Three-Phase Flow Modelling Using Pore-Scale Capillary Pressures and Relative Permeabilities for Mixed-Wet Media at the Continuum-Scale

When regions of three-phase flow arise in an oil reservoir, each of the flow parameters, i.e. capillary pressures and relative permeabilities, are generally functions of two phase saturations and depend on the wettability state. The idea of this work is to generate consistent pore-scale based three-phase capillary pressures and relative permeabilities. These are then used as input to a 1-D continuum core- or reservoir-scale simulator. The pore-scale model comprises a bundle of cylindrical capillary tubes, which has a distribution of radii and a prescribed wettability state. Contrary to a full pore-network model, the bundle model allows us to obtain the flow functions for the saturations produced at the continuum-scale iteratively. Hence, the complex dependencies of relative permeability and capillary pressure on saturation are directly taken care of. Simulations of gas injection are performed for different initial water and oil saturations, with and without capillary pressures, to demonstrate how the wettability state, incorporated in the pore-scale based flow functions, affects the continuum-scale displacement patterns and saturation profiles. In general, wettability has a major impact on the displacements, even when capillary pressure is suppressed. Moreover, displacement paths produced at the pore-scale and at the continuum-scale models are similar, but they never completely coincide.     

Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.     

An Analysis of Three-Phase Pore Occupancies and Relative Permeabilities in Porous Media with Variable Wettability

In three-phase flow, the macroscopic constitutive relations of capillary pressure and relative permeability as functions of saturation depend in a complex manner on the underlying pore occupancies. These three-phase pore occupancies depend in turn on the interfacial tensions, the pore sizes and the degree of wettability of the pores, as characterised by the cosines of the oil–water contact angles. In this work, a quasi-probabilistic approach is developed to determine three-phase pore occupancies in media where the degree of wettability varies from pore to pore. Given a set of fluid and rock properties, a simple but novel graphical representation is given of the sizes and oil–water contact angles underlying three-phase occupancies for every allowed combination of capillary pressures. The actual phase occupancies are then computed using the contact angle probability density function. Since a completely accessible porous medium is studied, saturations, capillary pressures, and relative permeabilities are uniquely related to the pore occupancies. In empirical models of three-phase relative permeability it is of central importance whether a phase relative permeability depends only on its own saturation and how this relates to the corresponding two-phase relative permeability (if at all). The new graphical representation of pore sizes and wettabilities clearly distinguishes all three-phase pore occupancies with respect to these saturation-dependencies. Different types of saturation-dependencies may occur, which are shown to appear in ternary saturation diagrams of iso-relative permeability curves as well, thus guiding empirical approaches. However, for many saturation combinations three-phase and two-phase relative permeabilities can not be linked. In view of the latter, the present model has been used to demonstrate an approach for three-phase flow modelling on the basis of the underlying pore-scale processes, in which three-phase relative permeabilities are computed only along the actual flow paths. This process-based approach is used to predict an efficient strategy for oil recovery by simultaneous water-alternating-gas (SWAG) injection.     

Volume Conservation of the Intermediate Phase in Three-Phase Pore-Network Models

Quasi-static rule-based network models used to calculate capillary dominated multi-phase transport properties in porous media employ equilibrium fluid saturation distributions which assume that pores are fully filled with a single bulk fluid with other fluids present only as wetting and/or spreading films. We show that for drainage dominated three-phase displacements in which a non-wetting fluid (gas) displaces a trapped intermediate fluid (residual oil) in the presence of a mobile wetting fluid (water) this assumption distorts the dynamics of three-phase displacements and results in significant volume errors for the intermediate fluid and erroneous calculations of intermediate fluid residual saturations, relative permeabilities and recoveries. The volume errors are associated with the double drainage mechanism which is responsible for the mobilization of waterflood residual oil. A simple modification of the double drainage mechanism is proposed which allows the presence of a relatively small number of partially filled pores and removes the oil volume errors.     

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.     

Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.     

