Numerical Solution of Equation for Dynamic,Spontaneous Imbibition with Variable Inlet Saturation and Interfacial Coupling Effects

In the oil industry, dynamic spontaneous imbibition plays an important role in several flow processes in porous media. A numerical approach is developed to simulate dynamic spontaneous imbibition with variable inlet saturation and interfacial coupling. The inclusion of interfacial coupling effects invalidates the assumption that the interfaces (fluid/fluid and fluid/solid) act in the same way. The one-dimensional numerical simulation model is developed using a Lagrangian formulation discretized in time and saturation. The solution of the partial differential equations utilizes an iteration process that includes two material balance criteria to ensure the validity of the variable inlet saturation. Furthermore, an error analysis, the validation of the model and a sensitivity study on the optimal number of time steps and saturation grid cells are undertaken. The numerical simulation solution represents an accurate approach to investigate the effect of fluid and rock properties on dynamic spontaneous imbibition.     

An Analytic Solution for the Frontal Flow Period in 1D Counter-Current Spontaneous Imbibition into Fractured Porous Media Including Gravity and Wettability Effects

Including gravity and wettability effects, a full analytical solution for the frontal flow period for 1D counter-current spontaneous imbibition of a wetting phase into a porous medium saturated initially with non-wetting phase at initial wetting phase saturation is presented. The analytical solution applicable for liquid–liquid and liquid–gas systems is essentially valid for the cases when the gravity forces are relatively large and before the wetting phase front hits the no-flow boundary in the capillary-dominated regime. The new analytical solution free of any arbitrary parameters can also be utilized for predicting non-wetting phase recovery by spontaneous imbibition. In addition, a new dimensionless time equation for predicting dimensionless distances travelled by the wetting phase front versus dimensionless time is presented. Dimensionless distance travelled by the waterfront versus time was calculated varying the non-wetting phase viscosity between 1 and 100 mPas. The new dimensionless time expression was able to perfectly scale all these calculated dimensionless distance versus time responses into one single curve confirming the ability for the new scaling equation to properly account for variations in non-wetting phase viscosities. The dimensionless stabilization time, defined as the time at which the capillary forces are balanced by the gravity forces, was calculated to be approximately 0.6. The full analytical solution was finally used to derive a new transfer function with application to dual-porosity simulation.     

Correction to: Time and Space Fractional Diffusion in Finite Systems

Additive manufacturing technology, or 3D printing, with silica sand has enabled the manufacture of porous rock analogues for the use in experimental studies of geomechanical properties of reservoir rocks. The accurate modelling of the fluid flow phenomena within a reservoir and improving the performance of hydrocarbon recovery require an understanding of physical and chemical interactions of the reservoir fluids and the rock matrix. Therefore, for the 3D printed samples to serve as rock analogues, flow properties have to be equivalent to the petrophysical properties of their natural counterparts, such as Berea sandstone. In this study, sandstones that were 3D printed with silica sand and Poly-Furfuryl alcohol (PFA) binder, were used to investigate interactions between porous media with different fluids. Wettability preference of 3D printed samples was characterized through contact angle measurements, as well as co-current and counter-current spontaneous imbibition experiments. Results indicated that 3D printed sandstones had mixed-wet characteristics due to the high preference of silica grains for polar fluids and the affinity PFA binder to the oleic phase. Printing configurations including binder saturation were found to greatly influence the wettability preference of the 3D printed analogue rocks as higher PFA concentrations resulted in more strongly oil-wet preferences. Efforts to optimize the printing process and challenges to control the wettability preferences of the 3D printed samples are also highlighted.

     

Theoretical Development of a Novel Equation for Dynamic Spontaneous Imbibition with Variable Inlet Saturation and Interfacial Coupling Effects

In oil recovery from fractured reservoirs, dynamic spontaneous imbibition (DSI) plays an important role. Conventional equations used for characterizing dynamic spontaneous imbibition neglect the effects of the driving forces acting across the wetting and non-wetting phases which are flowing in opposite directions. Such effects, defined as interfacial coupling effects (ICE), are known to cause a decrease in the calculated flow rate in drainage processes. Moreover, none of the numerical models have considered a variable inlet saturation (S*) for DSI. A new theoretical model has been developed using generalized transport equations to describe dynamic spontaneous imbibition for immiscible two-phase flow processes. The inclusion of interfacial coupling effects provides a more accurate way to describe dynamic spontaneous imbibition. Furthermore, the addition of variable inlet saturation allows one to establish whether the inlet-face saturation (S*) increases from the initial saturation to 1−Sro, or whether it can remain constant and equal to one minus the residual saturation to the non-wetting phase (1−Sro).     

