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

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

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

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

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.     

A Steady-State Upscaling Approach for Immiscible Two-Phase Flow

The paper presents a model for computing rate-dependent effective capillary pressure and relative permeabilities for two-phase flow, in 2 and 3 space-dimensions. The model is based on solving the equations for immiscible two-phase flow at steady-state, accounting for viscous and capillary forces, at a given external pressure drop. The computational performance of the steady-state model and its accuracy is evaluated through comparison with a commercial simulator ECLIPSE. The properties of the rate-dependent effective relative permeabilities are studied by way of computations using the developed steady-state model. Examples presented show the dependence of the effective relative permeabilities and capillary pressures, which incorporate the effects of fine scale wettability heterogeneity, on the external pressure drop, and thereby on the dimensionless macro-scale capillary number. The effective relative permeabilities converge towards the viscous limit functions as the capillary number tends to infinity. Special cases, when the effective relative permeabilities are rate-invariant, are also studied. The applicability of the steady-state upscaling algorithm in dynamic displacement situations is validated by comparing fine-gridded simulations in heterogeneous reservoirs against their homogenized counterparts. It is concluded that the steady-state upscaling method is able to accurately predict the dynamic behavior of a heterogeneous reservoir, including small scale heterogeneities in both the absolute permeability and the wettability.     

The Physical Origin of Interfacial Coupling in Two-Phase Flow through Porous Media

Recently developed transport equations for two-phase flow through porous media usually have a second term that has been included to account properly for interfacial coupling between the two flowing phases. The source and magnitude of such coupling is not well understood. In this study, a partition concept has been introduced into Kalaydjian's transport equations to construct modified transport equations that enable a better understanding of the role of interfacial coupling in two-phase flow through natural porous media. Using these equations, it is demonstrated that, in natural porous media, the physical origin of interfacial coupling is the capillarity of the porous medium, and not interfacial momentum transfer, as is usually assumed. The new equations are also used to show that, under conditions of steady-state flow, the magnitude of mobilities measured in a countercurrent flow experiment is the same as that measured in a cocurrent flow experiment, contrary to what has been reported previously. Moreover, the new equations are used to explicate the mechanism by which a saturation front steepens in an unstabilized displacement, and to show that the rate at which a wetting fluid is imbibed into a porous medium is controlled by the capillary coupling parameter, . Finally, it is argued that the capillary coupling parameter, , is dependent, at least in part, on porosity. Because a clear understanding of the role played by interfacial coupling is important to an improved understanding of two-phase flow through porous media, the new transport equations should prove to be effective tools for the study of such flow.     

Effect of hydrodynamic forces on capillary pressure and relative permeability

Because of the influence of hydrodynamic forces, the capillary pressure measured at static equilibrium may be different from that which pertains during flow. If such is the case, it may not be permissible to use steady-state relative permeabilities to predict unsteady-state flow. In this paper, the idea that the total flux of a given phase may be partitioned into several individual fluxes, together with a new pressure difference equation, is used to explore the possible impact that the hydrodynamic forces might have on capillary pressure and, as a consequence, relative permeability. This exploration reveals that, provided the pressure difference equation is implemented properly, capillarity has no impact on the relative permeability curves for the homogeneous, water-wet porous media considered. Moreover, it is demonstrated that, if the hydrodynamic effects are neglected, very little error is introduced into the analysis.     

Pore-by-Pore Modelling,Validation and Prediction of Waterflooding in Oil-Wet Rocks Using Dynamic Synchrotron Data

We predict waterflood displacement on a pore-by-pore basis using pore network modelling. The pore structure is captured by a high-resolution image. We then use an energy balance applied to images of the displacement to assign an average contact angle, and then modify the local pore-scale contact angles in the model about this mean to match the observed displacement sequence. Two waterflooding experiments on oil-wet rocks are analysed where the displacement sequence was imaged using time-resolved synchrotron imaging. In both cases the capillary pressure in the model matches the experimentally obtained values derived from the measured interfacial curvature. We then predict relative permeability for the full saturation range. Using the optimised contact angles distributed randomly in space has little effect on the predicted capillary pressures and relative permeabilities, indicating that spatial correlation in wettability is not significant in these oil-wet samples. The calibrated model can be used to predict properties outside the range of conditions considered in the experiment.