Vertical integration of three-phase flow equations for analysis of light hydrocarbon plume movement

A mathematical model is derived for areal flow of water and light hydrocarbon in the presence of gas at atmospheric pressure. Vertical integration of the governing three-dimensional, three-phase flow equations is performed under the assumption of local vertical equilibrium to reduce the dimensionality of the problem to two orthogonal horizontal directions. Independent variables in the coupled water and hydrocarbon areal flow equations are specified as the elevation of zero gauge hydrocarbon pressure (air-oil table) and the elevation of zero gauge water pressure (air-water table). Constitutive relations required in the areal flow model are vertically integrated fluid saturations and vertically integrated fluid conductivities as functions of air-oil and air-water table elevations. Closed-form expressions for the vertically integrated constitutive relations are derived based on a three-phase extension of the Brooks-Corey saturation-capillary pressure function. Closed-form Brooks-Corey relations are compared with numerically computed analogs based on the Van Genuchten retention function. Close agreement between the two constitutive models is observed except at low oil volumes when the Brooks-Corey model predicts lower oil volumes and transmissivities owing to the assumption of a distinct fluid entry pressure. Nonlinearity in the vertically integrated constitutive relations is much less severe than in the unintegrated relations. Reduction in dimensionality combined with diminished nonlinearity, makes the vertically integrated water and hydrocarbon model an efficient formulation for analyzing field-scale problems involving hydrocarbon spreading or recovery under conditions for which the vertical equilibrium assumption is expected to be a satisfactory approximation.     

Relative Permeability Calculations from Two-Phase Flow Simulations Directly on Digital Images of Porous Rocks

We present results from a systematic study of relative permeability functions derived from two-phase lattice Boltzmann (LB) simulations on X-ray microtomography pore space images of Bentheimer and Berea sandstone. The simulations mimic both unsteady- and steady-state experiments for measuring relative permeability. For steady-state flow, we reproduce drainage and imbibition relative permeability curves that are in good agreement with available experimental steady-state data. Relative permeabilities from unsteady-state displacements are derived by explicit calculations using the Johnson, Bossler and Naumann method with input from simulated production and pressure profiles. We find that the nonwetting phase relative permeability for drainage is over-predicted compared to the steady-state data. This is due to transient dynamic effects causing viscous instabilities. Thus, the calculated unsteady-state relative permeabilities for the drainage is fundamentally different from the steady-state situation where transient effects have vanished. These effects have a larger impact on the invading nonwetting fluid than the defending wetting fluid. Unsteady-state imbibition relative permeabilities are comparable to the steady-state ones. However, the appearance of a piston-like front disguises most of the displacement and data can only be determined for a restricted range of saturations. Relative permeabilities derived from unsteady-state displacements exhibit clear rate effects, and residual saturations depend strongly on the capillary number. We conclude that the LB method can provide a versatile tool to compute multiphase flow properties from pore space images and to explore the effects of imposed flow and fluid conditions on these properties. Also, dynamic effects are properly captured by the method, giving the opportunity to examine differences between steady and unsteady-state setups.     

Pore-scale Network Modeling of Three-Phase Flow Based on Thermodynamically Consistent Threshold Capillary Pressures. II. Results

We use the model described in Zolfaghari and Piri (Transp Porous Media, 2016) to predict two- and three-phase relative permeabilities and residual saturations for different saturation histories. The results are rigorously validated against their experimentally measured counterparts available in the literature. We show the relevance of thermodynamically consistent threshold capillary pressures and presence of oil cusps for significantly improving the predictive capabilities of the model at low oil saturations. We study systems with wetting and spreading oil layers and cusps. Three independent experimental data sets representing different rock samples and fluid systems are investigated in this work. Different disordered networks are used to represent the pore spaces in which different sets of experiments were performed, i.e., Berea, Bentheimer, and reservoir sandstones. All three-phase equilibrium interfacial tensions used for the simulation of three-phase experimental data are measured and used in the model’s validation. We use a fixed set of parameters, i.e., the input network (to represent the pore space) and contact angles (to represent the wettability state), for all experiments belonging to a data set. Incorporation of the MSP method for capillary pressure calculations and cusp analysis significantly improves the agreement between the model’s predictions of relative permeabilities and residual oil saturations with experimental data.     