Analysis of capillary interaction and oil recovery under ultrasonic waves

Although, the effects of ultrasonic irradiation on multiphase flow through porous media have been studied in the past few decades, the physics of the acoustic interaction between fluid and rock is not yet well understood. Various mechanisms may be responsible for enhancing the flow of oil through porous media in the presence of an acoustic field. Capillary related mechanisms are peristaltic transport due to mechanical deformation of the pore walls, reduction of capillary forces due to the destruction of surface films generated across pore boundaries, coalescence of oil drops due to Bjerknes forces, oscillation and excitation of capillary trapped oil drops, forces generated by cavitating bubbles, and sonocapillary effects. Insight into the physical principles governing the mobilization of oil by ultrasonic waves is vital for developing and implementing novel techniques of oil extraction. This paper aims at identifying and analyzing the influence of high-frequency, high-intensity ultrasonic radiation on capillary imbibition. Laboratory experiments were performed using cylindrical Berea sandstone and Indiana limestone samples with all sides (quasi-co-current imbibition), and only one side (counter-current imbibition) contacting with the aqueous phase. The oil saturated cores were placed in an ultrasonic bath, and brought into contact with the aqueous phase. The recovery rate due to capillary imbibition was monitored against time. Air–water, mineral oil–brine, mineral oil–surfactant solution and mineral oil-polymer solution experiments were run each exploring a separate physical process governing acoustic stimulation. Water–air imbibition tests isolate the effect of ultrasound on wettability, capillarity and density, while oil–brine imbibition experiments help outline the ultrasonic effect on viscosity and interfacial interaction between oil, rock and aqueous phase. We find that ultrasonic irradiation enhances capillary imbibition recovery of oil for various fluid pairs, and that such process is dependent on the interfacial tension and density of the fluids. Although more evidence is needed, some runs hint that wettability was not altered substantially under ultrasound. Preliminary analysis of the imbibition recoveries also suggests that ultrasound enhances surfactant solubility and reduce surfactant adsorption onto the rock matrix. Additionally, counter-current experiments involving kerosene and brine in epoxy coated Berea sandstone showed a dramatic decline in recovery. Therefore, the effectiveness of any ultrasonic application may strongly depend on the nature of interaction type, i.e., co- or counter-current flow. A modified form of an exponential model was employed to fit the recovery curves in an attempt to quantify the factors causing the incremental recovery by ultrasonic waves for different fluid pairs and rock types.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

Characteristics of developing vertical bubbly flow under normal and microgravity conditions

The axial development of the void fraction profile, interfacial area concentration and Sauter mean bubble diameter of adiabatic nitrogen-water bubbly flows in a 9 mm-diameter pipe were measured using stereo image processing under normal and microgravity conditions. The flow measurements were performed at four axial locations (axial distance from the inlet, z normalized by the pipe diameter, D, z/D = 5, 20, 40 and 60) and with various flows: superficial gas velocity of 0.00840-0.0298 m/s, and superficial liquid velocity of 0.138-0.914 m/s. The effect of gravity on radial distribution of bubbles and the axial development of two-phase flow parameters is discussed in detail based on the obtained database and visual observation. Following Serizawa-Kataoka’s phase distribution pattern criteria under normal gravity conditions, the phase distribution pattern map was developed. Similar to normal gravity two-phase flows, wall, core and intermediate void peak patterns are observed under microgravity conditions but a transition void distribution pattern is not observed in the current experimental conditions. The data obtained in the current experiment are expected to contribute to the benchmarking of CFD simulation of phase distribution pattern and interfacial area concentration in forced convective pipe flow under microgravity conditions.     

SPONTANEOUS CAPILLARY IMBIBITION MODEL IN TIGHT OIL RESERVOIRS 1)

部分致密油井压后关井一段时间,压裂液返排率普遍低于30%,但是致密油气井产量反而越高,这与压裂液毛细管力渗吸排驱原油有关。然而,致密油储层致密,物性差,渗流机理复杂,尚没有形成统一的自发渗吸模型。本文基于油水两相非活塞式渗流理论,建立了压后闷井期间压裂液在毛细管力作用下自发渗吸进入致密油储层的数学模型,采用数值差分方法进行求解,并分析了相关影响因素。结果显示渗吸体积、渗吸前缘移动距离与渗吸时间的平方根呈线性正相关关系,与经典Handy渗吸理论模型预测结果一致,说明毛细管力自发渗吸模型可靠性较高。数值计算结果表明毛细管水相扩散系数是致密储层自发渗吸速率的主控参数,毛细管水相扩散系数越高,自发渗吸速率越大。毛细管水相扩散系数随着含水饱和度先增加后减小;随着束缚水饱和度、油相和水相端点相对渗透率增加而增加;随着相渗特征指数、油水黏度比和残余油饱和度增加而减小。该研究有助于深入认识致密油储层压裂液渗吸机理,对优化返排制度、提高致密油井产量具有重要意义。     

