Radial Oil–Water Transport Toward an Extraction Well

We apply steady-state capillary-controlled upscaling in heterogeneous environments. A phase may fail to form a connected path across a given domain at capillary equilibrium. Moreover, even if a continuous saturation path exists, some regions of the domain may produce disconnected clusters that do not contribute to the overall connectivity of the system. In such cases, conventional upscaling processes might not be accurate since identification and removal of these isolated clusters are extremely important to the global connectivity of the system and the stability of the numerical solvers. In this study, we address the impact of percolation during capillary-controlled displacements in heterogeneous porous media and present a comprehensive investigation using random absolute permeability fields, for water-wet, oil-wet and mixed-wet systems, where J-function scaling is used to relate capillary pressure, porosity and absolute permeabilities in each grid cell. Important information is revealed about the average connectivity of the phases and trapping at the Darcy scale due to capillary forces. We show that in oil-wet and mixed-wet media, large-scale trapping of oil controlled by variations in local capillary pressure may be more significant than the local trapping, controlled by pore-scale displacement.     

New Trapping Mechanism in Carbon Sequestration

The modes of geologic storage of CO₂ are usually categorized as structural, dissolution, residual, and mineral trapping. Here we argue that the heterogeneity intrinsic to sedimentary rocks gives rise to a fifth category of storage, which we call local capillary trapping. Local capillary trapping occurs during buoyancy-driven migration of bulk phase CO₂ within a saline aquifer. When the rising CO₂ plume encounters a region (10⁻² to 10⁺¹m) where capillary entry pressure is locally larger than average, CO₂ accumulates beneath the region. This form of storage differs from structural trapping in that much of the accumulated saturation will not escape, should the integrity of the seal overlying the aquifer be compromised. Local capillary trapping differs from residual trapping in that the accumulated saturation can be much larger than the residual saturation for the rock. We examine local capillary trapping in a series of numerical simulations. The essential feature is that the drainage curves (capillary pressure versus saturation for CO₂ displacing brine) are required to be consistent with permeabilities in a heterogeneous domain. In this work, we accomplish this with the Leverett J-function, so that each grid block has its own drainage curve, scaled from a reference curve to the permeability and porosity in that block. We find that capillary heterogeneity controls the path taken by rising CO₂. The displacement front is much more ramified than in a homogeneous domain, or in a heterogeneous domain with a single drainage curve. Consequently, residual trapping is overestimated in simulations that ignore capillary heterogeneity. In the cases studied here, the reduction in residual trapping is compensated by local capillary trapping, which yields larger saturations held in a smaller volume of pore space. Moreover, the amount of CO₂ phase remaining mobile after a leak develops in the caprock is smaller. Therefore, the extent of immobilization in a heterogeneous formation exceeds that reported in previous studies of buoyancy-driven plume movement.     

Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

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.     

A Multi-Scale Approach to Model Two-Phase Flow in Heterogeneous Porous Media

The immiscible displacement of a wetting fluid by a non-wetting one in heterogeneous porous media is modeled using a multi-scale network-type analysis: (1) The pressure-controlled immiscible displacement of water by oil in pore-and-throat networks (1st length scale ~ 1?mm) is simulated as a capillary-driven process. (2) The pressure-controlled immiscible displacement in uncorrelated cubic lattices (2nd length scale ~ 1?cm) is simulated as a site percolation process governed by capillary and gravity forces. At this scale, each node represents a network of the previous scale. (3) The rate-controlled immiscible displacement of water by oil in cubic networks (3rd length scale ~ 10?cm), where each node represents a lattice of the previous scale, is simulated by accounting for capillary, gravity, and viscous forces. The multi-scale approach along with the information concerning the pore structure properties of the porous medium can be employed to determine the transient responses of the pressure drop and axial distribution of water saturation, and estimate the effective (up-scaled) relative permeability functions. The method is demonstrated with application to data of highly heterogeneous soils.     