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.     

Microtomography of Imbibition Phenomena and Trapping Mechanism

Water imbibition during the waterflooding process of oil production only sweeps part of the oil present. After water disrupts the oil continuity, most oil blobs are trapped in porous rock by capillary forces. Developing an efficient waterflooding scheme is a difficult task; therefore, an understanding of the oil trapping mechanism in porous rock is necessary from a microscopic viewpoint. The development of microfocused X-ray CT scanner technology enables the three-dimensional visualization of multiphase phenomena in a pore-scale. We scanned packed glass beads filled with a nonwetting phase (NWP) and injected wetting phase (WP) in upward and downward injections to determine the microscopic mechanism of immiscible displacement in porous media and the effects of buoyancy forces. We observed the imbibition phenomena for small capillary numbers to understand the spontaneous imbibition mechanism in oil recovery. This study is one of the first attempts to use a microfocused X-ray CT scanner for observing the imbibition and trapping mechanisms. The trapping mechanism in spontaneous imbibition is determined by the pore configuration causing imbibition speed differences in each channel; these differences can disrupt the oil continuity. Gravity plays an important role in spontaneous imbibition. In upward injection, the WP flows evenly and oil is trapped in single or small clusters of pores. In downward injection, the fingering phenomena determine the amount of trapped oil, which is usually in a network scale. Water breakthrough causes dramatic decrease in the oil extraction rate, resulting in lower oil production efficiency.     

Pore-Scale Network Modeling of Three-Phase Flow Based on Thermodynamically Consistent Threshold Capillary Pressures. I. Cusp Formation and Collapse

We present a pore-scale network model of two- and three-phase flow in disordered porous media. The model reads three-dimensional pore networks representing the pore space in different porous materials. It simulates wide range of two- and three-phase pore-scale displacements in porous media with mixed-wet wettability. The networks are composed of pores and throats with circular and angular cross sections. The model allows the presence of multiple phases in each angular pore. It uses Helmholtz free energy balance and Mayer–Stowe–Princen (MSP) method to compute threshold capillary pressures for two- and three-phase displacements (fluid configuration changes) based on pore wettability, pore geometry, interfacial tension, and initial pore fluid occupancy. In particular, it generates thermodynamically consistent threshold capillary pressures for wetting and spreading fluid layers resulting from different displacement events. Threshold capillary pressure equations are presented for various possible fluid configuration changes. By solving the equations for the most favorable displacements, we show how threshold capillary pressures and final fluid configurations may vary with wettability, shape factor, and the maximum capillary pressure reached during preceding displacement processes. A new cusp pore fluid configuration is introduced to handle the connectivity of the intermediate wetting phase at low saturations and to improve model’s predictive capabilities. Based on energy balance and geometric equations, we show that, for instance, a gas-to-oil piston-like displacement in an angular pore can result in a pore fluid configuration with no oil, with oil layers, or with oil cusps. Oil layers can then collapse to form cusps. Cusps can shrink and disappear leaving no oil behind. Different displacement mechanisms for layer and cusp formation and collapse based on the MSP analysis are implemented in the model. We introduce four different layer collapse rules. A selected collapse rule may generate different corner configuration depending on fluid occupancies of the neighboring elements and capillary pressures. A new methodology based on the MSP method is introduced to handle newly created gas/water interfaces that eliminates inconsistencies in relation between capillary pressures and pore fluid occupancies. Minimization of Helmholtz free energy for each relevant displacement enables the model to accurately determine the most favorable displacement, and hence, improve its predictive capabilities for relative permeabilities, capillary pressures, and residual saturations. The results indicate that absence of oil cusps and the previously used geometric criterion for the collapse of oil layers could yield lower residual oil saturations than the experimentally measured values in two- and three-phase systems.     

