[1] The macroscopic flow equations used to predict two-phase flow typically utilizes a capillary pressure-saturation relationship determined under equilibrium conditions. Theoretical reasoning, experimental evidence, and numerical modeling results have indicated that when one fluid phase replaces another fluid, this relationship may not be unique but may depend on the rate at which the phase saturations change in response to changes in phase pressures. This nonuniqueness likely depends on a variety of factors including soil-fluid properties and possibly physical scale. To quantify this dependency experimentally, direct measurements of equilibrium and dynamic capillary pressure-saturation relationships were developed for two Ottawa sands with different grain sizes using a 20 cm long column. A number of replicate air-water experiments were conducted to facilitate statistical comparison of capillary pressure-saturation relationships. Water and air pressures and phase saturations were measured at three different vertical locations in the sand column under different desaturation rates (1) to measure local capillary pressure-saturation relationships (static and dynamic); (2) to quantify the dynamic coefficient T , a measure of the magnitude of observed dynamic effects, as a function of water saturation for different grain sizes and desaturation rates; (3) to investigate the importance of grain size on measured dynamic effects; and (4) to assess the importance of sample scale on the magnitude of dynamic effects in capillary pressure. A comparison of the static and dynamic P c -S w relationships showed that at a given water saturation, capillary pressure measured under transient water drainage conditions is statistically larger than capillary pressure measured under equilibrium or static conditions, consistent with thermodynamic theory. The dynamic coefficient T , used in the expression relating the static and dynamic capillary pressures to the desaturation rate was dependant on porous media mean grain size but not on the desaturation rate. Results also suggest that the magnitude of the dynamic coefficient did not increase with the increased averaging volume considered in this study, as has been reported in the literature. This work suggests that dynamic effects in capillary pressure should be included in numerical models used to predict multiphase flow in systems when saturations change rapidly, particularly in fine-grained soil systems (e.g., CO 2 sequestration, enhanced oil recovery, air sparging for remediation).

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Experimental investigation of dynamic effects in capillary pressure: Grain size dependency and upscaling

[1] A series of primary drainage experiments was carried out in order to investigate nonequilibrium capillarity effects in two-phase ow through porous media. Experiments were performed with tetrachloroethylene (PCE) and water as immiscible uids in a sand column 21 cm long. Four drainage experiments were performed by applying large pressures on the nonwetting phase at the inlet boundary: 20, 30, 35, 38 kPa. Our results showed that the nonequilibrium local uids pressure difference-saturation curves are above the capillary pressure saturation curve. Moreover, the nonequilibrium pressure difference showed a nonmonotonic behavior with an overshoot that was more pronounced at higher injection

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Nonequilibrium capillarity effects in two‐phase flow through porous media at different scales

Evaporative salinization of soil is a common issue observed in arid and coastal regions. This process is driven by mass, momentum and energy exchange between the porous medium and the free-flow region. To analyze such coupled systems, we present a representative elementary volume-scale model concept for coupling non-isothermal multi-phase compositional porous-media flow and single-phase compositional laminar free flow. Our numerical results illustrate evaporation behavior from a porous medium initially saturated with NaCl solution, manifesting its influence on dissolved salt distribution, salt precipitation and porous-media properties. We show that the new model is capable to capture the evaporation physics for different stages of evaporative salinization and compare the numerical results to two different experimental datasets: (1) cumulative mass loss of water and dissolved salt during stage-1 of saline water evaporation and (2) evaporation rate for different stages of evaporative salinization. In addition, influence of the initial salt concentration on the saline water saturation vapor pressure and transition to stage-2 evaporation are analyzed and discussed.

Free-Flow–Porous-Media Coupling for Evaporation-Driven Transport and Precipitation of Salt in Soil

The main goal of this work is the development of a computational program for the quantitative assessment of the effects of entrained air in pipeline systems with respect to their operational safety. Likewise, two specific problems are investigated. (1) The effect of entrained air in form of pockets on hydraulic transients, during pump shutdown. It can be considered the most dangerous maneuver within a pumping pipeline. The computations corresponding to this study were evaluated by using the method of characteristics. (2) The numerical simulation of fluid transients caused by the shutdown of pumps, considering air pockets located at the high points of pumping pipeline systems and a water-air bubble mixture immediately downstream of the pockets. The constitutive equations – conservation of the gas mass, of the liquid mass, and the mixture momentum – yield a set of differential equations that will be solved by the method of characteristics. For the homogeneous model presented herein the two phases or components are treated as a single pseudofluid with average properties. It is assumed that there is no relative motion or slip between the phases, as well as for the momentum equation for the mixture. In the same way to the compressibility of the gas, the liquid compressibility and the pipe wall elasticity are included in the system of equations. The equation of energy is not used due to the moderate change in temperature of the mixture during the transient. In the case of negative impacts on the safety and operability of pipeline systems resulting from air entrainment, operational remediation measures will be suggested. 
Hier wurden bereits zielgerechte Modellversuche ausgefuhrt und Losungsvorschlage fur bauliche und betriebliche Anderungen ausgearbeitet. Letztere sollten nunmehr wissenschaftlich vertieft und zu allgemeinen Bemessungsregeln, moglichst auf analytischer Basis, ubergefuhrt werden. Vornehmlich stehen die Bewegungsvorgange von kompakten Lufteinschlussen und Wasser-Luft-Gemischen (Zweiphasenkomponenten-Stromung) in Pumpendruckleitungen und deren Verhalten beim plotzlichen Abschalten der Pumpenaggregate im Vordergrund.