     

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.     

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

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

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.     

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.     

Numerical Investigations of the Steady State Relative Permeability of a Simplified Porous Medium

The purpose of this paper is to investigate, by flow simulations in a uniform pore-space geometry, how the co and countercurrent steady state relative permeabilities depend on the following parameters: phase saturation, wettability, driving force and viscosity ratio. The main results are as follows: (i) with few exceptions, relative permeabilities are convex functions of saturation; (ii) the cocurrent relative permeabilities increase while the countercurrent ones decrease with the driving force; (iii) with one exception, phase 2 relative permeabilities decrease and phase 1 relative permeabilities increase with the viscosity ratio M=₁/₂; (iv) the countercurrent relative permeabilities are always less than the cocurrent ones, the difference being partly due to the opposing effect of the viscous coupling, and partly to different levels of capillary forces; (v) the pore-level saturation distribution, and hence the size of the viscous coupling, can be very different between the cocurrent and the countercurrent cases so that it is in general incorrect to estimate the full mobility tensor from cocurrent and countercurrent steady state experiments, as suggested by Bentsen and Manai (1993).(Now at AS Norske Shell, Norway.) e-mail:     

Pore Network Modeling of the Effects of Viscosity Ratio and Pressure Gradient on Steady-State Incompressible Two-Phase Flow in Porous Media

We perform steady-state simulations with a dynamic pore network model, corresponding to a large span in viscosity ratios and capillary numbers. From these simulations, dimensionless steady-state time-averaged quantities such as relative permeabilities, residual saturations, mobility ratios and fractional flows are computed. These quantities are found to depend on three dimensionless variables, the wetting fluid saturation, the viscosity ratio and a dimensionless pressure gradient. Relative permeabilities and residual saturations show many of the same qualitative features observed in other experimental and modeling studies. The relative permeabilities do not approach straight lines at high capillary numbers for viscosity ratios different from 1. Our conclusion is that this is because the fluids are not in the highly miscible near-critical region. Instead they have a viscosity disparity and intermix rather than forming decoupled, similar flow channels. Ratios of average mobility to their high capillary number limit values are also considered. Roughly, these vary between 0 and 1, although values larger than 1 are also observed. For a given saturation, the mobilities are not always monotonically increasing with the pressure gradient. While increasing the pressure gradient mobilizes more fluid and activates more flow paths, when the mobilized fluid is more viscous, a reduction in average mobility may occur.

     

Gas Condensate Relative Permeability for Well Calculations

This paper addresses several issues related to the modeling and experimental design of relative permeabilities used for simulating gas condensate well deliverability. Based on the properties of compositional flow equations, we make use of the fact that relative permeability ratio k _rg/k _ro is a purely thermodynamic variable, replacing saturation, when flow is steady-state. The key relation defining steady-state flow in gas condensate wells is relative permeability k _rg as a function of k _rg/k _ro. Consequently, determination of saturation and k _r as a function of saturation is not important for this specific calculation. Once the k _rg=f(k _rg/k _ro) relationship is experimentally established and correlated with capillary number (N _c), accurate modeling of condensate blockage is possible. A generalized model is developed for relative permeability as the function of k _rg/k _ro and capillary number. This model enables us to link the immiscible or rock curves with miscible or 'straight-line curves by a transition function dependent on the capillary number. This model is also extended to the case of high-rate, inertial gas flow within the steady-state condensate blockage regionand locally at the wellbore. We have paid particular attention to the effect of hysteresis on the relation k _rg=f(k _rg/k _ro), based on our observation that many repeated cycles of partial/complete imbibition and drainage occur in the near-well region during the life of a gas condensate well. Finally, the composite effect of condensate blockage is handled using a Muskat pseudopressure model, where relative permeabilities are corrected for the positive effect of capillary number dependence and the negative effect of inertial high velocity flow. Special steady-state experimental procedures have been developed to measure k _rg as a function of k _rg/k _ro and N _c. Saturations, though they can be measured, are not necessary. An approach for fitting steady-state gas condensate relative permeability data has been developed and used for modeling relative permeability curves.     

