NUPUS Workshop “Dynamic effects in the capillary pressure


Multiphase flow and transport processes in porous media are encountered in many real life problems, as appearing in civil and environmental engineering, geosciences and reservoir engineering, or biomechanics. The understanding of such processes requires formulating appropriate mathematical models. One fundamental question arising in the mathematical modeling of multi-phase flow in porous media is the relation between the pressure difference in the flowing phases (the capillary pressure) and the saturation of these phases. Standard models rely on equilibrium assumptions, leading to a strict dependency between these quantities, involving parameters that can be determined experimentally.

Experimental evidences have revealed the limitation of such approaches. In particular, the dynamics needs to be taken into account when elaborating suitable mathematical models for two phase porous media flows. This workshop will address different issues related to the dynamic effects in the capillary pressure: experimental results, mathematical models, and suitable numerical techniques.

Nonlinearities and Upscaling in Porous Media” (NUPUS) is a joint Dutch - German research training group that has started in 2007 and involves the universities in Stuttgart, Utrecht, Delft, Eindhoven and Wageningen. It offers scholarships for Ph’D and master students, as well as postdocs working on the development and implementation of methods and models to describe complex multiphase flow processes in rigid and deformable porous media in an interdisciplinary research environment.


Majid Hassanizadeh, Utrecht University

Fritz Stauffer, ETH Zürich

David DiCarlo, University of Texas at Austin

Paul Zegeling, Utrecht University

Sorin Pop, Eindhoven University of Technology

Barbara Wohlmuth, University of Stuttgart

Rudolf Hilfer, University of Stuttgart


July 17, 2009


9:45-18:30 hrs (lunch break 12:40-13:40)


Eindhoven University of Technology, Auditorium 12

Detailed program and abstracts

09:45-10:10 Welcome, coffee/tea

10:10-10:15 Opening by Rainer Helmig

10:15-10:45 Majid Hassanizadeh

Non-equilibrium capillarity effect; there is no dynamic capillary pressure
Non-equilibrium capillarity effects in multiphase flow through porous media have been investigated in recent years based on theoretical developments, experimental investigations, mathematical analysis, and/or computational approaches. Of course there are non-equilibrium effects in capillarity in porous media, but they are related to fluids pressure. If capillary pressure is concerned, there is only one capillary pressure. In this talk, thermodynamic and theoretical basis of capillary pressure at non-equilibrium will be described.

10:45-11:30 Fritz Stauffer

Dynamic effects in the capillary pressure: Experimental results from the seventies
Phenomena of transient flow in unsaturated flow in porous media were a topic of experimental investigations in the sixties and seventies (Topp et al., 1967, Vachaud et al., 1972, Stauffer, 1977, 1978) mainly with respect to the uniqueness of the capillary pressure-water content relationship (retention curve). Observed deviations in the capillary pressure of homogeneous porous media compared to static and (quasi-)steady state conditions were attributed to dynamic effects. Experiments with sand columns by Stauffer (1977, 1978) suggested a linear relationship between the increase in capillary pressure and the temporal rate of change in water saturation during fast drainage processes. Stauffer (1977, 1978) also reported on possible dynamic effects in the relative hydraulic conductivity function. Attempts to take into account dynamic effects in numerical modelling were undertaken by Stauffer (1977, 1982) using finite elements. The purpose of the presentation is to shed some light on motivation, circumstances and conditions of this relatively early phase of investigations on dynamic effects in the capillary pressure.

11:30-11:45 Coffee break

11:45-12:30 David DiCarlo

Can Dynamic Multi-Phase Flow Models Describe Preferential Flow?
Gravity driven preferential flow paths (fingers) have been shown to be the result of nonmonotonic saturation profiles observed during 1-D infiltration (also known as saturation overshoot). These nonmontonic profiles cannot be described by traditional unsaturated continuum flow models (Richards Equation). Instead, continuum models require additional dynamic extensions which are still open to much question. In this talk, we describe the extensive data set obtained on preferential flow and nonmonotonic behavior in well sorted and compare it to recently proposed dynamic multi-phase flow extensions.

12:40-13:40 lunch

13:45-14:30 Barbara Wohlmuth

Numerical simulation concepts for extended two-phase models
In the first part of the talk, we consider the influence of extended capillary pressure models on the numerical simulation. Standard models consider a stationary capillary pressure relation and continuity at material interfaces. However, recent experiments have shown disagreement between measurements and numerical solutions using such simple models. Firstly, to correct the non-physical behavior, we use a recently established saturation-dependent retardation term. Secondly, in the case of heterogeneous porous media, we apply a model with a capillary threshold pressure that controls the penetration process.

In the second part, we focus on the abstract framework of variational inequalities. Mathematically, we rewrite this model as inequality constraint at the interfaces which allows discontinuities in the saturation and pressure. To get a better understanding for the nature of inequalities, we consider three examples for variational inequalities in different application fields. Based on these settings, we show the need for abstract discretization concepts and solver strategies. A naive application of existing algorithms results quite often in spurious oscillations or even no convergence at all.

Several numerical examples demonstrate the efficiency and flexibility of the new algorithm in 2D and 3D and show the influence of the retardation term.

14:30-15:15 Paul Zegeling and Sorin Pop

Dynamic capillarity models: non-monotone travelling waves
We consider travelling wave solutions for an extended BuckleyLeverett (BL) equation describing two-phase flow in porous media. this equation includes a third order mixed derivatives term modeling dynamic effects in the capillary pressure. We focus on the existence of traveling waves in cases that are ruled out by standard two phase flow models. Such waves are possibly non monotone. A particular attention will be paid to the non-monotonic water saturation profiles that have been observed experimentally, under different flow rates.

15:15-15:45 Break

15:45-16:15 Rudolf Hilfer

Dynamic effects in the capillary pressure from a percolation perspective
Dynamic effects in capillary pressure saturation relations, saturation overshoot during gravity driven preferential flow, capillary desaturation curves, or dynamic and local switching between drainage and imbibition processes are experimental phenomena that require extensions of the existing standard theories and models for two phase flow in porous media. The presentation will discuss an extended approach based on percolation of fluid phases that addresses these problems.

16:15-17:30 Discussion on dynamic effects in the capillary pressure.
The discussions refer to the appropriate modeling of experimental results, appropriate mathematical, numerical and upscaling techniques.

17:30-18:30 Closing and drinks

For more information please contact the organizers, Rainer Helmig (University of Stuttgart,, Paul Zegeling (University of Utrecht, or Sorin Pop (Eindhoven University of Technology,

For the accommodation please check

Lunch, as well as drinks will be offered. Please confirm your attendance until Monday, July 13, to Sorin Pop, +31-40-2475516 (office)/+31-6-27574984 (mobile)

Eindhoven University of Technology

Den Dolech 2

5612 AZ Eindhoven

The Netherlands