Fluid Transport in the Subsurface

Hours per week: 28 (course) + 14 (excercise);
Credits: 4 ETC

Brief description:

Elementary groundwater courses usually cover water flow and
tracer transport. However, real subsurface flow can be far more
complex and include many simultaneous physical processes
(e.g., gravity, capillary effects, chemical reaction).

The course “Fluid Flow in the Subsurface” illustrates the physical
and mathematical aspects of the flow that are relevant for
environmental applications (the latter include, e.g., groundwater
recharge and pollution, management of coastal aquifers, risk
assessment of underground disposals of nuclear waste,
geological sequestration of carbon dioxide).

Specific goals:
– Get acquainted with the complex processes occurring in the subsurface
– Understand similarities and differences between the descriptions of different physical processes
– Be able to formulate and solve (numerically) flow problems

– Introduction on flow and transport in permeable rocks.
– The problem of scales: from pores to aquifers.
– Multiphase flow; density driven flow; reactive transport.
– Numerical techniques for modeling subsurface flow problems
– Applications to environmental problems.

Hydrogéologie (in french)

Houres par semaine: 28;
Crédits: 3 ETC

– Comprendre les concepts et les processus fondamentaux de l’écoulement souterrain
– Comprendre l’importance de la méthode quantitative en hydrogéologie
– Résolution de problèmes simples (écoulement 1D, puits, essai de pompage, etc.)

– Cycle hydrologique (cycle de l’eau)
– Eau souterraine et milieu poreux
– Concepts fondamentales en hydrogéologie (porosité; conductivité; aquifère; nappe…)
– Description mathématique de l’écoulement en milieu poreaux (Loi de Darcy, équation de continuité, …)
– Solutions analytiques de l’écoulement souterrain
– Puits et essai de pompage (Theis, Cooper-Jacob)
– Transport des polluants (équation de diffusion-advection)

J. Bear, Hydraulics of Groundwater, McGraw-Hill, New York, 1979
Domenico & Schwartz, Physical and Chemical Hydrogeology, J. Wilson & Sons, New York, 1990
G. de Marsily, Hydrogéologie quantitative, Masson, 1981

Coupled heat and mass transfer in field soils

Understanding and monitoring coupled heat and mass transfer in field-soils is relevant in many processes in which evaporation dynamics is crucial: soil-atmosphere interaction, groundwater recharge, vegetation growth. We investigate the role played by evaporation on soil-moisture dynamics to ameliorate the parametric models that describe field-soil response to diurnal forcing.

Our activity combines experiments, theory and modeling. Our theoretical analysis focuses on the practical effects of employing sparse data to estimate water and energy balance [2]. Our experiments concentrate on assessing the capability of different measurement techniques to monitor vadose-zone processes, and in particular the simultaneous transfer of moisture, heat and solutes. Our current research in this field focuses on the use of optical fibers [1] to measure both temperature and soil moisture (in the latter case the fiber is heated and the classical probe method is applied to infer thermal conductivity and water content). We also test Multi Functional Heat Pulse Probe and compare their performance with classical methods based on dielectric measurements [Ciocca, Sharma, Lunati, Parlange, 2013 –in preparation].

Contact: F. Ciocca, I. Lunati





Figure: (a) Comparison between soil moisture measured by traditional dielectric probes (green) and by the Heated Optical Fiber method using two different inversion procedures (red and blue) [1]. (b) Illustration of the Multi Function Heat Pulse Probe (MFHPP) that we are currently employing to measure thermal gradient in soils.


[1] Ciocca, F., I. Lunati, N. van de Giesen, and M.B. Parlange, Heated Optical Fiber for Distributed Soil-Moisture Measurements: A Lysimeter Experiment, Vadose Zone Journal, 11(4), 2012 doi:10.2136/vzj2011.0199 [pdf]

[2] Lunati, I., F. Ciocca, and M.B. Parlange, On the use of spatially discrete data to compute energy and mass balance, Water Resources Research, 48, W05542, 2012, doi:10.1029/2012WR012061

[3] Ciocca, F., I. Lunati, and M.B. Parlange, Effects of the water-retention curve on evaporation from dry soils, Geophysical Research Letters, 41(9), 3110–3116, 2014 doi:10.1002/2014GL059827

[4] Ciocca, F., Moisture and energy dynamics in fields soils: the influence of the diurnal cycle, PhD thesis, EPFL, 2013 [text]

Conceptual models for low permeability media (shale and clay)

Low-permeability media are becoming an extremely active research area. The interest in this geological materials is driven by their function as natural flow barriers to confine hazardous waste and, more recently, as economically exploitable unconventional hydrocarbon resources. Shale gas is expected to play a major role in US energy security. Also in Switzerland and Europe the attention to these unconventional resources is increasing. Understanding flow in low permeability media and the effects of fracking, which is used to increase the recovery rate, is essential to guarantee groundwater protection.

