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Advective trapping occurs when solute enters low velocity zones in heterogeneous porous media. Classical local modelling approaches combine the impact of slow advection and diffusion into a hydrodynamic dispersion coefficient and many temporally non-local approaches lump these mechanisms into a single memory function. This joint treatment makes parameterization difficult and thus prediction of large-scale transport a challenge. Here, we investigate the mechanisms of advective trapping and their impact on transport in media composed of a high conductivity background and isolated low permeability inclusions. Breakthrough curves show that effective transport changes from a streamtube-like behaviour to genuine random trapping as the degree of disorder of the inclusion arrangement increases. We upscale this behaviour using a Lagrangian view point, in which idealized solute particles transition over a fixed distance at random advection times combined with Poissonian advective trapping events. We discuss the mathematical formulation of the upscaled model in the continuous time random walk and mobile–immobile mass transfer frameworks, and derive a model for large-scale solute non-Fickian dispersion. These findings give new insight into transport in highly heterogeneous media., Data used for producing the figures can be downloaded from digital.csic.es (https://digital.csic.es/handle/10261/216991) and by solving the respective equations. The authors thank Professor T.R. Ginn and two anonymous reviewers for their comments on the paper. J.J.H. and M.D. acknowledge the support of the Spanish Ministry of Science and Innovation (project CEX2018-000794-S and project HydroPore PID2019-106887GB-C31). J.J.H. acknowledges the support of the European Social Fund and Spanish Ministry of Ministry of Science, Innovation and Universities through the ‘Ramón y Cajal’ fellowship (RYC-2017-22300). The authors thank P. Uszes for providing the code to measure the average distance between inclusions.

In urban water pipe networks, pipe leakage may lead to subsurface contamination or to reduced waste water treatment efficiency. The quantification of pipe leakage is challenging due to inaccessibility and unknown hydraulic properties of the soil. A novel physically-based model for three-dimensional numerical simulation of pipe leakage in variably saturated soil is presented. We describe the newly implemented coupling between the pipe flow simulator HYSTEM-EXTRAN and the groundwater flow simulator OpenGeoSys and its validation. We further describe a novel upscaling of leakage using transfer functions derived from numerical simulations. This upscaling enables the simulation of numerous pipe defects with the benefit of reduced computation times. Finally, we investigate the response of leakage to different time-dependent pipe flow events and conclude that larger pipe flow volume and duration lead to larger leakage while the peak position in time has a small effect on leakage.

International audience; We develop an efficient computational model for simulating fluid invasion patterns emerging in variable aperture fractures. This two-dimensional model takes into account the effect of capillary force on the fluid-fluid interfaces and viscous pressure drop in both fluid phases. The pressure distribution is solved at each time step based on mass balance and local cubic law, considering an imposed pressure jump condition at the fluid-fluid interface. This pressure jump corresponds to the Laplace pressure which includes both terms related to the out-of-plane (aperture-spanning) curvature and to the in-plane curvature. Simulating a configuration that emulates viscous fingering in two-dimensional random porous media confirms that the model accounts properly for the role of viscous forces. Furthermore, direct comparison with previously obtained experimental results shows that the model reproduces the observed drainage patterns in a rough fracture reasonably well. The evolutions of tip location, the inlet pressures, and the invading phase fractal dimensions are analyzed to characterize the transition from capillary fingering to viscous fingering regimes. A radial injection scenario of immiscible invasion is also studied with varying modified capillary number and viscosity ratio, showing displacement patterns ranging from capillary fingering to viscous fingering to stable displacement. Such simulations using two contact angles show that the invading phase becomes more compact when the wetting condition changes from strong to weak drainage, as already observed in 2-D porous media. The model can be used to bridge the gap in spatial scales of two-phase flow between pore-scale modeling approaches and the continuum Darcy-scale models.