Simulation of capillary pressure curves using bond correlated site percolation on a simple cubic network

A stochastic approach to network modelling has been used to simulate quasi-static immiscible displacement in porous media. Both number-based and volume-based network saturation results were obtained. Number-based results include: number-based saturation curves for primary drainage, secondary imbibition and secondary drainage, fluid distribution data, and cluster trapping history. Using pore structure data of porous media, it is possible to convert the number-based curves to capillary pressure — saturation relationships. Pore size distribution functions and pore shapes which are thought to closely represent Berea sandstone samples were used to predict the capillary curves. The physical basis of these calculations is a one-to-one correspondence between the cumulative node and bond index fractions in the network analysis, and the cumulative number-based distributions of pore body and pore throat diameters, respectively. The oil-water capillary pressure curve simulated for primary drainage closely resembles those measured experimentally. The agreement between the simulated and the measured secondary imbition and secondary drainage curves is less satisfactory.     

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.     

Microscale Visual Study of End Effects at Permeability Discontinuities

The physical effect of multiphase fluid distribution and flow at permeability boundaries has not been fully investigated, particularly at the pore scale (1–100 μm), although such behaviour can significantly affect the overall scaled-up reservoir trapping capacity and production performance. In this article, microscale physical models have been used to qualitatively study the pore scale flow events at permeability boundaries, both high to low and vice versa, to gain a better understanding of the role of these boundaries and water saturation on multiphase displacement behaviour at the pore scale. We have used etched glass models of stripes of large and small (a factor of two) pores with circular matrix. Capillary pressure, which is the controlling parameter is itself dependant on pore size and its spatial distribution, the magnitude of the interfacial tensions and the wettability between the fluids and the solid surface of the models. Sometimes, the only way the non-wetting fluid can penetrate the boundary is through a fortuitous leakage, whereby the presence of an initial saturation reduces the controlling capillary pressure. Examples are demonstrated including mechanisms of end-effects and how capillary boundary resistance (due to capillary forces) can be broken down and fluid movement across the boundary can develop. These micromodel experiments show vividly that connate water can assist in these processes, particularly oil trapping and leakage of water across a permeability boundary.     

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.     

Effective Relative Permeabilities and Capillary Pressure for One-Dimensional Heterogeneous Media

The paper presents an analytical construction of effective two-phase parameters for one-dimensional heterogeneous porous media, and studies their properties. We base the computation of effective parameters on analytical solutions for steady-state saturation distributions. Special care has to be taken with respect to saturation and pressure discontinuities at the interface between different rocks. The ensuing effective relative permeabilities and effective capillary pressure will be functions of rate, flow direction, fluid viscosities, and spatial scale of the heterogeneities.The applicability of the effective parameters in dynamic displacement situations is studied by comparing fine-gridded simulations in heterogeneous media with simulations in their homogeneous (effective) counterparts. Performance is quite satisfactory, even with strong fronts present. Also, we report computations studying the applicability of capillary limit parameters outside the strict limit.     

Percolation as a Basic Concept for Macroscopic Capillarity

The concepts of relative permeability and capillary pressure are crucial for the accepted traditional theory of two phase flow in porous media. Recently, a theoretical approach was introduced that does not require these concepts as input (Hilfer, Physica A, 359:119–128, 2006a; Phys. Rev. E, 73:016307, 2006b). Instead it was based on the concept of hydraulic percolation of fluid phases. This paper presents the first numerical solutions of the coupled nonlinear partial differential equations introduced in Hilfer (Phys. Rev. E, 73:016307, 2006b). Approximate numerical results for saturation profiles in one spatial dimension have been calculated. Long time limits of dynamic time-dependent profiles are compared to stationary solutions of the traditional theory. The initial and boundary conditions are chosen to model the displacement processes that occur when a closed porous column containing two immiscible fluids of different density is raised from a horizontal to a vertical position in a gravitational field. The nature of the displacement process may change locally in space and time between drainage and imbibition. The theory gives local saturations for nonpercolating trapped fluids near the endpoint of the displacement.     

