law fluid across fractal fibrous porous media in the presence of electric double layers (EDLs) is investigated based on the capillary bundle model. The acquired velocity distribution equation in a narrow capillary is then transformed into the form of series with appropriate Taylor approximation. After that, an analytical formula for dimensionless permeability is derived based on the generalized Darcy’s law. The effects of diverse parameters, including the fractal dimension of pore area, porosity, fractional order and Zeta potential on dimensionless permeability, are discussed. It can be seen from the results that lower fractional order has an amplification effect on dimensionless permeability with the change in Zeta potential. The results provide some theoretical guidance for revealing the seepage mechanism of a power-law fluid in charged porous media.

Facile Model for Predicting Sweat Mass and Concentration in Layer Melt Crystallization

Sweating is an important purification process involved in the layer melt crystallization technique. Modeling of the sweating process remains difficult due to the complicated association of mass balance, heat balance, and fluid flow through a dynamic crystal layer. A model of sweating was proposed in this work, attempting to give a facile prediction of the dynamic sweat mass and concentration based on the initial crystal layer structure other than empirical correlations. Heat balance equations were taken as the governing equations to calculate the crystal layer temperature and residual melt concentration. A curved capillary bundle model that obeys fractal distribution was adapted to describe the dynamic porous crystal layer. By combining mass balance, the nonlinear variation of melted mass with sweating time was estimated. Subsequently, para-xylene was used as the model material to validate the sweating model. A quasiequilibrium between solid and liquid was found at a slow sweating rate range in this work, which verified the model assumption and greatly simplify the calculation. It revealed that the melting amount of the crystal layer and the imbibition melt increased nonlinearly with the increase of temperature. The curved capillary bundle model was found to be able to give an accurate estimation of sweat mass according to the experimental data. The model has accurate predictions on the S-shaped curve of sweat mass and sweat concentration compared to the experimental sweating results.

Spontaneous imbibition model for micro–nano–scale pores in shale gas reservoirs considering gas–water interaction

Spontaneous imbibition is the main factor that reduces the fracturing efficiency and damages gas production in shale gas wells, which is regulated by the gas–water interaction. Water (fracturing fluid) is imbibed spontaneously due to the characteristics of the water–rock interface, especially in shale with micro–nano–scale pores, while imbibition is not impeded but prevented by the pore structure resistance and original free gas compression as well as the increased fraction from the desorption of adsorbed gas. To accurately predict the fluid loss in a shale gas reservoir during fracturing, the imbibition pressures of the gas–water phase were comprehensively analyzed in this study, and the increased gas pressure from the desorption of adsorbed gas was first considered with other imbibition pressures, such as gas pressure from free gas compression, displacement pressure, capillary pressure, and osmotic pressure. By substituting the gas–water phase pressures into the Lucas–Washburn equation with the fractal characteristic of the capillary bundle, an imbibition model for micro–nano-scale pores in shale gas reservoirs considering gas–water interaction is established. The proposed model is compared with the traditional imbibition models and verified through imbibition experiments, and a case application to a shale gas well in the Sichuan Basin is undertaken. The results prove that the new fractal imbibition model performs better than the other models for imbibition estimation in reservoir shale, and it can accurately predict the fluid loss during the hydraulic fracturing process for shale gas wells.

Mathematical model of dynamic imbibition in nanoporous reservoirs

The oil-water imbibition equation in the nano-scale pores considering the dynamic contact angle effect, nanoconfinement effect, inertia effect, and inlet end effect was established, and the relation between the friction coefficient of solid-oil-water three-phase contact line and the fluid viscosity in the interface zone was derived. In combination with the capillary bundle model and the lognormal distribution theory, the imbibition model of tight core was obtained and key parameters affecting imbibition dynamics were analyzed. The study shows that in the process of nanopore imbibition, the dynamic contact angle effect has the most significant impact on the imbibition, followed by nanoconfinement effect (multilayer sticking effect and slippage effect), and the inertia effect and inlet end effect have the least impact; in the initial stage of imbibition, the effect of inertial force decreases, and the effect of contact line friction increases, so the dynamic contact angle gradually increases from the initial equilibrium contact angle to the maximum and then remains basically stable; in the later stage of imbibition, the effect of contact line friction decreases, and the contact angle gradually decreases from the maximum dynamic contact angle and approaches the initial equilibrium contact angle; as the pore radius decreases, the dynamic contact angle effect increases in the initial stage of imbibition and decreases in the later stage of imbibition; as the oil-water interfacial tension increases, the imbibition power increases, and the dynamic contact angle effect increases; there is a critical value for the influence of interfacial tension on the imbibition dynamics. In improving oil recovery by imbibition in tight oil reservoir, interfacial tension too low cannot achieve good imbibition effect, and the best interfacial tension needs to be obtained through optimization.