Relative Permeability Estimation for Rich Gas-Condensate Reservoirs

This study addresses relative permeability prediction from well test data for low permeability, rich gas-condensate systems. Characteristic of these systems are high velocities and large pressure gradients within the near wellbore region. Within this region the relative permeabilties are rate sensitive and non-Darcy effects can be important. This study combines both the non-linear (in velocity) terms into a single effective relative permeability term. Effective relative permeabilities are estimated through non-linear regression with both synthetic and field data. Results show that a two-parameter simplified correlation is adequate for representing effective relative permeability. These parameters can be obtained by matching well test data. Mechanical skin was needed to match field data considered in this study. Non-Darcy effects can decrease the flowing bottom-hole pressures by about 480 kPa in high rate gas-condensate well tests. A well test design is proposed from which gas and condensate relative permeabilities can be estimated.     

3D pore network modeling and simulation for dynamic displacement of gas and condensate in wellbore region

Condensate dropout in near a wellbore region in gas condensate reservoir is the main reason of low well deliverability. Many researchers have studied gas and condensate relative permeabilities (RPs) in this region to find the condition for better deliverability. It is known that RP is a function of capillary number in low interfacial tension (IFT) systems such as gas-condensate. The positive dependency of RPs to velocity which is referred to as "positive coupling effect" is related to the simultaneous flow of gas and condensate associated with intermittent opening and closure of channels in porous media. The negative dependency of RPs to velocity named "negative inertia" is due to non-Darcy high-velocity flow. In this study, a 3D pore network modeling is developed to investigate fluids distribution in a gas-condense system at a pore scale to find out the effects of IFT and velocity on RPs. A new method is developed that applies a flash calculation in all throats in the network to estimate the amount of accumulated condensate in throats' corners at different time steps. A modified form of Poiseuille's law for polygonal cross-sectional throats is used to find and update pressure field and fluids distribution in the network and to determine the quantity of pushed out condensate from closed throats to neighboring throats. The displacement mechanism is considered to be determined by the volume of displaced phases in throats without applying any correlation. Simulation results indicate that gas and condensate RPs are increased by an increase in velocity. However, RPs sensitivity to velocity is reduced by increasing IFT which is in agreement with previous studies.     

Experimental study of two-phase flow in three sandstones. II. Capillary pressure curve measurement and relative permeability pore space capillary models

By means of the porous plate method and mercury porosimetry intrusion tests, capillary pressure curves of three different sandstones were measured. The testing results have been exploited jointly with three relative permeability models of the pore space capillary type (Burdine’s model type), these models are widely used and in rather distinct fields. To do so, capillary pressure has been correlated to saturation degree using six of the most popular relations encountered in the literature. Model predictions were systematically compared to the experimentally measured relative permeabilities presented in the first part of this work. Comparison indicated that the studied models underestimate the water relative permeability and over-estimate that of the non-wetting phase. Moreover, this modeling proves to be unable to locate the significant points that are the limits of fields of saturation where the variation of the relative permeabilities becomes consequent. We also showed that, if pore structure is modeled as a “bundle of capillary tubes”, model predications are independent of the capillary pressure curve measuring method.     

Relative permeabilities of a near critical binary fluid

Relative permeabilities were measured at very low interfacial tensions (IFT) for two-phase mixtures of methanol and hexane flowing through Clashach sandstone. These two components pass from a two- to a single-phase system as the temperature is increased above the critical solution temperature (CST). The interfacial tension between the coexisting phases approaches zero as the solution reaches miscibility. The phase behaviour of methanol and hexane mixtures has been well characterised allowing the calculation of relative permeabilities, saturations and capillary numbers. Flow data are reported for four different temperatures in the two-phase region (i.e., four values of IFT and capillary number). The capillary desaturation curve (CDC) for the strongly wetting methanol rich phase is also presented. In addition to the novel technique presented for measurement of relative permeability, the results indicate that relative permeabilities approach straight line functions very near the critical point. Furthermore, desaturation of the wetting phase was found to be dependent on the capillary number which, in turn, depends on the location of the mixture on the fluid phase diagram and the proximity to the critical temperature.     

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

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