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.     

The influence of surface roughness on cloud cavitation flow around hydrofoils

The aim of this study is to investigate experimentally the effect of surface roughness on cloud cavitation around Clark-Y hydrofoils. High-speed video and particle image velocimetry(PIV) were used to obtain cavitation patterns images(Prog. Aerosp. Sci. 37: 551–581, 2001), as well as velocity and vorticity fields. Results are presented for cloud cavitating conditions around a Clark-Y hydrofoil fixed at angle of attack of α = 8?for moderate Reynolds number of Re = 5.6 × 10~5. The results show that roughness had a great influence on the pattern, velocity and vorticity distribution of cloud cavitation. For cavitating flow around a smooth hydrofoil(A) and a rough hydrofoil(B), cloud cavitation occurred in the form of finger-like cavities and attached subulate cavities, respectively. The period of cloud cavitation around hydrofoil A was shorter than for hydrofoil B.Surface roughness had a great influence on the process of cloud cavitation. The development of cloud cavitation around hydrofoil A consisted of two stages:(1) Attached cavities developed along the surface to the trailing edge;(2) A reentrant jet developed, resulting in shedding and collapse of cluster bubbles or vortex structure. Meanwhile, its development for hydrofoil B included three stages:(1) Attached cavities developed along the surface to the trailing edge, with accumulation and rotation of bubbles at the trailing edge of the hydrofoil affecting the flow field;(2) Development of a reentrant jet resulted in the first shedding of cavities. Interaction and movement of flows from the pressure side and suction side brought liquid water from the pressure side to the suction side of the hydrofoil, finally forming a reentrant jet. The jet kept moving along the surface to the leading edge of the hydrofoil, resulting in large-scale shedding of cloud bubbles. Several vortices appeared and dissipated during the process;(3) Cavities grew and shed again.     

Sessile drop deformations under an impinging jet

The problem of steady axisymmetric deformations of a liquid sessile drop on a flat solid surface under an impinging gas jet is of interest for understanding the fundamental behavior of free surface flows as well as for establishing the theoretical basis in process design for the Aerosol \({{\rm Jet}^{\circledR}}\) direct-write technology. It is studied here numerically using a Galerkin finite-element method, by computing solutions of Navier–Stokes equations. For effective material deposition in Aerosol \({{\rm Jet}^{\circledR}}\) printing, the desired value of Reynolds number for the laminar gas jet is found to be greater than ~500. The sessile drop can be severely deformed by an impinging gas jet when the capillary number is approaching a critical value beyond which no steady axisymmetric free surface deformation can exist. Solution branches in a parameter space show turning points at the critical values of capillary number, which typically indicate the onset of free surface shape instability. By tracking solution branches around turning points with an arc-length continuation algorithm, critical values of capillary number can be accurately determined. Near turning points, all the free surface profiles in various parameter settings take a common shape with a dimple at the center and bulge near the contact line. An empirical formula for the critical capillary number for sessile drops with \({45^{\circ}}\) contact angle is derived for typical ranges of jet Reynolds number and relative drop sizes especially pertinent to Aerosol \({{\rm Jet}^{\circledR}}\) printing.     

Fast X-ray Micro-Tomography of Multiphase Flow in Berea Sandstone: A Sensitivity Study on Image Processing

Synchrotron-based fast micro-tomography is the method of choice to observe in situ multiphase flow and displacement dynamics on the pore scale. However, the image processing workflow is sensitive to a suite of manually selected parameters which can lead to ambiguous results. In this work, the relationship between porosity and permeability in response to systematically varied gray-scale threshold values was studied for different segmentation approaches on a dataset of Berea sandstone at a voxel length of 3  \(\upmu \) m. For validation of the image processing workflow, porosity, permeability, and capillary pressure were compared to laboratory measurements on a larger-sized core plug of the same material. It was found that for global thresholding, minor variations in the visually permissive range lead to large variations in porosity and even larger variations in permeability. The latter is caused by changes in the pore-scale flow paths. Pore throats were found to be open for flow at large thresholds but closed for smaller thresholds. Watershed-based segmentation was found to be significantly more robust to manually chosen input parameters. Permeability and capillary pressure closely match experimental values; for capillary pressure measurements, the plateau of calculated capillary pressure curves was similar to experimental curves. Modeling on structures segmented with hysteresis thresholding was found to overpredict experimental capillary pressure values, while calculated permeability showed reasonable agreement to experimental data. This demonstrates that a good representation of permeability or capillary pressure alone is not a sufficient quality criterion for appropriate segmentation, but the data should be validated with both parameters. However, porosity is the least reliable quality criterion. In the segmented images, always a lower porosity was found compared to experimental values due to micro-porosity below the imaging resolution. As a result, it is recommended to base the validation of image processing workflows on permeability and capillary pressure and not on porosity. Decane-brine distributions from a multiphase flow experiment were modeled in a thus validated \(\upmu \) -CT pore space using a morphological approach which captures only capillary forces. A good overall correspondence was found when comparing (capillary-controlled) equilibrium fluid distributions before and after pore-scale displacement events.     

