M.I.J. van Dijke

Saturation-dependencies of three-phase relative permeabilities in mixed-wet and fractionally wet systems

Abstract The saturation-dependency behaviour of three-phase relative permeabilities is of central importance for modelling three-phase displacement processes in porous media. In this paper, and in related work, a method has been developed to determine the saturation-dependencies of three-phase relative permeabilities. This method is suitable for all types of mixed-wet and fractionally wet porous systems that contain clusters of oil-wet and water-wet pores with constant but different oil–water contact angles, reflecting weakly wetted conditions. Based on the classification of all allowed pore occupancies in a completely accessible porous medium, saturation-dependencies of the corresponding relative permeabilities are derived. Furthermore, three-phase relative permeabilities that appear to depend only on their own saturations are either linked to the corresponding two-phase relative permeabilities or it is shown that such a link cannot be established. A comparison has been made with existing relative permeability models with respect to their saturation-dependencies.

The relation between interfacial tensions and wettability in three-phase systems: consequences for pore occupancy and relative permeability

Abstract In this paper, we consider the relationships between wettability and three-phase pore occupancy in simple systems of arbitrary wettability, defined in terms of the oil–water contact angle, θow, of a pore. When θow changes, the remaining contact angles θgw and θgo also vary, consistent with available limited experimental data and a theoretical constraint equation. This equation was derived originally by Bartell and Osterhof [Ind. Eng. Chem. 19 (11) (1927) 1277] and applied more recently by Zhou and Blunt [J. Contam. Hydrol. 25 (1997)]. The consequences of this theory on several aspects of three-phase flow in mixed-wet and fractionally wet systems are discussed, in particular the issues of fluid–fluid wetting order, capillary displacement invariants, pore occupancy and saturation-dependencies of three-phase relative permeabilities. The model of pore occupancy leads to quite complex saturation-dependencies of the corresponding three-phase relative permeabilities. In case of distributed contact angles in both the water-wet and oil-wet pore clusters, all of the three-phase relative permeabilities depend on two saturations in a complex manner for most saturation combinations. In the general case, where the three-phase relative permeabilities cannot be related simply to the underlying two-phase relative permeabilities, a process-based model based on the pore scale physics is required. Given the great importance of the constraint equation, we recommend further experimental measurements of three-phase interfacial tension-contact angle data to establish the actual relationship between the three contact angles.

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 θogw > 0 or cos θogw < 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.

Multi-phase flow modeling of air sparging

Air injection into groundwater (air sparging) in a homogeneous axially symmetric porous medium is modeled using a two-phase flow approach. A numerical method based on the mixed form of the Richards equation for both phases is presented. Furthermore two analytical approximations are discussed to explain the numerical results. One is a one-dimensional description explaining the occurrence of small air saturations. The other is a closed form approximation for the distribution of the air saturation in the resulting steady state. From the latter we can estimate the maximum radius of influence of air sparging, as a function of the physical parameters. The analytical approximation at steady state and the numerical results are in good agreement.

Stochastic Pore Network Generation from 3D Rock Images

Pore networks can be extracted from 3D rock images to accurately predict multi-phase flow properties of rocks by network flow simulation. However, the predicted flow properties may be sensitive to the extracted pore network if it is small, even though its underlying characteristics are representative. Therefore, it is a challenge to investigate the effects on flow properties of microscopic rock features individually and collectively based on small samples. In this article, a new approach is introduced to generate from an initial network a stochastic network of arbitrary size that has the same flow properties as the parent network. Firstly, we characterise the realistic parent network in terms of distributions of the geometrical pore properties and correlations between these properties, as well as the connectivity function describing the detailed network topology. Secondly, to create a stochastic network of arbitrary size, we generate the required number of nodes and bonds with the correlated properties of the original network. The nodes are randomly located in the given network domain and connected by bonds according to the strongest correlation between node and bond properties, while honouring the connectivity function. Thirdly, using a state-of-the-art two-phase flow network model, we demonstrate for two samples that the rock flow properties (capillary pressure, absolute and relative permeability) are preserved in the stochastic networks, in particular, if the latter are larger than the original, or the method reveals that the size of the original sample is not representative. We also show the information that is necessary to reproduce the realistic networks correctly, in particular the connectivity function. This approach forms the basis for the stochastic generation of networks from multiple rock images at different resolutions by combining the relevant statistics from the corresponding networks, which will be presented in a future publication.

Pore-scale modelling of three-phase flow in mixed-wet porous media: Multiple displacement chains

