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.     

On Barenblatt's Model of Spontaneous Countercurrent Imbibition

Water imbibition is a critical mechanism of secondary oil recovery from fractured reservoirs. Spontaneous imbibition also plays a significant role in storage of liquid waste by controlling the extent of rock invasion. In the present paper, we extend a model of countercurrent imbibition based on Barenblatt's theory of non-equilibrium two-phase flow by allowing the model's relaxation time to be a function of the wetting fluid saturation. We obtain two asymptotic self-similar solutions, valid at early and late times, respectively. At a very early stage, the time scale characterizing the cumulative volume of imbibed (and expelled) fluid is a power function with exponent between 1.5 and 1. At a later stage, the time scaling for this volume approaches asymptotically classical square root of time, whereas the saturation profile asymptotically converges to Ryzhik's self-similar solution. Our conclusions are verified against experiments. By fitting the laboratory data, we estimate the characteristic relaxation times for different pairs of liquids.     

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.     

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.     

Numerical Analysis of Imbibition Front Evolution in Fractured Sandstone under Capillary-Dominated Conditions

Fractures serve as primary conduits having a great impact on the migration of injected fluid into fractured permeable media. Appropriate transport properties such as relative permeability and capillary pressure are essential for successful simulation and prediction of multi-phase flow in such systems. However, the lack of a thorough understanding of the dynamics governing immiscible displacement in fractured media, limits our ability to properly represent their macroscopic transport properties. Previous experimental observations of imbibition front evolution in fractured rocks are examined in the present study using an automated history-matching approach to obtain representative relative permeability and capillary pressure curves. Predicted imbibition front evolution under different flow conditions resulted in an excellent agreement with experimental observations. Sensitivity analyses, in combination with direct experimental observation, allowed exploring the competing effects of relative permeability and capillary pressure on the development of saturation distribution and imbibing front evolution in fractured porous media. Results show that residual saturations are most sensitive to matrix relative permeability to oil, while the ratio of oil and water relative permeability, rock heterogeneity, boundary condition, and matrix–fracture capillary pressure contrast, affect displacement shape, speed, and geometry of the imbibing front.     

A Stochastic Approach to the Two-Phase Displacement Problem in Heterogeneous Porous Media

The problem of two-phase immiscible flow in heterogeneous porous media in the case of a horizontal displacement of some fluid by another, which is of practical importance in industrial oil recovery, is considered. Assuming that (a) the saturation jump on the displacement front is constant, (b) the log-permeability of the medium obeys Gaussian statistics, and (c) the case when the front is stable, the displacement front position and the saturation distribution are described analytically in terms of generalized functions. Note that in our analysis we do not assume that the front shape fluctuations are small, and in this respect our results may be regarded as exact. The assumption that the log-permeability fluctuations are small was only used in deriving the linear relation between the log-permeability of a porous medium and the total flow velocity (Nœtinger et al. in Fluid Dyn 41(5):830–842, 2006). By means of ensemble averaging, the mean saturation and saturation variance are found in the vicinity of the front. These characteristics are related to the variance of front displacements, which, in turn, can be calculated analytically. Next, a method for reconstructing the full solution for the saturation (rarefaction wave) is proposed. Such a full solution satisfies the mass conservation requirement. Finally, the theoretical predictions are compared with the results of numerical simulations carried out within the framework of Monte-Carlo method.     

High Resolution Imaging of Unstable,Forced Imbibition in Berea Sandstone

An experimental investigation is presented of immiscible, high-mobility ratio forced imbibition in a representative linear homogeneous sandstone. Water floods with mobility ratios (from 1 to 155) at various water injection rates were conducted. Fine-scale (order mm³) in situ water saturation history was collected via X-ray computed tomography (CT). Three-dimensional images were constructed of stable displacement and the initiation and growth of unstable water fingers. Interestingly, viscous fingers do not lead the displacement front by significant distances, counter to experience in miscible systems. In this homogeneous porous medium, both water (displacing phase) injection rate and oil (displaced phase) viscosity have an obvious effect on the stability of the water front. As the oil viscosity and displacement rate increase, the water front becomes less stable. In addition, the so-called shock mobility ratio, as computed from steady-state relative permeability, is found to be predictive regarding displacement front stability. When the shock mobility ratio is greater than 1, the displacement is always unstable. Steady-state relative permeability, however, is found to be a function of viscosity ratio for unstable displacements.     

