Relative permeability
Relative permeability is a dimensionless parameter in fluid mechanics and petroleum engineering that quantifies the conductance of a specific fluid phase through a porous medium when multiple immiscible fluid phases are present, defined as the ratio of the effective permeability of that phase to the absolute permeability of the medium.[1][2] Absolute permeability measures the medium's capacity to transmit a single saturating fluid, typically under single-phase conditions, while effective permeability accounts for the reduced flow due to interactions with other phases, such as oil, water, and gas in hydrocarbon reservoirs.[2] Values of relative permeability range from 0 (no flow) to 1 (single-phase flow), and their sum for all phases is typically less than or equal to 1 in multiphase systems.[2][3] This concept extends Darcy's law to multiphase flow scenarios, where the flow rate of each phase is proportional to its pressure gradient, viscosity, and relative permeability, allowing for the modeling of simultaneous fluid movement in porous rocks.[1][2] In reservoir engineering, relative permeability is crucial for simulating fluid displacement during production processes like waterflooding or gas injection, as it directly influences recovery factors and production rates by reflecting how saturation levels affect phase mobilities.[3] It varies with fluid saturation, wettability of the rock surface, pore geometry, and saturation history (e.g., drainage vs. imbibition), often visualized through relative permeability curves that plot these functions against saturation.[2] Relative permeability data are obtained experimentally through core flooding tests on rock samples under controlled conditions, providing empirical curves essential for reservoir simulation models.[3] These measurements reveal heterogeneities across reservoirs, influenced by factors like fluid chemistry and rock properties, which must be integrated with capillary pressure data for accurate predictions of hydrocarbon volumes and flow behavior over the reservoir's lifecycle.[3] In practice, relative permeability endpoints—such as the residual saturation where a phase's relative permeability reaches zero—define critical thresholds for fluid trapping and mobility, impacting enhanced oil recovery strategies.[2]Fundamentals
Definition
Relative permeability, denoted as k_{ri}, is defined as the ratio of the effective permeability of a fluid phase k_i to the absolute permeability k of the porous medium, mathematically expressed as k_{ri} = \frac{k_i}{k}, where $0 \leq k_{ri} \leq 1.[4] This dimensionless quantity quantifies the reduction in a phase's ability to flow through the medium due to the presence of other immiscible phases, serving as a key parameter in extending Darcy's law from single-phase to multiphase flow.[4] The concept of relative permeability was introduced by Morris Muskat in the late 1930s, building on earlier work in porous media flow to address multiphase systems.[5] Muskat's seminal book, The Flow of Homogeneous Fluids Through Porous Media (1937), formalized this extension, enabling the modeling of simultaneous flow of multiple fluids.[5] Relative permeability is primarily applied in porous media such as rocks, soils, and filters, where multiple immiscible fluids—like oil and water or gas and liquid—coexist and interact during flow processes.[4]Physical Interpretation
Relative permeability emerges at the pore scale from the competition between immiscible fluid phases for limited pore space within a porous medium, where the presence of one phase obstructs the flow paths of the other, thereby reducing the effective conductivity for each phase relative to single-phase flow. This competition is governed by interfacial tension, which creates menisci that trap portions of the non-preferred phase in smaller pores or dead-end spaces, limiting connectivity and overall flow efficiency. Phase occupancy further influences this process, as the distribution of fluids across pore throats and bodies determines the available pathways; for instance, the non-wetting phase tends to occupy larger pores during drainage, fragmenting the wetting phase's network and hindering its mobility.[6][7] Wettability, the preferential affinity of the solid surface for one fluid phase over another, profoundly affects phase distribution and the resulting flow hindrance in multiphase systems. In water-wet rocks, where the solid prefers water, the wetting phase spreads in thin films along pore walls, maintaining connectivity even at low saturations and allowing it to access a larger fraction of the pore space, which enhances its relative permeability compared to the non-wetting oil phase. Conversely, in oil-wet rocks, the solid's preference for oil leads to the wetting oil occupying central pore bodies and larger throats, displacing water to corners and films, which restricts water's flow paths and reduces its relative permeability while potentially improving oil mobility. This wettability-driven redistribution alters the effective pore network available to each phase, directly impacting the hindrance imposed on flow.