Effective Rheology of Two-Phase Flow in Three-Dimensional Porous Media: Experiment and Simulation

We present an experimental and numerical study of immiscible two-phase flow of Newtonian fluids in three-dimensional (3D) porous media to find the relationship between the volumetric flow rate (Q) and the total pressure difference (\(\Delta P\)) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold (\(P_t\)), making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium. In this regime, Q depends quadratically on an excess pressure drop (\(\Delta P-P_t\)). While increasing the flow rate, there is a transition, beyond which the overall flow is Newtonian and the relationship is linear. In our experiments, we build a model porous medium using a column of glass beads transporting two fluids, deionized water and air. For the numerical study, reconstructed 3D pore networks from real core samples are considered and the transport of wetting and non-wetting fluids through the network is modeled by tracking the fluid interfaces with time. We find agreement between our numerical and experimental results. Our results match with the mean-field results reported earlier.     

An Experimental and Numerical Study of Relative Permeability Estimates Using Spatially Resolved $$T_1$$- z NMR

Relative permeability is a key characteristic describing flow properties of petroleum reservoirs, aquifers and water retention of soils. Various laboratory methods, typically categorised as steady-state, unsteady-state and centrifuge are used to measure relative permeability and may lead to different results. In recent years, 1D MRI, NMR \(T_2\) and \(T_1\) profiling have been applied for the characterisation of rock cores. It has been shown that spatially resolved NMR in conjunction with centrifuge technique may provide high-quality capillary pressure curves. Combining Burdine and Brooks–Corey models enables estimation of relative permeability from capillary pressure curves. This approach assumes a strong relationship between capillary pressure and relative permeability known to be complex. Here we compare a generalised approach of Green, which relies on saturation profiles set by various capillary drainage techniques, to a NMR relaxation approach. Comparisons are performed experimentally and numerically using three sandstone rocks to test the influence of rock morphology. The numerical part includes simulation of a centrifuge capillary drainage by applying morphological drainage transforms on high-resolution 3D tomograms. \(T_1\) responses along the sample are simulated using a random walk technique. The NMR relaxation-based approach is then compared to LBM simulated relative permeability and to experiment. The study confirms the applicability of NMR relaxation methods for relative permeability estimation of water-wet rocks and validates a numerical approach against experiment.     

Estimation of Shale Intrinsic Permeability with Process-Based Pore Network Modeling Approach

A multi-scale pore network model is developed for shale with the process-based method (PBM). The pore network comprises three types of sub-networks: the \(\upmu \)m-scale sub-network, the nm-scale pore sub-network in organic matter (OM) particles and the nm-scale pore sub-network in clay aggregates. Process-based simulations mimic shale-forming geological processes and generate a \(\upmu \)m-scale sub-network which connects interparticle pores, OM particles and clay aggregates. The nm-scale pore sub-networks in OM and clay are extracted from monodisperse sphere packing. Nm-scale throats in OM and clay are simplified to be cylindrical and cuboid-shaped, respectively. The nm-scale pore sub-networks are inserted into selected OM particles and clay aggregates in the \(\upmu \)m-scale sub-network to form an integrated multi-scale pore network. No-slip permeability is evaluated on multi-scale pore networks. Permeability calculations verify that shales permeability keeps decreasing when nm-scale pores and throats replace \(\upmu \)m-scale pores. Soft shales may have higher porosity but similar range of permeability with hard shales. Small compaction leads to higher permeability when nm-scale pores dominate a pore network. Nm-scale pore networks with higher interconnectivity contribute to higher permeability. Under constant shale porosity, the shale matrix with cuboid-shaped nm-scale throats has lower no-slip permeability than that with cylindrical throats. Different from previous reconstruction processes, the new reconstruction process first considers the porous OM and clay distribution with PBM. The influence of geological processes on the multi-scale pore networks is also first analyzed for shale. Moreover, this study considers the effect of OM porosities and different pore morphologies in OM and clay on shale permeability.     