Pore-scale Simulation of Water Alternate Gas Injection

We use a three-dimensional mixed-wet random network model representing Berea sandstone to compute displacement paths and relative permeabilities for water alternating gas (WAG) flooding. First we reproduce cycles of water and gas injection observed in previously published experimental studies. We predict the measured oil, water and gas relative permeabilities accurately. We discuss the hysteresis trends in the water and gas relative permeabilities and compare the behavior of water-wet and oil-wet media. We interpret the results in terms of pore-scale displacements. In water-wet media the water relative permeability is lower during water injection in the presence of gas due to an increase in oil/water capillary pressure that causes a decrease in wetting layer conductance. The gas relative permeability is higher for displacement cycles after first gas injection at high gas saturation due to cooperative pore filling, but lower at low saturation due to trapping. In oil-wet media, the water relative permeability remains low until water-filled elements span the system at which point the relative permeability increases rapidly. The gas relative permeability is lower in the presence of water than oil because it is no longer the most non-wetting phase.     

Approximate Semi-analytical Solution for Injection–Falloff–Production Well Test: An Analytical Tool for the In Situ Estimation of Relative Permeability Curves

An injection–falloff–production test (IFPT) was originally proposed in Chen et al. (in: SPE conference paper, 2006. doi: 10.2118/103271-MS, SPE Reserv Eval Eng 11(1):95–107, 2008) as a well test for the in situ estimation of two-phase relative permeability curves to be used for simulating multiphase flows in porous media. Hence, we develop an approximate semi-analytical solution for the two-phase saturation distribution in an oil–water system during the flowback period of an IFPT according to the mathematical theory of waves. In fact, we show that the weak solution we construct for the saturation equation for the flowback period satisfies the Oleinik entropy condition and hence is unique. In addition, we allow the governing relative permeabilities during the flowback period to be different from the relative permeabilities during injection. Using the saturation solution with the steady-state pressure theory of Thompson and Reynolds, we obtain a solution for the wellbore pressure during the flowback period. By comparing results from our solution with those from a commercial numerical simulator, we show that our approximate semi-analytical solution yields accurate saturation profiles and bottom hole pressures history. The use of very small time steps and a highly refined radial grid is necessary to generate a good solution from a reservoir simulator. The approximate analytical pressure solution developed is used as a forward model to match pressure and water flow rate data from an IFPT in order to estimate reservoir rock absolute permeability and skin factor in conjunction with in situ imbibition and drainage water–oil relative permeabilities.     

The inverse problem of determining the filtration function and permeability reduction in flow of water with particles in porous media

Deep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion of reservoir production zones, fines migration in oil fields, industrial filtering, bacteria, viruses or contaminants transport in groundwater etc. The basic features of the process are particle capture by the porous medium and consequent permeability reduction. Models for deep bed filtration contain two quantities that represent rock and fluid properties: the filtration function, which is the fraction of particles captured per unit particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities. These quantities cannot be measured directly in the laboratory or in the field; therefore, they must be calculated indirectly by solving inverse problems. The practical petroleum and environmental engineering purpose is to predict injectivity loss and particle penetration depth around wells. Reliable prediction requires precise knowledge of these two coefficients. In this work we determine these quantities from pressure drop and effluent concentration histories measured in one-dimensional laboratory experiments. The recovery method consists of optimizing deviation functionals in appropriate subdomains; if necessary, a Tikhonov regularization term is added to the functional. The filtration function is recovered by optimizing a non-linear functional with box constraints; this functional involves the effluent concentration history. The permeability reduction is recovered likewise, taking into account the filtration function already found, and the functional involves the pressure drop history. In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.     

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.     