We have recently started to work on simple models to predict flow in fractured micro- and nano-porosity media. In the latter, self-diffusion is the process that controls gas velocity and recovery rate. We are proposing a simple dual-capillary model to estimate the relevant time scales [1]. The model can be used both in shale-gas applications and to investigate the rate of overpressure dissipation in nuclear waste disposals.




Figure: (a) Hydraulic fracturing from horizontal wells creates several primary-fracture planes connected by a network of secondary fractures that originate from microseismic events. (b) A dimensionless effective diffusion coefficient (red curve) can be introduced that contains all relevant gas-transport mechanisms in fractured nanopororous shales; when plotted as a function of a dimensionless density divided by the Knudsen number, this coefficient contains a single parameter that depends on the porous geometry [1].


[1] Lunati, I., and S.H. Lee, A dual-tube model for gas dynamics in fractured nanoporous shale formations, Journal of Fluid Mechanics, 757, 943-971, 2014 doi:10.1017/jfm.2014.519 [preprint]

[2] Lee, S.H., C.L. Jensen, and I. Lunati, A statistical dual-tube model to analyze gas production from shale formations, Paper SPE 173260-MS, SPE Reservoir Simulation Symposium, Houston, TX., February 23-25, 2015 doi:10.2118/173260-MS

Machine learning for risk assessment in hydrogeology applications

In most geoenergy applications, the heterogeneity structure of the aquifer reservoirs uncertain. Monte Carlo methods are widely used to estimate uncertainty propagation to the quantities of interest. To reduce the computational costs (which can become prohibitive for realistically complex problems) fast, approximate models (proxies) can be used. In our research, we combine Functional Data Analysis (FDA) with Principal Component Analysis (PCA) to build error models that allow improving the uncertainty estimate [1,2] (see also Figure).

In addition, we are testing a new strategy to generate stochastic realizations of aquifers and reservoirs. Starting from a geological training image, we use pattern recognition and local flow solutions to generate realizations that are already equipped with information about local fluxes. Our approach offers significant advantages in terms of computational costs because only an upscaled problem remains to be solved in each realization while the small scale details of the flow field are preserved [Josset, Lunati, Stubhaar, Renard, 2013 –in preparation].

Contact: L. Josset, I. Lunati





Figure: We use data clustering techniques to construct error models and improve uncertainty estimation: (a) Cumulative Distribution Function (CDF) obtained with the an approximate model (the Multiscale Finite Volume method, MsFV, in this case); (b) a more accurate CDF estimate is obtained by applying a global error model (GEM), which is constructed from the difference between the approximate and the exact responses for a subset of realizations [1].


[1] Josset, L., and I. Lunati, Local and global error models for improving uncertainty quantification, Mathematical Geosciences, 45(5), 601–620, 2013 doi:10.1007/s11004-013-9471-4

[2] Josset, L., D. Ginsbourger, and I. Lunati, Functional error modeling for uncertainty quantification in hydrogeology, Water Resources Research, 51 (2), 1050–1068, 2015 doi:10.1002/2014WR016028

[3] Josset, L., V. Demyanov, A.H. Elsheikh, and I. Lunati, Accelerating Monte Carlo Markov chains with proxy and error models, 2014 (under review)

Multiphase flow in deformable media

Our objective is to understand the key mechanisms that dictate fluid propagation in deformable media (e.g., competition between capillary/viscous flow and medium deformation). These phenomena are relevant applications such as sediment transport and geomorphology, fluidize bed filters, hydrates formation, triggering of landslides, migration of highly pressurized gases in hazardous waste disposals or in hydrocarbon reservoirs.

We are developing a framework to model recently performed experiments which includes air injection in dry unconsolidated granular material, granular Rayleigh taylor instability, and air injection in a wet granular media. One of the main difficulties consists in finding an effective set of constitutive relationships that are able to describe all possible flow regimes.

