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.     

Relative Permeability Analysis of Tube Bundle Models,Including Capillary Pressure

The analytical equations for calculating two-phase flow, including local capillary pressures, are developed for the bundle of parallel capillary tubes model. The flow equations that are derived were used to calculate dynamic immiscible displacements of oil by water under the constraint of a constant overall pressure drop across the tube bundle. Expressions for averaged fluid pressure gradients and total flow rates are developed, and relative permeabilities are calculated directly from the two-phase form of Darcy's law. The effects of pressure drop and viscosity ratio on the relative permeabilities are discussed. Capillary pressure as a function of water saturation was delineated for several cases and compared to a steady-state mercury-injection drainage type of capillary pressure profile. The bundle of serial tubes model (a model containing tubes whose diameters change randomly at periodic intervals along the direction of flow), including local Young-Laplace capillary pressures, was analyzed with respect to obtaining relative permeabilities and macroscopic capillary pressures. Relative permeabilities for the bundle of parallel tubes model were seen to be significantly affected by altering the overall pressure drop and the viscosity ratio; relative permeabilities for the bundle of serial tubes were seen to be relatively insensitive to viscosity ratio and pressure, and were consistently X-like in profile. This work also considers the standard Leverett (1941) type of capillary pressure versus saturation profile, where drainage of a wetting phase is completed in a step-wise steady fashion; it was delineated for both tube bundle models. Although the expected increase in capillary pressure at low wetting-phase saturation was produced, comparison of the primary-drainage capillary pressure curves with the pseudo-capillary pressure profiles, that are computed directly using the averaged pressures during the displacements, revealed inconsistencies between the two definitions of capillary pressure.     

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.     

Percolation theory approach to transport phenomena in porous media

The percolation theory approach to static and dynamic properties of the single- and two-phase fluid flow in porous media is described. Using percolation cluster scaling laws, one can obtain functional relations between the saturation fraction of a given phase and the capillary pressure, the relative permeability, and the dispersion coefficient, in drainage and imbibition processes. In addition, the scale dependency of the transport coefficient is shown to be an outcome of the fractal nature of pore space and of the random flow pattern of the fluids or contaminant.     

Bulk Flow Regimes and Fractional Flow in 2D Porous Media by Numerical Simulations

We investigate a two-dimensional network simulator that models the dynamics of two-phase immiscible bulk flow where film flow can be neglected. We present a method for simulating the detailed dynamical process where the two phases are allowed to break up into bubbles, and bubbles are allowed to merge together. The notions of drainage and imbibition are not adequate to describe this process since there is no clear front between the fluids. In fact, the simulator is constructed so that one can study the behaviour of the system far from inlets and outlets, where the two fluids have been mixed together so much that all initial fronts have broken up. The simulator gives the fractional flow as a function of the saturation of each of the fluids. For the case of two fluids with equal viscosity, we classify flow regimes that are parametrized by the capillary number.     

Uniqueness of Specific Interfacial Area–Capillary Pressure–Saturation Relationship Under Non-Equilibrium Conditions in Two-Phase Porous Media Flow

The capillary pressure?Csaturation (P ^c?CS ^w) relationship is one of the central constitutive relationships used in two-phase flow simulations. There are two major concerns regarding this relation. These concerns are partially studied in a hypothetical porous medium using a dynamic pore-network model called DYPOSIT, which has been employed and extended for this study: (a) P ^c?CS ^w relationship is measured empirically under equilibrium conditions. It is then used in Darcy-based simulations for all dynamic conditions. This is only valid if there is a guarantee that this relationship is unique for a given flow process (drainage or imbibition) independent of dynamic conditions; (b) It is also known that P ^c?CS ^w relationship is flow process dependent. Depending on drainage and imbibition, different curves can be achieved, which are referred to as ??hysteresis??. A thermodynamically derived theory (Hassanizadeh and Gray, Water Resour Res 29: 3389?C3904, 1993a) suggests that, by introducing a new state variable, called the specific interfacial area (a ^nw, defined as the ratio of fluid?Cfluid interfacial area to the total volume of the domain), it is possible to define a unique relation between capillary pressure, saturation, and interfacial area. This study investigates these two aspects of capillary pressure?Csaturation relationship using a dynamic pore-network model. The simulation results imply that P ^c?CS ^w relation not only depends on flow process (drainage and imbibition) but also on dynamic conditions for a given flow process. Moreover, this study attempts to obtain the first preliminary insights into the global functionality of capillary pressure?Csaturation?Cinterfacial area relationship under equilibrium and non-equilibrium conditions and the uniqueness of P ^c?CS ^w?Ca ^nw relationship.     

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).     

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.     

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。     

A Mathematical Model for Hysteretic Two-Phase Flow in Porous Media

We develop a mathematical model for hysteretic two-phase flow (of oil and water) in waterwet porous media. To account for relative permeability hysteresis, an irreversible trapping-coalescence process is described. According to this process, oil ganglia are created (during imbibition) and released (during drainage) at different rates, leading to history-dependent saturations of trapped and connected oil. As a result, the relative permeability to oil, modelled as a unique function of the connected oil saturation, is subject to saturation history. A saturation history is reflected by history parameters, that is by both the saturation state (of connected and trapped oil) at the most recent flow reversal and the most recent water saturation at which the flow was a primary drainage. Disregarding capillary diffusion, the flow is described by a hyperbolic equation with the connected oil saturation as unknown. This equation contains functional relationships which depend on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes) and stationary discontinuities (connecting different saturation histories). The entropy condition for travelling waves is generalized to include admissible shock waves which coincide with flow reversals. It turns out that saturation history generally has a strong influence on both the type and the speed of the waves from which the solution is constructed.     