[8][9] Hysteresis in relative permeability arises from the path-dependent nature of multiphase displacement processes, where the saturation history—specifically drainage (non-wetting phase invasion) versus imbibition (wetting phase invasion)—leads to different phase configurations and trapping at the same saturation. During drainage, the non-wetting phase advances through larger pores, leaving behind connected wetting-phase films, which results in higher relative permeability for the non-wetting phase compared to imbibition, where capillary forces trap disconnected non-wetting phase ganglia in smaller pores, reducing its mobility and altering the available flow paths for the wetting phase. This trapping mechanism causes the relative permeability curves to diverge between drainage and imbibition paths, reflecting irreversible changes in phase occupancy and connectivity due to the sequence of saturation alterations.[10][11]Mathematical Framework
Formulation
The formulation of relative permeability arises from extending Darcy's law, originally developed for single-phase flow, to describe multiphase fluid transport in porous media.[12] In multiphase flow, the volumetric flux \mathbf{q}_i of phase i is given by the extended Darcy's law: \mathbf{q}_i = -\frac{k_{ri} k}{\mu_i} \nabla P_i, where k is the absolute permeability of the porous medium, k_{ri} is the relative permeability of phase i (a dimensionless scalar between 0 and 1), \mu_i is the dynamic viscosity of phase i, and \nabla P_i is the pressure gradient in phase i. This equation accounts for the reduced conductance of each phase due to the presence of other immiscible phases occupying the pore space.[12] For a two-phase system, such as oil and water in a reservoir, separate flux equations apply to each phase: \mathbf{q}_o = -\frac{k_{ro} k}{\mu_o} \nabla P_o, \quad \mathbf{q}_w = -\frac{k_{rw} k}{\mu_w} \nabla P_w, where subscripts o and w denote oil and water, respectively, and the relative permeabilities k_{ro} and k_{rw} depend on the saturation S_j of the respective phases (with S_o + S_w = 1). These functions capture how the effective pathway for each phase diminishes as the other phase's saturation increases. The concept extends to general multiphase flow involving n immiscible phases, where the relative permeability of phase i is expressed as k_{ri} = f(S_1, S_2, \dots, S_n), subject to the constraint \sum_{j=1}^n S_j = 1. This functional dependence reflects the complex interactions among phases in sharing the porous medium's conductance.Key Assumptions
The formulation of relative permeability extends Darcy's law to multiphase flow in porous media under several core assumptions that enable the independent application of the law to each phase while simplifying pore-scale complexities. A primary assumption is steady-state flow, where fluid saturations and velocities remain constant over time and uniform across the medium, allowing equilibrium conditions for measurement and modeling. The fluids are treated as immiscible, with no significant mass transfer between phases, and the flow is assumed horizontal, neglecting gravitational segregation effects that could otherwise alter saturation distributions. The porous medium is idealized as homogeneous and isotropic, implying uniform pore structure and properties without directional variations in permeability.[13] Each phase is presumed continuous and interconnected within the porous matrix above its irreducible saturation, below which flow ceases due to entrapment. Standard models further neglect interphase momentum transfer, known as viscous coupling, and adsorption of fluid components onto the solid matrix, assuming negligible interaction beyond saturation-dependent hindrance.[4] These assumptions limit applicability in certain scenarios; for instance, in heavy oil reservoirs with high viscosity contrasts, unmodeled viscous coupling can result in apparent relative permeabilities exceeding 1, as observed in depletion tests where momentum transfer enhances phase mobility beyond independent flow predictions.[14] Similarly, capillary-induced hysteresis—arising from path-dependent saturation changes during drainage and imbibition—is often not fully captured, leading to discrepancies in dynamic simulations where wetting history affects permeability curves.[11]Normalization and Parameters
Endpoints
