非牛顿流体两相渗流问题的摄动解

本文用奇异摄动法结合正则摄动法求解了考虑毛管力因素时多孔介质中弱非牛顿流体的两相驱替问题,得到了分流函数和湿相饱和度的渐近解析解。所得结果同数值解和经典的牛顿流体两相渗流结果进行了比较,并着重讨论了非牛顿因素的影响。     

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.     

双相介质P波波动方程小波域多尺度模拟

相比于单相介质理论而言,双相介质理论更接近实际地层的真实情况,因此在地球物理勘探、地震工程和岩土动力学等领域有着广泛的应用。传统的波动方程数值解法由于本身固有的不足不利于求解诸如双相介质波动方程等复杂的非线性和不规则性问题;而小波方法则由于自身良好的特性可以用来构建解决此类问题的自适应性算法。本文详细推导了双相介质P波波动方程的有限差分矩阵表示形式,利用小波变换将其转移到小波域,设置阈值形成更为稀疏的迭代矩阵以构建自适应算法,从而达到减少计算量,增加地震波场数值模拟灵活性和准确性的目的。地球物理勘探的数值模拟实例验证了方法的有效性。     

裂缝性低渗透油藏流-固耦合理论与数值模拟

根据裂缝性低渗油藏的储层特征,建立适合裂缝性砂岩油藏渗流的等效连续介质模型。将渗流力学与弹塑性力学相结合,建立裂缝性低渗透油藏的流-固耦合渗流数学模型,并给出其数值解.通过数值模拟对一实际井网开发过程中孔隙度、渗透率的变化以及开发指标进行计算,并和刚性模型以及双重介质模型的计算结果进行了分析比较.     

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.     

Dynamics of the Water–Oil Front for Two-Phase, Immiscible Flow in Heterogeneous Porous Media. 2 – Isotropic Media

We study the evolution of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. Our analysis takes into account the viscous coupling between the pressure field and the saturation map. Although most of previously published stochastic homogenization approaches for upscaling two-phase flow in heterogeneous porous media neglect this viscous coupling, we show that it plays a crucial role in the dynamics of the front. In particular, when the mobility ratio is favorable, it induces a transverse flux that stabilizes the water–oil front, which follows a stationary behavior, at least in a statistical sense. Calculations are based on a double perturbation expansion of equations at first order: the local velocity fluctuation is defined as the sum of a viscous term related to perturbations of the saturation map, on one hand, plus the perturbation induced by the heterogeneity of the permeability field with a base-state saturation map, on the other hand. In this companion paper, we focus on flows in isotropic media. Our results predict the dynamics of the water–oil front for favorable mobility ratios. We show that the statistics of the front reach a stationary limit, as a function of the geostatistics of the permeability field and of the mobility ratio evaluated across the front. Results of numerical experiments and Monte-Carlo analysis confirm our predictions.     

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.     

Dynamics of the Water–Oil Front for Two-Phase, Immiscible Flow in Heterogeneous Porous Media. 1 – Stratified Media

We study the evolution of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. Our analysis takes into account the viscous coupling between the pressure field and the saturation map. Although most of previously published stochastic homogenization approaches for upscaling two-phase flow in heterogeneous porous media neglect this viscous coupling, we show that it plays a crucial role on the dynamics of the front. In particular, when the mobility ratio is favorable, the viscous coupling induces a transverse flux that stabilizes the water–oil front, which follows a stationary behavior, at least in a statistical sense. Calculations are based on a double perturbation expansion of equations at first order: the local velocity fluctuation is defined as the sum of a viscous term related to perturbations of the saturation map, on one hand, plus the perturbation induced by the heterogeneity of the permeability field with a base-state saturation map, on the other hand. In this first paper, we focus on flows in stratified reservoirs, with stratification parallel to the mean flow. Our results allow to predict the evolution of large Fourier mode of the front, and the emergence of a stationary front, for favorable mobility ratios. Numerical experiments confirm our predictions. Our approach is applied to downscaling. Extension of our theory to isotropic media is presented in the companion paper.     

Investigation of Generalized Relative Permeability Coefficients for Electrically Assisted Oil Recovery in Oil Formations

Evaluation of relative permeability coefficients is one of the key steps in reliable simulation of two-phase flow in porous media. An extensive body of work exists on evaluation of these coefficients for two-phase flow under pressure gradient. Oil transport under an applied electrical gradient in porous media is also governed by the principles of two-phase flow, but is less understood. In this paper, relative permeability coefficients under applied electric field are evaluated for a specific case of two- phase fluid flow in water-wet porous media, where the second fluid phase is oil. It is postulated that the viscous drag on the oil phase, exerted by the electro-osmotic flow of the water phase, is responsible for the transport of oil in the absence of a pressure gradient. Reliable prediction of the flow patterns necessitates accurate representation and determination of the relative permeability coefficients under the electrical gradient. The contribution of each phase to the flow is represented mathematically, and the relative permeability coefficients are evaluated through electro-osmotic flow measurements conducted on oil bearing rock cores.     

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.     

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.     