Capillary Pressure at the Imbibition Front During Water–Oil Counter-Current Spontaneous Imbibition

Counter-current spontaneous imbibition (COUCSI) in porous media is driven by capillary forces. Capillary action results in a high capillary imbibition pressure at the imbibition front and a low capillary drainage pressure at the outlet face. It is the difference between these two pressures that draws in the wetting phase and pushes out the non-wetting phase. A technique for measuring the capillary pressure at an imbibition front under restricted flow conditions has been developed and applied to Berea sandstone with a range of permeabilities. In the experiments, brine was the wetting phase and refined oil was the non-wetting phase. One end face of a sandstone core was butted to a short section of a finer-pored rock. The composite core surface was then sealed, apart from the end face of the low-permeability segment. A connection to a pressure transducer was set in the opposite end face of the core. Initially, the main core segment was filled with oil. In most cases, the finer-pored segment was filled with brine. Imbibition was started by immersing the core in brine. The purpose of the finer-pored segment was to prevent the escape of non-wetting phase from the open face. For some tests there was an initial period of co-current spontaneous imbibition (COCSI) created by allowing production of non-wetting phase through an outlet tapping in the sealed end face. The outlet was then connected to the transducer and the imbibition changed to COUCSI. There followed an increase in the monitored end pressure to a maximum as fluid redistributed within the core. For the tests in which the fine-pored segments were pre-saturated with brine, even without an initial period of co-current imbibition, limited invasion of the main core segment by brine resulted in an asymptotic rise of the end pressure to a maximum as the imbibition front dispersed. To confirm that the dispersing front did not reach the dead end of the core, the distance of advance of the wetting liquid was detected by a series of electrodes. The maximum value of the end pressure provides an estimate of the capillary pressure at an imbibition front for COUCSI. The maximum capillary pressure generated by the invading fluids ranged from 6.6 kPa to 42 kPa for sandstone with permeabilities between 1.050 (μm)² and 0.06 (μm)².     

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.     

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.

     

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。     

Fracture prediction for piezoelectric ceramics based on the electric field saturation concept

The electrical field saturation model is applied to the fracture prediction of piezoelectric materials containing electrically impermeable cracks. This model is analogously similar to the electric displacement saturation model that available in the literature. An electrical field saturation strip near the crack front is introduced in the analytical model. The stress intensity factor K and the energy release rate G are obtained in closed-form. It is found that fracture predictions based on K and G criteria are identical. Fracture predictions based on the electric field saturation model and the electric displacement model are also found to be the same.     

基于核磁共振孔隙划分的致密油藏自发渗吸原油可动性研究

自发渗吸驱油是致密油藏提高采收率的有效手段,但不同的孔隙划分方法会导致原油可动性精细定量表征存在差异性.基于此,以鄂尔多斯盆地延长组致密油藏为研究对象,开展了四种典型致密岩心的自发渗吸驱油实验,利用基于核磁共振分形理论的流体分布孔隙精细划分方法,区分了致密砂岩岩心孔隙类型,明确了不同类型岩心孔隙结构对原油可动性和自发渗吸驱油速率的控制特征.研究结果表明不同类型岩心自发渗吸模拟油动用程度介于22.07%～33.26%,核磁共振T2谱双峰型岩心自发渗吸模拟油动用程度高于单峰型岩心;不同类型致密砂岩岩心中流体分布孔隙可初步划分出P1和P2两种类型, P1类型孔隙则可进一步划分出P1-1, P1-2和P1-3三种亚类型孔隙;致密砂岩岩心中P1和P2类孔隙中模拟油均有不同程度的动用, P1类孔隙作为致密岩心中主要孔隙,尤其是P1类孔隙中P1-2和P1-3类孔隙的数量决定了自发渗吸模拟油动用程度;P1-1, P1-2和P1-3类孔隙结构差异性对自发渗吸模拟油动用程度起决定性作用,较小尺寸孔径孔隙较大的孔隙结构差异性不仅提升了自发渗吸模拟油动用程度,而且提升了自发渗吸驱油速率;流体可动性指数较高的P...     