Capillary Hysteresis in Neutrally Wettable Fibrous Media: A Pore Network Study of a Fuel Cell Electrode

Hysteresis in the saturation versus capillary pressure curves of neutrally wettable fibrous media was simulated with a random pore network model using a Voronoi diagram approach. The network was calibrated to fit experimental air-water capillary pressure data collected for carbon fibre paper commonly used as a gas diffusion layer in fuel cells. These materials exhibit unusually strong capillary hysteresis, to the extent that water injection and withdrawal occur at positive and negative capillary pressures, respectively. Without the need to invoke contact angle hysteresis, this capillary behaviour is re-produced when using a pore-scale model based on the curvature of a meniscus passing through the centre of a toroid. The classic Washburn relation was shown to produce erroneous results, and its use is not recommended when modelling fibrous media. The important effect of saturation distribution on the effective diffusivity of the medium was also investigated for both water injection and withdrawal cases. The findings have bearing on the understanding of both capillarity in fibrous media and fuel cell design.     

Gas Migration in Highly Water-Saturated Opalinus Clay Microfractures Using a Two-Phase TRT LBM

Hydrogen gas migration modeling through water-saturated engineering barriers and the host rock of a deep geological repository for radioactive waste is of concern for safety assessment of such facilities. A two-phase two-relaxation-time lattice Boltzmann model using the Rothman and Keller approach was parallelized on graphic processing units to simulate hydrogen gas migration in a 3D image obtained by X-ray microtomography of Opalinus clay microfractures. A dimensional analysis combined with a grid refinement analysis was carried out to set the model parameters to reproduce the realistic viscous, capillary and inertial forces of the natural system. Relative permeabilities curves were first calculated in a simple regular fracture with different initial two-phase configurations. We observed that segmented gas flow configurations led to a drop in the relative gas permeability by two orders of magnitude as compared to parallel flow configuration. The model was then applied to 4\(\times \) refined 3D images. For lower water saturation values (\(0.5 \le S_\mathrm{w} < 0.7\)), hydrogen gas migrated through continuous gas paths oriented in the flow direction. At high water saturation values (\(S_\mathrm{w}\ge 0.7\)), the relative gas permeability dropped to zero because the hydrogen phase segmented into gas pockets that were stuck in local narrow throats of the clay fracture. The study pointed out that the high capillary forces prevented the gas bubbles from distorting themselves to pass through these narrow paths.     

Ethylene–air detonation in water spray

Detonation experiments are conducted in a 52 \(\hbox {mm}\) square channel with an ethylene–air gaseous mixture with dispersed liquid water droplets. The tests were conducted with a fuel–air equivalence ratio ranging from 0.9 to 1.1 at atmospheric pressure. An ultrasonic atomizer generates a polydisperse liquid water spray with droplet diameters of 8.5–12 \(\upmu \hbox {m}\), yielding an effective density of 100–120 \(\hbox {g}/\hbox {m}^{3}\). Pressure signals from seven transducers and cellular structure are recorded for each test. The detonation structure in the two-phase mixture exhibits a gaseous-like behaviour. The pressure profile in the expansion fan is not affected by the addition of water. A small detonation velocity deficit of up to 5 % was measured. However, the investigation highlights a dramatic increase in the cell size (\(\lambda \)) associated with the increase in the liquid water mass fraction in the two-phase mixture. The detonation structure evolves from a multi-cell to a half-cell mode. The analysis of the decay of the post-shock pressure fluctuations reveals that the ratio of the hydrodynamic thickness over the cell size (\(x_{{\mathrm {HT}}}/{\lambda }\)) remains quite constant, between 5 and 7. A slight decrease of this ratio is observed as the liquid water mass fraction is increased, or the ethylene–air mixture is made leaner.     