In relative permeability analysis for multiphase flow in porous media, particularly oil-water systems, the endpoints define the boundary conditions of the relative permeability curves at extreme phase saturations, serving as essential parameters for model calibration and simulation. The relative permeability to oil at irreducible water saturation, denoted as K_{rot}, represents the value of the oil relative permeability (k_{ro}) when the water saturation (S_w) equals the irreducible water saturation (S_{wir}). At this point, water is immobile and occupies the smallest pores without contributing to flow, allowing oil to achieve near-maximum mobility relative to single-phase conditions. Similarly, the relative permeability to water at residual oil saturation, K_{rwr}, is the value of the water relative permeability (k_{rw}) when the oil saturation (S_o) reaches the residual oil saturation (S_{orw}), where oil ganglia become trapped and cease to flow, enabling water to dominate the pore space.[15][16] These critical saturations mark the thresholds of phase mobility: S_{wir} is the minimum water saturation at which water flow effectively stops due to capillary forces binding it in place, while S_{orw} is the minimum oil saturation remaining after imbibition or displacement processes, reflecting trapping mechanisms like snap-off and bypassing. In water-wet systems, common in many sandstone reservoirs, S_{wir} typically ranges from 0.15 to 0.30, and S_{orw} from 0.20 to 0.40, depending on rock wettability, pore geometry, and fluid properties; these values directly impact initial fluid distribution and ultimate recovery potential.[15][17] Endpoints are generally normalized relative to the absolute permeability (k) of the rock or the effective single-phase permeability under reservoir conditions, ensuring dimensionless consistency in multiphase models. For instance, K_{rot} is often approximately 0.8 to 1.0 in water-wet oil-water systems, indicating that irreducible water reduces oil mobility by only a modest amount, while K_{rwr} typically falls in the range of 0.2 to 0.4, as residual oil continues to impede water flow significantly even at high water saturations. These ranges arise from empirical correlations, such as those linking K_{rot} to S_{wir} via K_{rot} \approx 1.31 - 2.62 S_{wir} + 1.1 S_{wir}^2 for S_{wir} between 0.2 and 0.5, highlighting the influence of initial water saturation on endpoint values.[15][18]Saturation Scaling
Saturation scaling in relative permeability involves normalizing the actual fluid saturations to a standard range, typically between 0 and 1, to facilitate consistent modeling and comparison across diverse rock types. For the water phase in a two-phase oil-water system, the normalized water saturation S_{wn} is defined as S_{wn} = \frac{S_w - S_{wir}}{1 - S_{wir} - S_{orw}}, where S_w is the actual water saturation, S_{wir} is the irreducible water saturation, and S_{orw} is the residual oil saturation to waterflood.[19] This transformation maps the mobile saturation range—bounded by the irreducible and residual endpoints—onto a unit interval, enabling the relative permeability curves k_{rw} and k_{row} to be expressed as functions of S_{wn} rather than the absolute S_w.[19] The primary purpose of this normalization is to scale relative permeability curves to a universal form, mitigating the influence of rock-specific residual saturations that vary due to differences in pore structure, wettability, and fluid properties across formations. By doing so, it allows for the direct comparison, averaging, and upscaling of datasets from laboratory experiments or core samples to field-scale simulations, improving the accuracy of reservoir performance predictions. For instance, normalized curves can be grouped by wettability states (e.g., water-wet or oil-wet) and rock types, facilitating uncertainty quantification and the generation of representative inputs for multiphase flow models. In multi-phase systems, such as three-phase oil-water-gas flows, similar normalization principles are extended to account for all mobile phases. The normalized saturation for each phase is adjusted relative to the total mobile saturation window, incorporating gas saturation S_g alongside water and oil, to preserve the functional dependence of relative permeabilities on phase interactions.[20] This approach ensures that saturation paths and hysteresis effects are captured consistently, even as residual saturations for multiple phases (e.g., S_{org} for residual oil to gas) influence the scaling.[20]Classical Models
Corey Model
The Corey model, developed by A. T. Corey in 1954, represents a foundational parametric approach to describing two-phase relative permeability curves through simple power-law expressions. It defines the relative permeability for the non-wetting phase, such as oil, ask_{ro} = K_{rot} (1 - S_{wn})^{N_o}
and for the wetting phase, such as water, as
k_{rw} = K_{rwr} S_{wn}^{N_w},
where K_{rot} and K_{rwr} are the endpoint relative permeabilities at residual saturations, N_o and N_w are empirical exponents that control the curve shapes (typically ranging from 2 to 4), and S_{wn} denotes the normalized wetting-phase saturation.[21] The model's parameters consist of the endpoints K_{rot} (maximum oil relative permeability) and K_{rwr} (maximum water relative permeability), with the shape factors N_o and N_w influencing the curvature; in oil-wet systems, N_o < N_w reflects the preferential flow of the wetting oil phase.[22][23] This formulation offers significant advantages, including ease of implementation in numerical reservoir simulations due to its minimal parameter requirements, and its effectiveness in fitting empirical two-phase data from diverse sandstone and carbonate reservoirs.[21]