Acyclic pore model and its applications in imbibition efficiency calculation

The classical pore structure conceptual model, such as a bundle of capillaries and lattice model is challenged for applicability in unconventional reservoirs, due to the micro/nano pore scale in unconventional reservoirs, with most of them below 100nm. Meanwhile, it is important to obtain an accurate evaluation of imbibition efficiency in unconventional reservoirs since imbibition is the main method for sweeping oil in matrix. This paper improves the acyclic pore model which is more practical for unconventional reservoirs. An imbibition efficiency calculation method is proposed, the effects of bifurcation number, pore throat radii corresponding to a mercury saturation of 50% (r50) and wettability is studied. The mechanism about how the pore structure affects imbibition performance is revealed. Results show that the increment of bifurcation number can decelerate the imbibition time in the branch level with smaller diameter, enlarges the imbibition accessible pore size as well. As r50 and hydrophilic saturation increase, wetting fluid volume imbibed and driving force for imbibition is also enhanced, which results in an improvement of imbibition efficiency.

Development of an installation for studying the flow of liquid in microcapillaries

The article is dedicated to the creation of an installation for studying the flow of fluids through microcapillaries. This installation is of scientific interest for the development of digital models of the oil fluid flow through the pore space of the rock. Elaboration of the hydrodynamic modeling parameters will allow more accurate selection of technology for the extraction of hydrocarbon reserves from the bowels. The purpose of the work is to develop an installation and a technology for studying flows in microcapillaries. To achieve this goal, a selection of tomographic images of core samples was collected and three main channel shapes were identified, based on the images, in accordance with which microcapillaries were made from cylindrical glass tubes. Fluids are selected for pumping through the created capillaries. The installation for studying the flow of liquid in microcapillaries was assembled, the testing of the installation made it possible to identify and analyze its limitations and features of exploitation. The improvement of the installation provided more extensive data on the processes occurring in the microcapillary. A technique for conducting experiments and data recording has been worked out, automated data collection from a differential pressure sensor with the possibility of visual control using a personal computer has been implemented. Further modernization of the developed installation may allow conducting research on a bundle of capillaries of various shapes, simulating a network of channels of a real rock. The result of this work is the creation of an experimental installation that makes it possible to study the flow of liquids in microcapillaries of various shapes. Comparison of the results between series of experiments showed a high degree of repeatability, which eliminates the errors of the measurements. The obtained data were analyzed and compared with theoretical calculations.

Study on the Water Blocking Capacity and Influencing Factors of Hydrophobic Gravel Beds.

Excess water production is a common and extremely serious problem during the process of reservoir development, especially in the middle and later stages of unconsolidated sandstone reservoirs. High water content ratios will negatively affect the recovery efficiency and equipment safety of oil and gas wells. In this study, a hydrophobic quartz sand proppant was prepared with 1,1,1,3,3, 3-hexamethyldisilazane (HMDS) for gravel packing technology. The capillary resistance of hydrophobic quartz sand beds to water was used to achieve water inhibition and oil enhancement. Then, a theoretical capillary bundle model of a quartz sand bed was developed to predict the capillary resistance and water blocking capacity of the quartz sand bed. An evaluation of the experiment results shows that the activation index is up to 96% and the water contact angle is up to 151° after treatment by 0.6 wt % HMDS, which indicates that the quartz sand after modification has strong hydrophobicity. The calculation results demonstrate that the capillary resistance (CR) and the water blocking capacity (WBC) decrease with increasing quartz sand particle size and porosity of the quartz sand bed. As the water contact angle increases, the CR and WBC are also increased. In addition, to make the WBC of hydrophobic quartz sand bed δ > 20%, the parameters of the quartz sand bed must be satisfied: particle size dg < 50 μm, porosity ϕ < 30%, water contact angle θ > 150°, and production pressure difference Δp < 1 MPa. The results of this study could provide theoretical guidance for the construction parameters of hydrophobic gravel packing in high water-cut oil and gas fields.