Capillary Pressure at a Saturation Front During Restricted Counter-Current Spontaneous Imbibition with Liquid Displacing Air

The ultimate driving force for counter-current spontaneous imbibition of a fluid into a porous material is the capillary pressure developed under dynamic conditions at the imbibition front. This is a difficult variable to measure. We report experiments using restricted counter-current spontaneous imbibition to find the maximum capillary pressure developed during imbibition of a light mineral oil (and brine) into initially air-filled sandstone core samples with one end-face open. The production of air from the core was prevented by covering its open face with a low permeability core segment set against the main test segment. The location of the imbibition front and the pressure resulting from compression of air ahead of the imbibition front were monitored. In some cases, in order to achieve stabilized gas pressures with the front still advancing through the core, the air in the core was compressed at the start of the imbibition test. The subsequently measured stabilized air pressures dropped only slightly as imbibition slowed. The measured pressures are directly related to the effective capillary pressures that drive spontaneous imbibition. After spontaneous imbibition ceased, the pressure was released by flow of air through the sealed end of the core and further spontaneous imbibition occurred in co-current mode. Comparison of the stabilized pressures with previously published oil/brine imbibition results showed close agreement after compensation for the difference in interfacial tension.     

A Dynamic Network Model for Two-Phase Flow in Porous Media

We present a dynamic model of immiscible two-phase flow in a network representation of a porous medium. The model is based on the governing equations describing two-phase flow in porous media, and can handle both drainage, imbibition, and steady-state displacement. Dynamic wetting layers in corners of the pore space are incorporated, with focus on modeling resistivity measurements on saturated rocks at different capillary numbers. The flow simulations are performed on a realistic network of a sandpack which is perfectly water-wet. Our numerical results show saturation profiles for imbibition in agreement with experiments. For free spontaneous imbibition we find that the imbibition rate follows the Washburn relation, i.e., the water saturation increases proportionally to the square root of time. We also reproduce rate effects in the resistivity index for drainage and imbibition.     

Effect of dynamic contact angle variation on spontaneous imbibition in porous materials

We investigate the influence of contact angle variations on spontaneous imbibition of moisture in porous materials. While the contact angle is typically assumed constant when modelling the moisture transfer in porous media, experimental findings put this assumption into question. It has been shown that during imbibition the contact angle notably rises with increasing meniscus velocity. This phenomenon resultantly affects the moisture retention curve, the relation linking the local capillary pressure to the local moisture saturation, which in turn impacts the imbibition rate and moisture distribution. This study investigates these dynamic effects via a pore network technique as well as a continuum approach. It is shown that the impacts of pore-scale contact angle variations on the imbibition process can be reproduced at the continuum scale through a modified moisture retention curve including a dynamic term. Complementarily a closed-form equation expressing the dynamic capillary pressure in terms of local saturation and saturation rate is derived. The continuum approach is then finally employed to predict measured moisture saturation profiles for imbibition in Berea sandstone and diatomite found in literature, and a fair agreement between simulated and measured outcomes is observed.

     

Pore Scale Modeling of Rate Effects in Imbibition

We use pore scale network modeling to study the effects of flow rate and contact angle on imbibition relative permeabilities. The model accounts for flow in wetting layers that occupy roughness or crevices in the pore space. Viscous forces are accounted for by solving for the wetting phase pressure and assuming a fixed conductance in wetting layers. Three-dimensional simulations model granular media, whereas two-dimensional runs represent fracture flow.We identify five generic types of displacement pattern as we vary capillary number, contact angle, and initial wetting phase saturation: flat frontal advance, dendritic frontal advance, bond percolation, compact cluster growth, and ramified cluster growth. Using phase diagrams we quantify the range of physical properties under which each regime is observed. The work explains apparently inconsistent experimental measurements of relative permeability in granular media and fractures.     