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.     

Two-phase jet flows in porous media

An analytical model describing the development of the filtration instability of the displacement front of fluids with different viscosities in a porous medium with account for capillary forces is proposed. A set of laboratory experiments on viscous fluid displacement from a porous medium is carried out. To describe the observable flows the model deals with the characteristic profile of the mean water saturation along the flow rather than with the curves of relative phase permeabilities of the fluids. The analytical model developed well describes the results of the laboratory modeling and the data of an actual oil field operation.     

Theoretical and experimental aspects of vacuum impregnation of porous media using transparent etched networks

Vacuum impregnation is a process method in which air and native solution are removed from the porous space of a given porous material and replaced by an external solution. Vacuum impregnation is divided into two steps: Firstly, the porous material is immersed in a liquid solution and exposed to subatmospheric pressure for a given time to ensure that air trapped in the porous materials will be removed; secondly, atmospheric pressure is re-established and the external solution penetrates the pore structure of the porous material. The objective of this study was to describe the hydrodynamic mechanisms involved in vacuum impregnation of porous materials as a function of capillary number and viscosity ratio. To achieve the objectives proposed in the present study, a transparent glass micromodel 7.7 cm × 7.4 cm was first constructed using the photolithographic technique. In addition, a stainless steel vacuum tank was built. The tank top was covered with a transparent reinforced glass plate. The whole system was connected to a vacuum pump, and a conventional video camera was adapted to record the experiments. Liquid saturation was determined through the image analysis process. Capillary number and viscosity ratio were determined for the drainage and imbibition processes. For the systems studied, we conclude that transport mechanisms ranged between stable displacement and capillary fingering during the vacuum step (drainage) while transport mechanisms ranged between continuous capillary and discontinuous capillary domains during the atmospheric step (imbibition). Earlier work indicated that our proposed process should be even more efficient for realistically large systems.     

Strip-electromechanical model solution for piezoelectric plate cut along two semi-permeable collinear cracks

A strip electric saturation and mechanical yielding model solution is proposed for a piezoelectric plate cut along two equal collinear semi-permeable mode-I cracks with electrical polarization reaching a saturation limit and normal stress reaching a yield stress along a line segment in front of the cracks. By using Stroh formalism and complex variable technique, we derived the analytical solution for the field quantities. Three different situations are investigated when developed electrical saturation zone is bigger/smaller or equal to the developed mechanical yield zone. Numerical results show that the effect of different electric boundary conditions on the crack opening displacement and crack opening potential drop is significant and should not be ignored. The influence of electric load displacement on the energy release rate is also investigated for PZT-4, PZT-5H and BaTiO₃ ceramics, and it may assists for the correct choosing of ceramic for specific job.     

Analysis of wet pressing of paper: the three-phase model. Part I: constant air density

In this study, we consider a one-dimensional three-phase model describing wet pressing of paper. Part I is devoted to the simplified case in which air is assumed incompressible. In Part II we drop this assumption. The model is formulated in terms of water saturation and void ratio and it uses a material coordinate to describe spatial dependence. It also involves cross or matching conditions between the wet paper and the felt. In mathematical terms, we end up with a coupled system of equations: a nonlinear diffusion equation and a first order hyperbolic equation. We present some analytical observations to explain the essential behaviour of the model and we carry out numerical experiments using an upwind and a front tracking method.     

An exact solution of crack problems in piezoelectric materials

IntroductionInrecentyearscrackproblemsinpiezoelectricmaterialhavereceivedmuchattention.Manytheoreticalanalyseshavebeengivenby[1～16].Itshouldbe,however,notedthatalltheaboveanalysesarebasedonaso-calledimpermeablecrackassumphon,i.e.thecrackfacesareassumedtobeimpermeabletoelectricfield,sotheelectricdisplacementvanishesinsidethecrack.Usingthisassumption,onewillobtainthefollowingresultS[2'3'5,6'9'16]=whentheelectricloadsaresolelyaPPliedatinLfinity,theelectricdisplacementissquare-rootsingularatthe…