Contact: R. Ruiz-Baier, I. Lunati


Multiphysics modeling and hybrid algorithms

Multiscale methods are suitable to describe nonlinear physicochemical processes in large geological formations and can be used as a platform for multiphysics modeling. They offer an excellent interpretative framework for field-scale studies in which complex processes can be selectively considered only in the subregion of interest. Currently, we have used the Multiscale Finite Volume (MsFV) method as a framework to build hybrid models that combine Darcy-scale and pore-scale description of multiphase flow in a single algorithm. These methods can be used to assess the limits of validity of current continuum-scale models or to perform simulations that solve pore-scale flow only in a portion of the computational domain, while Darcy flow is considered elsewhere.

Contact: P. Tomin, I. Lunati

M = 3.0 M = 0.2
FS m3_fs m02_fs
AdMS m3_ms m02_ms
Average saturation profile fig_m3_front fig_m02_front

Comparison between fine-scale simulations (FS) and adaptive hybrid multiscale simulations (AdMS) of pore-scale flow with interfaces; the multiscale algorithm (which couple local pore-scale solutions by means of a Darcy-like coarse-scale solution) is able to correctly capture the transition from stable to unstable flow and the statistics of the front morphology (see average saturation profile).


Adaptive resolution for two-phase flow: macroscale description is applied in stable regions, while the front region is accurately resolved solving the full set of equations at pore scale. The hybrid MsFV method is used to couple the two regions. The sub-problems are solved locally until there is no substantial feedback to the global scale.



[1] P. Tomin, I. Lunati. Spatiotemporal adaptive multiphysics simulations of drainage-imbibition cycles, Computational Geosciences, 2015. doi:10.1007/s10596-015-9521-8 [text]

[2] P. Tomin, I. Lunati. Local-global splitting for spatiotemporal-adaptive multiscale methods. Journal of Computational Physics, V. 280, P. 214-231, 2015. doi:10.1016/j.jcp.2014.09.022 [text]

[3] P. Tomin, I. Lunati. Spatiotemporal adaptive multiscale multiphysics simulations of two-phase flow. 14th European Conference on the Mathematics of Oil Recovery, Catania, Sicily, Italy, 8–11 September 2014. doi:10.3997/2214-4609.20141845

[4] P. Tomin, I. Lunati. Hybrid Multiscale Finite Volume Method for Two-Phase Flow in Porous Media. Journal of Computational Physics, V. 250, P. 293–307, 2013. doi:10.1016/j.jcp.2013.05.019 [text]

[5] P. Tomin, A. Ferrari, R. Kuenze, and I. Lunati. A Framework for Hybrid Simulations of Two-phase Flow in Porous Media. 13th European Conference on the Mathematics of Oil Recovery, Biarritz, France, 10–13 September 2012. earthdoc.org

Adaptive multiscale modeling of complex subsurface flow

Geological formations are characterized by a hierarchy of spatial and temporal scales, which cannot be explicitly described due to prohibitive computational costs. Due to the lack of scale separations in critical flow regimes, traditional upscaling techniques are in general unable to describe nonlinear/nonequilibrium processes.

Multiscale numerical methods offer an alternative: they allow to remember fine-scale details where and if needed, and to correctly capture the large-scale response of the system. Our research focuses on the Multiscale Finite Volume Methods (MsFVM), which we have developed to improve the computational efficiency of large-reservoir simulators [3, 2,4]. Currently, we explore the possibility of using the MsFVM to capture flow instability and non-equilibrium processes (e.g., gravity fingers [1,6] due to the dissolution rate of supercritical CO2 in deep aquifers, reaction or phase transition kinetics). In this context, the method can also be used as a downscaling technique, which offer peculiar advantages [1,5] .

Contact: R. Künze, I. Lunati







Figure: Adaptive multiscale simulations of saltwater-freshwater instability with different front-detection criteria and grid refinements; shown is the normalized salt concentration (red corresponds to the highest concentration) at the same time step obtained with different adaptive criteria. From left to right the adaptive criterion becomes looser and fine-scales details of the flow are modeled in a smaller portion of the domain. A looser adaptive criterion leads to a partially coarsened description of the flow, but the overall behavior is still captured correctly–(b) and (c) [1].