Method of Negative Saturations for Modeling Two-phase Compositional Flow with Oversaturated Zones

We examine the two-phase flow through porous media of multicomponent partially miscible fluids. The composition of both the phases is variable in space and time and is assumed to be in local thermodynamic equilibrium. One of the basic problems in modeling such systems is related to the appearance of single-phase zones occupied by the fluid which is over- (or under-) saturated, i.e., it is significantly remote from the equilibrium two-phase region. In an oversaturated zone, the two-phase flow equations degenerate and can no longer be used, which provokes serious numerical problems. We propose to describe the two-phase and oversaturated single-phase zone by a uniform system of classic two-phase equations while extending the concept of the phase saturation so that it may be negative and higher than one. Physically this means that we consider the oversaturated single-phase states as the pseudo two-phase states which are characterized by a negative saturation of the imaginary phase. Such an extension of the concept of the phase saturation requires the development of some consistence conditions that ensure the equivalence between the pseudo two-phase equations and the true single-phase flow model in the oversaturated zones. This method allows using the existing numerical simulators of two-phase flow for modeling single-phase zones by adding a simple plug-in with no modification of the structure of the simulators. The method is illustrated by several examples of hydrogene-water flow in a waste radioactive storage and of CO₂ injection in an oil reservoir.     

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.     

The Local Analysis of Changing Force Balances in Immiscible Incompressible Two-Phase Flow

The balance of viscous, capillary and gravity forces strongly affects two-phase flow through porous media and can therefore influence the choice of appropriate methods for numerical simulation and upscaling. A strict separation of the effects of these various forces is not possible due to the nature of the nonlinear coupling between the various terms in the transport equations. However, approximate prediction of this force balance is often made by calculation of dimensionless quantities such as capillary and gravity numbers. We present an improved method for the numerical analysis of simulations which recognises the changing balance of forces – in both space and time – in a given domain. The classical two-phase transport equations for immiscible incompressible flow are expressed in two forms: (i) the convection–diffusion-gravity (CDG) formulation where convection and diffusion represent viscous and capillary effects, respectively, (ii) the oil pressure formulation where the viscous effects are attributed to the product of mobility difference and the oil pressure gradient. Each formulation provides a different perspective on the balance of forces although the two forms are equivalent. By discretising the different formulations, the effect of each force on the rate of change of water saturation can be calculated for each cell, and this can be analysed visually using a ternary force diagram. The methods have been applied to several simple models, and the results are presented here. When model parameters are varied to determine sensitivity of the estimators for the balance of forces, the CDG formulation agrees qualitatively with what is expected from physical intuition. However, the oil pressure formulation is dominated by the steady-state solution and cannot be used accurately. In addition to providing a physical method of visualising the relative magnitudes of the viscous, gravity and capillary forces, the local force balance may be used to guide our choice of upscaling method.     

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.     

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.     

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.     

An Experimental Study of Three-Phase Trapping in Sand Packs

The trapped saturations of oil and gas are measured as functions of initial oil and gas saturation in water-wet sand packs. Analogue fluids—water, octane and air—are used at ambient conditions. Starting with a sand-pack column which has been saturated with brine, oil (octane) is injected with the column horizontal until irreducible water saturation is reached. The column is then positioned vertically and air is allowed to enter from the top of the column, while oil is allowed to drain under gravity for varying lengths of time. At this point, the column may be sliced and the fluids analyzed by gas chromatography to obtain the initial saturations. Alternatively, brine is injected through the bottom of the vertical column to trap oil and gas, before slicing the columns and measuring the trapped or residual saturations by gas chromatography and mass balance. The experiments show that in three-phase flow, the total trapped saturations of oil and gas are considerably higher than the trapped saturations reported in the literature for two-phase systems. It is found that the residual saturation of oil and gas combined could be as high as 23 %, as opposed to a maximum two-phase residual of only 14 %. For very high initial gas saturations, the residual gas saturation, up to 17 %, was also higher than for two-phase displacement. These observations are explained in terms of the competition between piston-like displacement and snap-off. It is also observed that less oil is always trapped in three-phase flow than in two-phase displacement, and the difference depends on the amount of gas present. For low and intermediate initial gas saturations, the trapped gas saturation rises linearly with initial saturation, followed by a constant residual, as seen in two-phase displacements. However, at very high initial gas saturations, the residual saturation rises again.     

Interfaces of Phase Transition and Disappearance and Method of Negative Saturation for Compositional Flow with Diffusion and Capillarity in Porous Media

The specific case of interfaces separating a single-phase fluid and a two-phase continuum appears in the theory of compositional flow through porous media. They are usually called the interfaces of phase transition (PT-interfaces) or the interfaces of phase disappearing (PD-interfaces). The principle of equivalence is proved which shows that a single-phase multi-component fluid may be replaced by an equivalent fictitious two-phase fluid having specific properties. The equivalent properties are such that the extended saturation of a fictitious phase is negative. This principle enables us to develop the uniform system of two-phase equations in the overall domain in terms of the extended saturation (the NegSat model), and to apply the direct numerical simulation. In the case with diffusion, the uniform NegSat model contains a new term proportional to the gradient of saturation in the relation for flow velocity. The canonical NegSat model represents a transport equation with discontinuous nonlinearities. The qualitative analysis of this model shows that the PT-interfaces represent the shocks of the extended saturation, or, in some cases, can transform into weak shocks. The diffusion and capillarity do not destroy necessarily the shocks, but change their velocity. The analytical technique is developed which allows capturing PT-shocks. The method is illustrated by several examples of miscible gas injection in oil reservoir. In two-dimensional case, the effects of multiple shock collisions in heterogeneous media are automatically modeled. In the case of immiscible fluids and a classic interface, the suggested method converges to the VOF-method.     

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。