Analytical Solutions for Linear Counter-Current Spontaneous Imbibition in the Frontal Flow Period

Analytical solutions have been derived in this article for counter-current spontaneous imbibition in a semi-infinite linear porous medium. The derivation is based on six fundamental equations, i.e., the existence of only two fluid phases, the continuity of flow, the generalized Darcy’s law for the wetting phase, the generalized Darcy’s law for the nonwetting phase, the relation between the capillary pressure generated by an interface and the difference in pressure across the interface between the nonwetting and wetting phases, and the conservation of mass. The mechanisms revealed by the solutions are simple and clear, and most of them are newly discovered. The solutions specify flow characteristics in three aspects. First, flow decelerates according to the square root of time. Second, the imbibed wetting phase volume is proportional to the square root of the product of porosity, suction capacity, and piston efficiency, while the sweeping distance is proportional to the square root of the ratio of suction capacity to the product of porosity and piston efficiency. Third, the volume of imbibed wetting phase is related to the area of the sample cross section, while sweeping distance does not. Suction capacity and piston efficiency are two measurable physical properties during counter-current spontaneous imbibition. An important step has been taken to extend the solutions to the cases where the saturation profiles are characterized by truncated fronts common in the productive reservoirs. It has been proven that the results calculated by analytical algorithms and the corresponding high-precision numerical methods match each other very closely. Previous study has already shown that the numerical simulation results can be highly consistent with the experimental performance measures. Therefore, with the advantages of clarity, accuracy, and speed, the analytical solutions provided in this article are very useful in increasing mechanism comprehension, enhancing engineering calculations and promoting wettability determination.     

A Parametric Model for Predicting Relative Permeability-Saturation-Capillary Pressure Relationships of Oil–Water Systems in Porous Media with Mixed Wettability

A parametric two-phase, oil–water relative permeability/capillary pressure model for petroleum engineering and environmental applications is developed for porous media in which the smaller pores are strongly water-wet and the larger pores tend to be intermediate- or oil-wet. A saturation index, which can vary from 0 to 1, is used to distinguish those pores that are strongly water-wet from those that have intermediate- or oil-wet characteristics. The capillary pressure submodel is capable of describing main-drainage and hysteretic saturation-path saturations for positive and negative oil–water capillary pressures. At high oil–water capillary pressures, an asymptote is approached as the water saturation approaches the residual water saturation. At low oil–water capillary pressures (i.e. negative), another asymptote is approached as the oil saturation approaches the residual oil saturation. Hysteresis in capillary pressure relations, including water entrapment, is modeled. Relative permeabilities are predicted using parameters that describe main-drainage capillary pressure relations and accounting for how water and oil are distributed throughout the pore spaces of a porous medium with mixed wettability. The capillary pressure submodel is tested against published experimental data, and an example of how to use the relative permeability/capillary pressure model for a hypothetical saturation-path scenario involving several imbibition and drainage paths is given. Features of the model are also explained. Results suggest that the proposed model is capable of predicting relative permeability/capillary pressure characteristics of porous media mixed wettability.     

Wetting Phase Bridges Establish Capillary Continuity Across Open Fractures and Increase Oil Recovery in Mixed-Wet Fractured Chalk

The effect of fractures on oil recovery and in situ saturation development in fractured chalk has been determined at near neutral wettability conditions. Fluid saturation development was monitored both in the matrix and in the fractures and the mechanisms of fracture crossing were determined using high spatial resolution MRI. Capillary continuity across open oil-filled fractures was verified by imaging the water bridges established within the fracture. Despite an alternate escape fracture for the water, separate water bridges were shown to be stable for the entire duration of the experiments. The established capillary contact resulted in oil recovery exceeding the spontaneous imbibition potential in the outlet-isolated cores by ca. 10% PV. This is explained by viscous recovery provided by water bridges across open fractures. The size of the bridges seemed to be controlled by the wettability of the rock and not by the differential pressure applied across the open fracture.     

Effect of Organosilicon-Acrylic Emulsion Treatment on Wettability of Porous Media

To investigate the influence of the organosilicon-acrylic on wetting properties of porous media, contact angle tests were performed on two different sandstones. In addition, the effectiveness of the emulsion on wettability alteration of porous media was validated by capillary rise and spontaneous imbibition tests. The results of wettability tests showed that the wettability of two sandstones was altered from water-wet to gas-wet after treatment with the emulsion. The principle that the critical radius of pore throats and wettability of porous media affect liquids flow was derived analytically and verified experimentally. Coreflood results demonstrated that the latex resulted in increasing the water permeability through altering the rock wettability to gas-wetting, then decreasing the friction drag between liquids and rocks surface. Thereby, the emulsion treatment could increase the flowback rate of trapped liquids. Experimental results were in good agreement with the theoretical analysis. In conclusion, all results indicated that the emulsion could alter the wettability from water-wet to intermediate gas-wet and enhance water permeability in porous media. It was extrapolated that the emulsion had the tremendous potential to be applied in field conditions, enhancing gas productivity through the cleanup of trapped water in the vicinity of the wellbore.     