Pore-Scale Flow in Surfactant Flooding

Recovery of oil from the blocks of an initially oil-wet, naturally fractured, reservoir as a result of counter-current flow following introduction of aqueous wettability-altering surfactant into the fracture system is considered, as an example of a practical process in which phenomena acting at the single pore-scale are vital to the economic displacement of oil at the macroscopic scale. A Darcy model for the process is set up, and solutions computed illustrating the recovery rate controlling role of the bulk diffusion of surfactant. A central ingredient of this model is the capillary pressure relation, linking the local values of the pressure difference between the oleic and aqueous phases, the aqueous saturation and the surfactant concentration. Using ideas from single capillary models of oil displacement from oil-wet tubes by wettability-altering surfactant, we speculate that the use of a capillary pressure function, with dependences as assumed, may not adequately represent the Darcy scale consequences of processes acting at the single pore-scale. Multi-scale simulation, resolving both sub-pore and multi-pore flow processes may be necessary to resolve this point. Some general comments are made concerning the issues faced when modelling complex displacement processes in porous media starting from the pore-scale and working upwards.     

Influence of Viscous Fingering on Dynamic Saturation–Pressure Curves in Porous Media

We report on results from primary drainage experiments on quasi-two-dimensional porous models. The models are transparent, allowing the displacement process and structure to be monitored in space and time during primary drainage experiments carried out at various speeds. By combining detailed information on the displacement structure with global measurements of pressure, saturation and the capillary number Ca, we obtain a scaling relation relating pressure, saturation, system size and capillary number. This scaling relation allows pressure–saturation curves for a wide range of capillary numbers to be collapsed on the same master curve. We also show that in the case of primary drainage, the dynamic effect in the capillary pressure–saturation relationship observed on partially water saturated soil samples might be explained by the combined effect of capillary pressure along the invasion front of the gaseous phase, and pressure changes caused by viscous effects in the wetting fluid phase.     

Pore Scale Modeling of Rate Effects in Imbibition

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

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.     

Enhancing Immiscible Fluid Displacement in Porous Media by Capillary Pressure Discontinuities

Fluid displacement in porous media plays an important role in many industrial applications, including biological filtration, carbon capture and storage, enhanced oil recovery, and fluid transport in fuel cells. The displacement front is unstable, which evolves from smooth into ramified patterns, when the mobility (ratio of permeability to viscosity) of the displacing fluid is larger than that of the displaced one; this phenomenon is called viscous fingering. Viscous fingering increases the residual saturation of the displaced fluid, considerably impairing the efficacy of fluid displacement. It is of practical importance to develop suitable methods to improve fluid displacement. This paper presents an experimental study on applying the discontinuity of capillary pressure to improve immiscible fluid displacement in drainage for which the displacing fluid (air) wets the porous media less preferentially than does the displaced fluid (silicone oil). The concept involves using a heterogeneous packing system, where the upstream region features large pores and small capillary pressure, and the downstream region features small pores and large capillary pressure. The increase in capillary pressure prevents fingering from directly crossing the media interface, thus enhancing the displacement. The experimental apparatus was a linear cell comprising porous media between two parallel plates, and glass beads of 0.6 and 0.125 mm diameter were packed to compose the heterogeneous porous media. The time history of the finger flow was recorded using a video camera. Pressure drops over the model from the inlet to the outlet were measured to compare viscous pressure drops with capillary pressures. The results show that the fluid displacement was increased by the capillary discontinuities. The optimal displacement was determined through linear regression by adjusting the relative length of the large- and small-pore region. The results may assist in the understanding of fingering flow across the boundaries of different grain-sized bands for the gas and oil reservoir management, such as setting the relative location of the injection and production wells. The findings may also serve as a reference for industrial applications such as placing the grain bands in an adequate series to improve the displacement efficacy in biological filtration.     

The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media

We consider the one-dimensional two-phase flow including capillary effects through a heterogeneous porous medium. The heterogeneity is due to the spatial variation of the absolute permeability and the porosity. Both these quantities are assumed to be piecewise constant. At interfaces where the rock properties are discontinuous, we derive, by a regularisation technique, conditions to match the values of the saturation on both sides. There are two conditions: a flux condition and an extended pressure condition. Applying these conditions we show that trapping of the wetting phase may occur near heterogeneities. To illustrate the behaviour of the saturation we consider a time-dependent diffusion problem without convection, a stationary convection-diffusion problem, and the full time-dependent convection-diffusion problem (numerically). In particular the last two problems explicitly show the trapping behaviour.     

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.