[1] Künze, R., and I. Lunati, An adaptive multiscale method for density-driven instabilities, Journal of Computational Physics, 231, 5557–5570, 2012, doi:10.1016/j.jcp.2012.02.025

[2] Künze, R., I. Lunati, and S.H. Lee, A Multilevel Multiscale Finite Volume method, Journal of Computational Physics, 255, 502–520, 2013 doi:10.1016/j.jcp.2013.08.042

[3] Lunati, I., M. Tyagi, and S.H. Lee, An iterative Multiscale Finite-Volume algorithm con- verging to the exact solution, Journal of Computational Physics, 230(5), 1849-1864, 2011, doi:10.1016/j.jcp.2010.11.036

[4] Hajibeygi, H., S.H. Lee, and I. Lunati, Accurate and Efficient Simulation of Multiphase Flow in a Heterogeneous Reservoir by Using Error Estimate and Control in the Multiscale Finite-Volume Framework, SPE Journal, 17 (4), 1071–1083, SPE-141954-PA, 2012 doi:10.2118/141954-PA

[5] Tomin, P., and I. Lunati, A Hybrid Multiscale Method for Two-Phase Flow in Porous Media, Journal of Computational Physics, 250, 293–307, 2013 doi:10.1016/j.jcp.2013.05.019

[6] Künze, R., P. Tomin, and I. Lunati, Local modelling of instability onset for global finger evolution, Advances in Water Research, 70, 148–159, 2014 doi:10.1016/j.advwatres.2014.05.003

[7] Künze, R., Multiscale Descriptions of Density Driven Instabilities, PhD Thesis, University of Lausanne, 2014 [text]

Investigation of pore-scale processes and limits of validity of Darcy’s law

Although geological formations are described at field scale as a continuum, the physicochemical processes take place in a highly discontinuous pore geometry and are controlled by interface processes. As a result, the standard description of multiphase flow (based on generalized Darcy’s law) fails to effectively capture a wide range of relevant phenomena, such as instability, residual trapping, hysteresis, intermittent flow.

To investigate the limits of existing models, we perform direct numerical simulations of pore-scale flow in presence of interface. We solve the full Navier-Stokes in complex pore geometries and model the evolution of fluid-fluid interfaces by the Volume Of Fluid (VOF) methods. This approach offer indubitable advantages with respect to traditional pore-network models. VOF can be applied to several problems –including microfluidics and wetting– and allows sub-pore resolution, which captures coalescence and break up [1, 2], hysteresis of the contact angle [3], film and corner flow [4].

The results of the simulations are used as virtual experiments in which all relevant quantities can be ”measured“ to investigate the effects of spatial averaging. In recent work [1, 2] we have demonstrated that volume averaged quantities can be misleading and that capillary pressure is better defined from Helmotz free energy variations. We have also designed and run a set of laboratory experiments (performed at the University of Rennes, France, in collaboration with Geosciences Rennes) to compare observations and simulations in case of unstable flow regimes [5].

Contact: A. Ferrari, I. Lunati





Figure: (a) Comparison between experiments and simulations (red is the simulated nonwetting fluid distribution) [5]; (b) Comparison between different definitions of the macroscopic capillary pressure during a drainage experiment: in yellow a definition based on free-energy variation that has proven more accurate than other definitions (i.e., top-bottom pressure difference and difference between phase averaged pressures) [1, 4]


[1] Ferrari A., Lunati I., Direct numerical simulations of interface dynamics to link capillary pressure and total surface energy, Advances in Water Resources, 57, 19-31, 2013, doi:10.1016/j.advwatres.2013.03.005

[2] Ferrari, A., and I. Lunati, Direct simulation of interface dynamics: linking capillary pressure, interfacial area and surface energy, CMWR XIX – Computational Methods in Water Resources, Urbana-Champaign, IL, June 17-22, 2012 [pdf link]

[3] Lunati, I., and D. Or, Gravity driven slug motion in capillary tubes, Physics of Fluids, 21(5), 052003, 2009, doi:10.1063/1.3125262

[4] Ferrari, A., and I. Lunati, Inertial effects during irreversible meniscus reconfiguration in angular pores, Advances in Water Resources, 74, 1–13, 2014, doi:10.1016/j.advwatres.2014.07.009

[5] Ferrari, A., J. Jimenez-Martinez, T. LeBorgne, Y. Méheust, and I. Lunati, Challenges in modeling unstable multiphase flow experiments in micromodels, Water Resources Research, 51(3), 1381–1400, 2015, doi:10.1002/2014WR016384

[6] Andrea Ferrari, Pore-scale modeling of two-phase flow instabilities in porous media, PhD Thesis [text]