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.     

Pore-throat size correlation from capillary pressure curves

Void spaces in porous media can be considered as three-dimensional networks consisting of bulges (pores) connected by constrictions (throats). Computer simulations of drainage-imbibition processes show that the critical end points of wetting-phase and nonwetting-phase saturation, in drainage and imbibition respectively, and the form of simulated relative permeability curves all were significantly different for uncorrelated and correlated pore-throat models. Since these models were identical except for the arrangement of throats in relation to pores, the degree of pore-throat size correlation appears to be an important property influencing flow and fluid displacement. Examples of uncorrelated and correlated pore-throat structures in rocks are presented and it is shown that this property, although difficult to quantify by direct observation, can be evaluated from capillary pressure curves.     

Effects of Organic Base Chemistry on Interfacial Tension,Wettability, and Capillary Pressure in Multiphase Subsurface Waste Systems

The presence of surfactants may have profound effects on the transport of organic contaminants in multiphase systems. It is a common practice, however, to model the subsurface migration of liquids independently of the aqueous phase composition. As such, transport in these systems may not be adequately characterized. This study investigates the impact of pH on interfacial tension, wettability, and the drainage capillary pressure–saturation relationship in air–water–quartz and oxylene–water–quartz systems containing dodecylamine, an organic base. In these systems, three mechanisms, speciation, partitioning, and sorption, are important in determining the interfacial tension and contact angle, and consequently, important in determining the capillary pressure. By adjusting the pH above and below, the pK_a of the base, the relative importance of these mechanisms was altered. Below dodecylamine's pK_a of 10.6, the base was primarily in a cationic form resulting in minimal partitioning into the nonaqueous liquid and greater sorption at the quartz surface. Above the pK_a, the base was primarily in a neutral form which did not sorb to the quartz, and, furthermore, partitioned into the organic liquid phase where its surface activity was minimized. The combination of these processes caused the capillary pressure to change in a manner consistent with porescale theory of capillarity. The utility in this approach lies in the possibility of predicting transport properties in multiphase systems while incorporating the direct effects of solution chemistry.     

Decomposing <Emphasis Type="Italic">J</Emphasis>-function to Account for the Pore Structure Effect in Tight Gas Sandstones

The J-function predicts the capillary pressure of a formation by accounting for its transport properties such as permeability and porosity. The dependency of this dimensionless function on the pore structure is usually neglected because it is difficult to implement such dependency, and also because most clastic formations contain mainly one type of pore structure. In this paper, we decompose the J-function to account for the presence of two pore structures in tight gas sandstones that are interpreted from capillary pressure measurements. We determine the effective porosity, permeability, and wetting phase saturation of each pore structure for this purpose. The throats, and not the pores, are the most important parameter for this determination. We have tested our approach for three tight gas sandstones formations. Our study reveals that decomposing the J-function allows us to capture drainage data more accurately, so that there is a minimum scatter in the scaled results, unlike the traditional approach. This study can have major implications for understanding the transport properties of a formation in which different pore structures are interconnected.     

The motion of viscous liquid column with finite length in a vertical straight capillary tube

This paper deals with the motion of viscous liquid column with finite length and two free surfaces in a vertical straight capillary tube. It is assumed that fluid is Newtonian. Linearizing the boundary conditions, analytic expressions in the form of infinite series have been obtained for velocity, piessure and free surface at low Reynolds number. The numerical calculation is carried out for a set of cylinder’s length of water and blood. It has been revealed that there are considerable circulating currents at the upper and lower meniscuses. Its maximum velocity is about 57% of the average velocity of the mainstream. Iner-tial effect is also studied in this paper. Using the time-dependent method in finite difference techniques, numerical solution of the corresponding nonlinear equation at Re<24.5 is computed. Comparing it with analytic exact solution at low Reynolds number shows that inertial effect is negligible provided Re<24.5.     