Investigation on the effects of entrained air in pipelines

In standard multi-phase flow models on porous media, a capillary pressure saturation relationship developed under static conditions is assumed. Recent experiments have shown that this static relationship cannot explain dynamic effects as seen for example in outflow experiments. In this paper, we use a static capillary pressure model and a dynamic capillary pressure model based on the concept of Hassanizadeh and Gray and examine the behavior with respect to material interfaces. We introduce a new numerical scheme for the one-dimensional case using a Lagrange multiplier approach and develop a suitable interface condition. The behavior at the interface is discussed and verified by various numerical simulations.

Dynamic capillary effects in heterogeneous porous media

This study contributes to the understanding of dynamic effects in the capillary pressure-saturation relationship on the local scale and macroscale as well as giving an assessment as to when these effects might need to be taken into account for the evaluation of two-phase flow processes. 

In a general form, the rate-dependent Pc-Sw relationship as analysed here postulates that the difference between the non-wetting and wetting phase pressure minus the equilibrium capillary pressure is a linear function of the rate of change of saturation, introducing the factor of proportionality tau. This extended Pc-Sw relationship is examined in this study under three aspects.

In order to identify flow regimes where dynamic effects might be of importance, a dimensional analysis of the two-phase balance equations being closed with the extended Pc-Sw relationship is carried out. In the dimensionless equations the well-known capillary and gravitational number, Ca and Gr, evolve. Moreover, the new dimensionless number Dy appears as a factor. Dy quantifies the ratio of the 'dynamic capillary force' to the viscous force. Relating Dy to Ca or Gr yields DyC, the ratio of the dynamic to the capillary equilibrium force, and DyG, the ratio of the dynamic capillary to the gravitational force. The influence of the dynamic capillary force decreases with increasing characteristic length scale in relation to the equilibrium capillary, viscous or gravitational force. Moreover, with increasing transient flow velocity, the dynamic capillary force gains in importance.
 
A second part of this study deals with the determination of the coefficient tau on the basis of primary drainage laboratory experiments performed by GeoDelft, The Netherlands and numerical experiments. Using the laboratory experiments tau was calculated to vary between 11.0 kPas and 154.7 kPas depending on the water saturation. The coefficient increases with decreasing water saturation. For the determination of tau using numerical experiments first homogeneous domains were considered in order to test the averaging approach. The coefficient tau was found to scale proportionally to the porosity, the (saturation-weighted average) viscosity, and the squared averaging length, as well as inversely proportional to the intrinsic permeability. Parameters that describe the equilibrium Pc-Sw relationship have a minor influence on the coefficient tau. The dependence of tau on the averaging length poses a problem as its magnitude is thus not clearly bounded, making it impossible to define an REV. As a consequence, either the averaging length should relate to a bounded property or the averaging of the phase pressures as applied in this study has to be reconsidered. Moreover, two heterogeneous domains were studied, one being a simple pattern of two fine sand lenses in a coarse sand and the other a spatially-correlated random field. An influence resulting from the simple heterogeneity pattern on the coefficient is only noticeable in case the fine sand lenses are drained. The heterogeneity stemming from the spatially-correlated random field does not affect the magnitude of the coefficient.
 
Finally, the impact on the numerical solution of the balance equations being closed with an extended Pc-Sw relationship is assessed with simulations of imbibition processes. The dimensionless numbers Dy and DyC can be consulted to assess the impact on the solution. The front width should be applied as the characteristic length scale. It is observed that for a dominance of the dynamic capillary over the viscous force, the rate of change of saturation is dampened and thus a retardation effect with respect to the cumulative mass occurs. Locally, accelerating and slowing down of the displacement can flatten a front in comparison to a reference case. Moreover, for a high influx of wetting phase a process reversal from imbibition to drainage and back occurred. If additionally the equilibrium capillary forces are diminished, oscillations in the solution might occur for a large viscosity ratio. 
Diese Arbeit tragt zum Verstandnis von dynamischen Effekten in der Kapillardruck-Sattigungsbeziehung auf der lokalen und der Makroskala bei. Sie gibt eine Einschatzung unter welchen Bedingungen diese Effekte berucksichtigt werden sollten, wenn Zweiphasefluss-Prozesse betrachtet werden.