超临界CO_2-水两相流与CO_2毛细捕获:微观孔隙模型实验与数值模拟研究

CO_2毛细捕获机制是CO_2地质封存中的关键科学问题,然而有关孔隙尺度下(微米极)超临界CO_2毛细捕获的研究较少.采用高压流体-显微镜-微观模型实验装置,开展超临界CO_2条件(8.5 MPa,45?C)下CO_2驱替水(排水)和水驱替CO_2(吸湿)实验,采用高分辨率照相机采集CO_2水两相流运动图像,并借助光学显微镜直接观测孔隙尺度下CO_2毛细捕获特征.同时,采用计算流体动力学方法对实验过程进行三维数值模拟.数值模拟不仅反映了实验过程中两相流驱替锋面的推进过程,还刻画了孔隙尺度下被捕获的CO_2液滴/团簇三维空间形态特征.最后,基于数值模拟给出了CO_2初始饱和度与残余饱和度曲线,即毛细捕获曲线,并对比分析了3种毛细捕获曲线预测模型(即Jurauld模型、Land模型和Spiteri模型)的优劣.分析表明,Jurauld模型的描述能力稍优于Land模型,Spiteri模型的描述能力较弱.由于Land模型只需单个参数,且参数具有明确的物理意义,因此在实际工程中,建议优先采用Land模型.     

Numerical Simulation of a CT-Scanned Counter-Current Flow Experiment

Counter-current flow occurs in many reservoir processes and it is important to understand and model these processes in order to operate them effectively. Both drainage and imbibition processes exist simultaneously during counter-current flow. It has thus proven difficult to model this type of flow using conventional techniques because of the impossibility of assigning a single capillary pressure curve applicable over the entire sample. In the current paper, a new saturation-history-dependent approach has been developed to simulate a counter-current flow experiment done with an X-ray CT scanner. Hysteresis in both capillary pressure and relative permeabilities is considered during simulation. Capillary hysteresis loop and relative permeabilities are extracted through history matching and a family of scanning curves is constructed connecting the two branches of the capillary hysteresis loop. Each gridblock of the sample is assigned a different scanning curve according to the local saturation history. History-dependent modeling of the experiment reproduced two-dimensional saturation distributions over time with good accuracy, which cannot be obtained with traditional simulation using only one capillary pressure curve.     

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 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.     

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P ^c–S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P ^c–S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P ^c–S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.     

Fluid Flow Across Capillary Heterogeneities,Experiments and Simulations

A comparative study of numerical modelling and laboratory experiments of two-phase immiscible displacements in a 33 cm × 10 × 3 cm thick cross-bedded reservoir model is reported. Dynamic two-dimensional fluid saturation development was obtained from experiments by use of a nuclear tracer imaging technique and compared to numerical predictions using a full-field black oil simulator.The laboratory cross-bedded reservoir model was a sandpack consisting of two strongly waterwet sands of different grain sizes, packed in sequential layers. The inlet and outlet sand consisted of low permeable, high capillary, sand while the central crosslayer with a dip angle of 30° was a high permeable, low capillary, sand. Results on moderate contrasts in permeability and capillary heterogeneities in the cross-bedded reservoir model at different mobility ratios and capillary number floods temporarily showed a bypass of oil, resulting in a prolonged two-phase production. The final remaining oil saturations, however, were as for isolated samples. Hence, permanently trapped oil was not observed.Simulations of waterfloods, using a commercial software package, displayed correct water breakthrough at low flow rate and unity viscosity ratio, but failed in predicting local saturation development in detail, probably due to numerical diffusion.The simulator was used to test several cases of heterogeneity contrasts, and influence from different relative permeability curves. Further, by altering the capillary pressure at the outlet, the end effects were proven important.     

Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs

In the absence of capillarity the single-component two-phase porous medium equations have the structure of a nonlinear parabolic pressure (equivalently, temperature) diffusion equation, with derivative coupling to a nonlinear hyperbolic saturation wave equation. The mixed parabolic-hyperbolic system is capable of substaining saturation shock waves. The Rankine-Hugoniot equations show that the volume flux is continuous across such a shock. In this paper we focus on the horizontal one-dimensional flow of water and steam through a block of porous material within a geothermal reservoir. Starting from a state of steady flow we study the reaction of the system to simple changes in boundary conditions. Exact results are obtainable only numerically, but in some cases analytic approximations can be derived. When pressure diffusion occurs much faster than saturation convection, the numerical results can be described satisfactorily in terms of either saturation expansion fans, or isolated saturation shocks. At early times, pressure and saturation profiles are functionally related. At intermediate times, boundary effects become apparent. At late times, saturation convection dominates and eventually a steady-state is established. When both pressure diffusion and saturation convection occur on the same timescale, initial simple shock profiles evolve into multiple shocks, for which no theory is currently available. Finally, a parameter-free system of equations is obtained which satisfactorily represents a particular case of the exact equations.