Abstract A three-dimensional (3D) pore-scale network simulator is presented for modelling capillary-dominated three-phase flow in porous media where the wettability varies from pore to pore. The physics of weakly wetted systems has been included by allowing a pore to have any contact angle and, indeed, for each pore to have a different contact angle from a chosen distribution. An important complication arising from weakly wetting conditions is the absence of wetting films, which strongly reduces the continuity of the phases throughout the network. This reduction in phase continuity implies that during water-alternating-gas (WAG) injection processes a large number of phase clusters may form, that are disconnected from both inlet and outlet. Mobilisation of these clusters can happen through so-called multiple displacement chains, which involve a string of different phase clusters between inlet and outlet. We have explored the impact of these multiple displacement chains on WAG flow processes and the underlying three-phase flow mechanisms in a mixed-wet porous medium with the larger pores oil-wet. Assuming total absence of wetting films, we have varied the connectivity of the network (co-ordination number and dimension), the size of the network, the allowed maximum length of the displacement chains and the number of WAG cycles. The results are presented not only in terms of saturations and oil recovery, but also through statistics per flood of the length and type of displacement chains, the pore occupancy and through snapshots of the actual flood distributions (2D). From the simulations we conclude that for highly connected networks a steady state is reached after only a few WAG cycles, during which oil production ceases. However, in this state oil continues to be moved around within the network as a result of multiple displacement chains. The maximum allowed chain length has a substantial effect on the WAG saturation path, although the presence of longer chains during higher order WAG cycles is reduced for smaller networks and for networks with higher connectivity. For the investigated wettability state of the porous medium, four prevailing types of displacements emerge during each water flood and four different types emerge during each gas flood. When suppressing chains longer than two displacements in each flood one type disappears. Finally, we have listed some ways in which the predictions from the simulations may be tested experimentally both in core material and also in 2D micromodels. The latter type of experiments are particularly attractive, since lengths and types of displacement chains can be observed and counted directly.

Three-phase capillary entry conditions in pores of noncircular cross-section

In this work we challenge the assumption that the capillary entry pressures for displacements in three-phase flow are the same as those in two-phase flow. Using an energy balance, as derived by R.P. Mayer and R.A. Stowe (J. Colloid Interface Sci. 20 (1965) 893-911) and H.M. Princen (J. Colloid Interface Sci. 30 (1969) 69-75; 30 (1969) 359-371; 34 (1970) 171-184) for two-phase flow, we derive a general formula for determination of the capillary entry pressures for piston-like displacement of two bulk phases in a pore where a third phase may also be present. The method applies to capillaries of angular cross-section and uniform but arbitrary wettability. To use this method we have determined all possible underlying phase occupancies in cross-sections on either side of the main terminal meniscus, in particular the presence of corner arc menisci (AMs). Indeed, the capillary entry pressures for piston-like displacements depend on the pressure in the remaining third phase if the cross-sectional fluid configurations contain this phase. This dependence only vanishes when layers of the intermediate-wetting phase completely separate the wetting and the non-wetting phases. The complexity of the corresponding equations and the quantitative effects are studied using two different geometries, the equilateral triangle and the rhombus. The main difference is that the latter geometry has unequal corners, which may carry different AMs. We have carried out a limited sensitivity study with respect to the effect of wettability, the spreading coefficient of the intermediate-wetting phase, and the aspect ratio of the principal radii of the rhombus.

Modeling of air sparging in a layered soil: Numerical and analytical approximations

Air sparging in an aquifer below a less permeable horizontal layer is modeled using a two-phase flow approach. Supported by numerical simulations, we show that a steady state situation is reached. For an analysis of the steady state, we distinguish three different flow regimes, which occur between the well screen and the unsaturated zone. Just below the interface that separates the high and the low permeable layers a regime with almost hydrostatic capillary pressures develops. We use this observation to derive an ordinary differential equation for the pressure at the interface, which leads to an approximation of the air flow pattern just below and within the low permeable layer. The approximation provides an estimate for the radius of influence as a function of the physical parameters. The agreement between the analytical approximation and the numerical steady state results is almost perfect when heterogeneity is increased. With a few modifications the analysis applies also to a dense non-aqueous phase liquid (DNAPL) spill above a less permeable layer. Comparison with an illustrative numerical simulation shows that the analytical approximation provides a good estimate of the radial spreading of the DNAPL flow on top of and within the low permeable layer.

Pore-scale modeling of multiphase flow and transport: achievements and perspectives

When Irvin Fatt wrote his classical paper on pore-network modeling (Fatt 1956), he would probably not have thought that this field would become one of the largest fields of research in the porous media discipline. Pore-scale modeling has found its way as an expanding field of research for understanding the physics of flow and transport in porous media. In addition, it is becoming a valuable tool for prediction of petrophysical properties as part of the so-called Digital Rock Physics approaches, thus supplementing and replacing expensive and time consuming laboratory experiments. The recent popularity of pore-level modeling can also be attributed to advances in visualization of the pore space, to very high image resolution, and to the steady increase in computing power. This has made it possible to deal with a multitude of processes in the pore space and interactions with the solid phase (van Dijke and Piri 2007). The focus of this special issue of Transport in Porous Media is to provide an overview of some recent developments of various techniques for pore-scale modeling of multiphase flow and reactive transport

Analytical approximation to characterize the performance of in situ aquifer bioremediation

The performance of in situ bioremediation to remove organic contaminants from contaminated aquifers depends on the physical and biochemical parameters. We characterize the performance by the contaminant removal rate and the region where biodegradation occurs, the biologically active zone (BAZ). The numerical fronts obtained by one-dimensional in situ bioremediation modeling reveal a traveling wave behavior: fronts of microbial mass, organic contaminant and electron acceptor move with a constant velocity and constant front shape through the domain. Hence, only one front shape and a linear relation between the front position and time is found for each of the three compounds. We derive analytical approximations for the traveling wave front shape and front position that agree perfectly with the traveling wave behavior resulting from the bioremediation model. Using these analytical approximations, we determine the contaminant removal rate and the BAZ. Furthermore, we assess the influence of the physical and biochemical parameters on the performance of the in situ bioremediation technique.