Die hier analysierte Kapillardruck-Sattigungsbeziehung postuliert, dass der Unterschied zwischen dem Druck der nichtbenetzenden und der benetzenden Phase minus des Gleichgewichtskapillardrucks eine lineare Funktion der Veranderung der Sattigung uber die Zeit ist. Dabei wird der Koeffizient tau eingefuhrt. Diese erweiterte Pc-Sw Beziehung wird in der Arbeit unter drei Aspekten analysiert.

Um Flussregime zu identifizieren, bei denen dynamische Effekte von Bedeutung sein konnten, wird eine Dimensionsanalyse basierend auf den Zweiphasen-Erhaltungsgleichungen, welche mit der erweiterten Pc-Sw Beziehung geschlossen wurden, durchgefuhrt. In den dimensionslosen Gleichungen treten die bekannten Zahlen, Kapillarzahl Ca und Gravitationszahl Gr, auf. Dazu kommt eine neue Zahl, hier dynamische Zahl Dy genannt, welche das Verhaltnis von 'dynamischen Kapillarkraften' zu viskosen Kraften quantifiziert. Setzt man Dy ins Verhaltnis zu Ca und Gr erhalt man DyC (dynamische zu Gleichgewichts-Kapillarkraft) bzw. DyG (dynamische Kapillarkraft zu Gravitationskraft). Der Einfluss der dynamischen Kapillarkraft nimmt mit zunehmender charakteristischer Lange ab im Vergleich zu den Gleichgewichts-Kapillarkraften, den viskosen Kraften und und den Gravitationskraften. Auserdem nimmt ihr Einfluss mit zunehmender (transienter) Stromungsgeschwindigkeit zu.

Der zweite Teil der Arbeit beschaftigt sich mit der Bestimmung des Koeffizienten tau auf der Grundlage von Laborexperimenten von GeoDelft (Niederlande) und numerischen Experimenten. Mit Hilfe der Laborexperimente wurden tau-Werte in Abhangigkeit der Wassersattigung zwischen 11.0 kPas und 154.7 kPas bestimmt. Die Werte nehmen mit abnehmender Wassersattigung zu. Fur die Bestimmung von tau auf der Grundlage von numerischen Experimenten wurden zunachst homogene Gebiete betrachtet, um den Mittelungsansatz zu uberprufen. Der Koeffizient verhalt sich direkt proportional zur Porositat, zur (sattigungsgewichteten, gemittelten) Viskositat und zur Mittelungslange zum Quadrat. Des Weiteren skaliert er umgekehrt proportional mit der intrinsischen Permeabilitat. Ein Einfluss der Parameter, die die Gleichgewicht Pc-Sw Beziehung beschreiben, konnte kaum festgestellt werden. Die Abhangigkeit von tau von der Mittelungslange bereitet ein Problem, da dadurch die Grose von tau nicht klar begrenzt ist und so kein REV bestimmt werden kann. Deshalb sollte entweder die Mittelungslange zu einer klar definierten Grose in Beziehung stehen oder die Mittelung der Drucke sollte uberdacht werden. Des Weiteren wurden zwei heterogene Gebiete untersucht, eins in dem zwei Feinsandlinsen in einen Grobsand eingebettet sind und eins ein raumlich-korreliertes Zufallsfeld. Ein Einfluss der einfachen Geometrie auf den Koeffizienten konnte nur beobachtet werden, wenn die Feinsandlinsen drainiert wurden. Die Heterogenitat von dem raumlich-korrelierten Feld hat die Werte von tau nicht beeinflusst.

Zum Schluss wird der Einfluss der Einbeziehung der erweiterten Pc-Sw Beziehung auf das Ergebnis von numerischen Simulationen eines Imbibitionsprozesses untersucht. Die dimensionslosen Zahlen Dy und DyC konnen herangezogen werden, um den Einfluss auf die Losung abzuschatzen. Als charakteristische Lange sollte dazu die Frontbreite eingesetzt werden. Es kann beobachtet werden, dass bei einer Dominanz der dynamischen Kapillarkrafte uber die viskosen Krafte die Sattigungsrate gedampft wird und so ein Retardationseffekt entsteht, beispielsweise wenn man die kumulative Masse im System betrachtet. Lokal kann die Front im Vergleich zu einem Referenzbeispiel abflachen. Fur eine hohe Zuflussrate uber den Rand konnte eine Prozessumkehr von Imbibition zu Drainage und zuruck beobachtet werden. Wenn zusatzlich der Einfluss der Gleichgewichtskapillarkrafte minimiert wird, kann es bei hohen Viskositatsverhaltnissen zu Oszillationen in der Losung kommen.

Sabine Manthey

Papers

Two-phase flow processes with dynamic effects in porous media - parameter estimation and simulation