A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure

This article describes a semi-analytical model for two-phase immiscible flow in porous media. The model incorporates the effect of capillary pressure gradient on fluid displacement. It also includes a correction to the capillarity-free Buckley–Leverett saturation profile for the stabilized-zone around the displacement front and the end-effects near the core outlet. The model is valid for both drainage and imbibition oil–water displacements in porous media with different wettability conditions. A stepwise procedure is presented to derive relative permeabilities from coreflood displacements using the proposed semi-analytical model. The procedure can be utilized for both before and after breakthrough data and hence is capable to generate a continuous relative permeability curve unlike other analytical/semi-analytical approaches. The model predictions are compared with numerical simulations and laboratory experiments. The comparison shows that the model predictions for drainage process agree well with the numerical simulations for different capillary numbers, whereas there is mismatch between the relative permeability derived using the Johnson–Bossler–Naumann (JBN) method and the simulations. The coreflood experiments carried out on a Berea sandstone core suggest that the proposed model works better than the JBN method for a drainage process in strongly wet rocks. Both methods give similar results for imbibition processes.     

Effect of fluids properties on non-equilibrium capillarity effects: Dynamic pore-network modeling

We have developed a Dynamic Pore-network model for Simulating Two-phase flow in porous media (DYPOSIT). The model is applicable to both drainage and imbibition processes. Employing improved numerical and geometrical features in the model facilitate a physically-based pore-scale simulator. This computational tool is employed to perform several numerical experiments (primary and main drainage, main imbibition) to investigate the current capillarity theory. Traditional two-phase flow formulations state that the pressure difference between the two phase is equal to the capillary pressure, which is assumed to be a function of saturation only. Many theoretical and experimental studies have shown that this assumption is invalid and the pressure difference between the two fluids is not only equal to the capillary pressure but is also related to the variation of saturation with time in the domain; this is referred to as the non-equilibrium capillarity effect. To date, non-equilibrium capillarity effect has been investigated mainly under drainage. In this study, we analyze the non-equilibrium capillarity theory under drainage and imbibition as a function of saturation, viscosity ratio, and effective viscosity. Other aspects of the dynamics of two-phase flow such as trapping and saturation profile are also studied.     

Gas cluster growth by solute diffusion in porous media. Experiments and automaton simulation on pore network

The study presented in this paper deals with the liquid–gas phase change by pressure decline of supersaturated CO₂ solutions in 2D porous media. The growth of the gas phase is studied experimentally and numerically as a function of supersaturation, wettability and gravity. Experiments are performed on a transparent etched network (micromodel) and simulations with a specific numerical automaton.In the experiments, the nucleation process, i.e. the occurrence of the gas bubbles, as well as the growth of these bubbles are visualised and analysed by means of a micro video camera and an image processing apparatus. The observations confirm the heterogeneous nature of nucleation and the disordered growth pattern of the gas phase. The analysis of the growth rate of a single gas cluster shows that this phenomenon is different from the compact growth of an isolated single bubble in the bulk. As previously predicted, the bubble growth by mass transfer and volume expansion in porous media is characterised by a pattern of the invasion percolation type under normal laboratory conditions.Numerical simulations of the growth pattern and the growth rate of a single gas cluster are performed with a numerical automaton. Based on a pore network modelling technique and a set of hypotheses derived from the observations, this automaton is first validated by comparing the numerical results with the experiments. Then, the automaton is used to conduct a sensitivity study. In particular, the influences of the Jakob number, pressure decline rate, Bond number, wettability and characteristics of the microstructure are investigated.     

Effect of gravity on axial development of bubbly flow at low liquid Reynolds number

In view of the great importance of two geometrical parameters such as void fraction and interfacial area concentration to the accurate two-phase flow analysis at microgravity conditions, axial developments of flow parameters such as void fraction, interfacial area concentration, bubble Sauter mean diameter, and bubble number density were measured in bubbly flow at microgravity and low liquid Reynolds number conditions where the gravity effect on the flow parameters were pronounced. A total of seven data sets were acquired in the flow range of the void fraction from 1.01% to 3.36% and the liquid Reynolds number from 1,400 to 4,750. The measurements were also performed in the similar flow range at normal gravity conditions. The transport mechanisms of the flow parameters are discussed in detail based on the data measured at normal and microgravity conditions, and the drift-flux model developed at microgravity conditions are compared with the measured data.An erratum to this article can be found at