Calculation Model of Relative Permeability in Tight Sandstone Gas Reservoir with Stress Sensitivity

During the development of tight gas reservoir, the irreducible water saturation, rock permeability, and relative permeability change with formation pressure, which has a significant impact on well production. Based on capillary bundle model and fractal theory, the irreducible water saturation model, permeability model, and relative permeability model are constructed considering the influence of water film and stress sensitivity at the same time. The accuracy of this model is verified by results of nuclear magnetic experiment and comparison with previous models. The effects of some factors on irreducible water saturation, permeability, and relative permeability curves are discussed. The results show that the stress sensitivity will obviously reduce the formation permeability and increase the irreducible water saturation, and the existence of water film will reduce the permeability of gas phase. The increase of elastic modulus weakens the stress sensitivity of reservoir. The irreducible water saturation increases, and the relative permeability curve changes little with the increase of effective stress. When the minimum pore radius is constant, the ratio of maximum pore radius to minimum pore radius increases, the permeability increases, the irreducible water saturation decreases obviously, and the two-phase flow interval of relative permeability curve increases. When the displacement pressure increases, the irreducible water saturation decreases, and the interval of two-phase flow increases. These models can calculate the irreducible water saturation, permeability and relative permeability curves under any pressure in the development of tight gas reservoir. The findings of this study can help for better understanding of the productivity evaluation and performance prediction of tight sandstone gas reservoirs.

Dynamic streaming potential coupling coefficient in porous media with different pore size distributions

SUMMARY Seismoelectric signals are generated by electrokinetic coupling from seismic wave propagation in fluid-filled porous media. This process is directly related to the existence of an electrical double layer at the interface between the pore fluid and minerals composing the pore walls. The seismoelectric method attracts the interest of researchers in different areas, from oil and gas reservoir characterization to hydrogeophysics, due to the sensitivity of the seismoelectric signals to medium and fluid properties. In this work, we propose a physically based model for the dynamic streaming potential coupling coefficient (SPCC) by conceptualizing a porous medium as a bundle of tortuous capillaries characterized by presenting different pore size distributions (PSD). The results show that the dynamic streaming potential coupling coefficient is a complex function depending on the properties of pore fluid, mineral–pore fluid interfaces, microstructural parameters of porous media and frequency. Parameters influencing the dynamic SPCC are investigated and explained. In particular, we show that the PSD affects the transition frequency as well as the shape of the SPCC response as a function of frequency. The proposed model is then compared with published data and previous models. It is found that the approach using the lognormal distribution is in very good agreement with experimental data as well as with previous models. Conversely, the approach that uses the fractal distribution provides a good match with published data for sandstone samples but not for sand samples. This result implies that the fractal PSD may not be pertinent for the considered sand samples, which exhibit a relatively narrow distribution of pore sizes. Our proposed approach can work for any PSD, for example, including complex ones such as double porosity or inferred from direct measurements. This makes the proposed models more versatile than models available in literature.

A NOVEL FRACTAL MODEL FOR THE INVASION DEPTH OF FLUID THROUGH THE TORTUOUS CAPILLARY BUNDLE WITH ROUGHENED SURFACES IN POROUS MEDIA

In this paper, a novel fractal model for the invasion depth of fluid through the tortuous capillary bundle with roughened surfaces in porous media is proposed. The capillary pressure effect is considered in the proposed model. The proposed model is expressed as a function of structure parameters of porous media, including the relative roughness, the fractal dimension for pore size distribution, the contact angle, the density, the gravitational acceleration, the tortuosity and porosity. The invasion depth can be quantitatively characterized by the proposed model. By using the fractal theory, the effect of relative roughness on the invasion depth is discussed. It is observed that the invasion depth decreases with increasing the relative roughness of the tortuous capillary bundle with roughened surfaces. In addition, it is found that the invasion depth increases with the increase of the invasion time. The proposed model predictions are in good agreement with the available experimental data. Each parameter in our model has clear a distinct physical meaning, which may contribute to comprehend the better understanding of seepage mechanisms.

Constructing the pore-throat capillary bundle model using MICP data

Constructing the pore-throat capillary bundle model using MICP data