Study of fracture permeability using Lattice Gas Automata

We study the problem of flow permeability of fracture joints using Lattice-Gas Automata simulations. We model the fracture as a rough channel bounded by a self-affine surface. Changing the surface roughness exponent, rough walls having different microstructures are obtained. Different relative roughnesses — defined as the height of the largest surface asperity divided by the mean aperture — are obtained pulling apart the two surfaces that constitute the rough walls of the channel. We calculate the macroscopic variables volume flow rate and pressure difference using microscopic balances. In the low Reynolds number regime the pressure difference and the flow rate are linearly related (the behavior is described by Darcy's law). In this regime, we study the effect of geometry on the permeability. We have found that permeability is independent of the surface roughness exponentH and it is fully determined in terms of the relative roughness and mean aperture of the fracture joint. For larger Reynolds numbers a transition to a regime in which pressure difference and flow rate are not longer linearly related is observed. This transition is observed in a domain of Reynolds numbers for which the behavior in a smooth channel remains linear. We discuss this transition.     

Foam Drainage in Porous Media

In this paper we present a simple analysis of liquid drainage in foams confined in porous media. First we derive the equation for the evolution of the liquid saturation using general mass and momentum conservation arguments and phenomenological relations between the transport parameters and liquid saturation. We find an unusual foam drainage equation in which the determinant terms express the competition between the external force field, represented here by the gravity field, and capillary pressure gradient. We present analytical solutions of the drainage equation in three cases: (a) gravity forces are dominant over capillary forces, (b) capillary forces are dominant over gravity forces, and (c) capillary and gravity forces are comparable in order of magnitude.     

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.     

Analysis of the USBM wettability test

In this paper, we analyse the capillary pressure curves obtained by the centrifuge method in order to perform the USBM wettability test. The physical displacement mechanisms present both in the porous plate and in the centrifuge method, are described for different cases of wettability of the pore surface.The wetting fluid is defined as the fluid being at the lower pressure while displacing the other fluid, and this displacement is defined as imbibition. On the other hand, the process in which the fluid under the lower pressure is the displaced fluid is defined as drainage. The capillary pressure is defined as the positive pressure difference between the two fluids. By adhering to these definitions, there is a unique and consistent terminology for the same physical process: the displacement of oil by water in an oil wet system and the displacement of water by oil in an water wet system are both designated as drainage.An important result is that the centrifuge method is limited to the determination of drainage capillary pressure curves for strongly oil or water wet samples. There is no capillary equilibrium possible when a water wet sample is centrifuged under water because the wetting phase is under higher pressure than the nonwetting phase; the resulting forced imbibition curve should not be called a capillary pressure curve. For samples with bicontinuous fractional wettability, the curves obtained by the centrifuge method correspond to combination displacement, i.e. a combination of equilibrium drainage and forced imbibition coupled with blob mobilisation.     

Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems

A simple process-based model of three-phase displacement cycles for both spreading and non-spreading oils in a mixed-wet capillary bundle model is presented. All possible pore filling sequences are determined analytically and it is found that the number of pore occupancies that are permitted on physical grounds is actually quite restricted. For typical non-spreading gas/oil/water systems, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils. These two cases correspond to whether water or gas is the intermediate-wetting phase in oil-wet pores as determined by the corresponding contact angles, that is, cos ^o _gw > 0 or cos ^o _gw < 0, respectively. Analysis of the derived pore occupancies leads to the establishment of a number of relationships showing the phase dependencies of three-phase capillary pressures and relative permeabilities in mixed-wet systems. It is shown that different relationships hold in different regions of the ternary diagram and the morphology of these regions is discussed in terms of various rock/fluid properties. Up to three distinct phase-dependency regions may appear for a non-spreading oil and this reduces to two for a spreading oil. In each region, we find that only one phase may be specified as being the intermediate-wetting phase and it is only the relative permeability of this phase and the capillary pressure between the two remaining phases that depend upon more than one saturation. Given the simplicity of the model, a remarkable variety of behaviour is predicted. Moreover, the emergent three-phase saturation-dependency regions developed in this paper should prove useful in: (a) guiding improved empirical approaches of how two-phase data should be combined to obtain the corresponding three-phase capillary pressures and relative permeabilities; and (b) determining particular displacement sequences that require additional investigation using a more complete process-based 3D pore-scale network model.