A Novel Relative Permeability Model Based on Mixture Theory Approach Accounting for Solid–Fluid and Fluid–Fluid Interactions

A novel model is presented for estimating steady-state co- and counter-current relative permeabilities analytically derived from macroscopic momentum equations originating from mixture theory accounting for fluid–fluid (momentum transfer) and solid–fluid interactions (friction). The full model is developed in two stages: first as a general model based on a two-fluid Stokes formulation and second with further specification of solid–fluid and fluid–fluid interaction terms referred to as \(R_{{i}}\) (i =  water, oil) and R, respectively, for developing analytical expressions for generalized relative permeability functions. The analytical expressions give a direct link between experimental observable quantities (end point and shape of the relative permeability curves) versus water saturation and model input variables (fluid viscosities, solid–fluid/fluid–fluid interactions strength and water and oil saturation exponents). The general two-phase model is obeying Onsager’s reciprocal law stating that the cross-mobility terms \(\lambda _\mathrm{wo}\) and \(\lambda _\mathrm{ow}\) are equal (requires the fluid–fluid interaction term R to be symmetrical with respect to momentum transfer). The fully developed model is further tested by comparing its predictions with experimental data for co- and counter-current relative permeabilities. Experimental data indicate that counter-current relative permeabilities are significantly lower than corresponding co-current curves which is captured well by the proposed model. Fluid–fluid interaction will impact the shape of the relative permeabilities. In particular, the model shows that an inflection point can occur on the relative permeability curve when the fluid–fluid interaction coefficient \(I>0\) which is not captured by standard Corey formulation. Further, the model predicts that fluid–fluid interaction can affect the relative permeability end points. The model is also accounting for the observed experimental behavior that the water-to-oil relative permeability ratio \(\hat{{k}}_{\mathrm{rw}} /\hat{{\mathrm{k}}}_{\mathrm{ro}} \) is decreasing for increasing oil-to-water viscosity ratio. Hence, the fully developed model looks like a promising tool for analyzing, understanding and interpretation of relative permeability data in terms of the physical processes involved through the solid–fluid interaction terms \(R_{{i}}\) and the fluid–fluid interaction term R.     

Phase distribution and intrapore salt exchange during drilling mud invasion of an oil- and gas-bearing formation

As a result of drilling mud filtrate invasion of a formation saturated with oil, gas and natural water, the distribution of the immiscible phases and the electrophysical characteristics of the near-well zone change as compared with its initial state. Taking this change into account is necessary for successful interpretation of electrical well logging data. In this paper, on the basis of the equations of immiscible fluid flow through a porous medium and the system of transfer equations with account for instantaneous salt exchange between the filtrate and the natural water inside the pores, the regions of initial formation fluid saturations for which in the invasion zone the displacement fronts retain their relative position are determined.     

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。     

致密砂岩逆向渗吸作用距离实验研究

中国致密油储量丰富, 但多数致密储层波及效率低, 衰竭开发效果较差. 逆向渗吸是致密油藏注水开发过程中的一种重要的提高采收率途径, 目前许多学者主要针对致密油藏渗吸采收率及其影响因素开展研究, 而对于渗吸作用距离(表征致密油藏渗吸作用范围)研究较少. 本文采用CT在线扫描装置建立了致密岩心逆向渗吸作用距离量化方法, 明确了逆向渗吸的作用范围, 进一步研究了流体压力、含水饱和度、岩心渗透率和表面活性剂对逆向渗吸作用距离的影响, 阐明了逆向渗吸作用距离与渗吸采收率的关系, 为提高致密油藏采收率提供指导. 研究结果表明, 渗透率为0.3 mD的致密岩心逆向渗吸作用距离尺度仅为1.25 ~ 1.625 cm; 5 MPa条件下渗透率为0.302 mD的岩心逆向渗吸作用距离为1.375 cm. 在本实验条件下, 流体压力和初始含水饱和度对致密岩心逆向渗吸作用距离的影响较小, 而渗透率和表面活性剂对致密岩心逆向渗吸作用距离的影响显著, 渗透率为0.784 mD的岩心逆向渗吸作用距离相较于渗透率为0.302 mD的岩心提高2.63倍. 逆向渗吸作用距离是渗吸采收率表征的重要参数, 决定了逆向渗吸作用的波及范围.