In this paper, we present an upscaled model for horizontal immiscible displacement in highly heterogeneous media. This type of heterogeneity can be found, for instance, in fractured rock, which consists of two flow domains, a mobile fracture and a virtually immobile matrix. We derive an upscaled double-continuum model capable of predicting flow in mobile-immobile domains using homogenization theory. The model consists of a flow equation for the saturation of displacing fluid in the fracture domain, and a capillary flow equation for saturation in the matrix, which are coupled via a source term. By linearizing capillary counter current flow in the matrix domain, we combine this system of equations into a non-local single-equation model for the fracture saturation, which can be interpreted as a multi-rate mass-transfer (MRMT) model for immiscible displacement. We discuss this simplification and the parametrization of the upscaled model equation from local hydraulic parameters obtained from rock samples and from knowledge of the average flow properties of the fracture network. We demonstrate its performance for predicting two-phase flow by considering a single fracture with imbibition into a rectangular matrix domain. The upscaled model is parameterized directly from geometry and hydraulic parameters of matrix and fracture of the reference model, which means that no parameters need to be fitted. We compare the detailed and upscaled models in terms of breakthrough curves for the displaced fluid at a control plane within the medium. Both the detailed numerical simulations and the upscaled model show a preasymptotic t - 1 / 2 scaling and a breakoff at the characteristic time scale for filling the matrix by counter current flow. © 2013 Elsevier Ltd., M. D. and J. C. acknowledge the support of the FP7 EU project PANACEA (Grant No. 282900), and the Spanish Ministry of Economy and Competitivity through the project HEART (CGL2010-18450).

We study modeling of two-phase flow in highly heterogeneous fractured and porous media. The flow behaviour is strongly influenced by mass transfer between a highly permeable (mobile) fracture domain and less permeable (immobile) matrix blocks. We quantify the effective two-phase flow behavior using a multirate rate mass transfer (MRMT) approach. We discuss the range of applicability of the MRMT approach in terms of the pertinent viscous and capillary diffusion time scales. We scrutinize the linearization of capillary diffusion in the immobile regions, which allows for the formulation of MRMT in the form of a non-local single equation model. The global memory function, which encodes mass transfer between the mobile and the immobile regions, is at the center of this method. We propose two methods to estimate the global memory function for a fracture network with given fracture and matrix geometry. Both employ a scaling approach based on the known local memory function for a given immobile region. With the first method, the local memory function is calculated numerically, while the second one employs a parametric memory function in form of truncated power-law. The developed concepts are applied and tested for fracture networks of different complexity. We find that both physically based parameter estimation methods for the global memory function provide predictive MRMT approaches for the description of multiphase flow in highly heterogeneous porous media. © 2016 Elsevier Ltd., This work was supported by the compute cluster, which is funded by the Leibniz University Hanover, the Lower Saxony Ministry of Science and Culture (MWK) and by the German Research Association (DFG) under the grant NE 34 824 10-1. We gratefully acknowledge the help of Bernd Flemisch from the Uni- versity of Stuttgart with the Dumux model. M.D. acknowledges the funding from the European Research Council through the project MHetScale (Grant agreement no. 617511).

A well-controlled 3-D experiment with pre-defined block heterogeneities is conducted, where neutron tomography is used to map 3-D water distribution after two successive drainage steps. The material and hydraulic properties of the two sands are first measured in the laboratory with multistep outflow experiments. Additionally, the pore structure of the sands is acquired by means of image analysis of synchrotron tomography data and the structure is used for pore-scale simulation of one- and two-phase flow with Lattice-Boltzmann methods. This gives us another set of material and hydraulic parameters of the sands. The two sets of hydraulic properties (from the lab scale and from the pore scale) are then used in numerical simulations of the 3-D experiment. The paper discusses critical aspects and benchmarks for experimental measurements of 3-D water distribution in heterogeneous porous media. Additionally, we discuss possibilities as well as difficulties and limitations in the determination of hydraulic properties of materials using two conceptually different approaches (pore scale and lab scale). We then test with the numerical simulations how good can predictions on flow and water content in structured media be when using these state-of-the-art methods for the determination of hydraulic properties. Based on the numerical simulations, we discuss which parameters are more difficult to predict and which of the two approaches (lab scale or pore scale) enables better predictions.

This paper discusses multi-step drainage experiments in two heterogeneously packed sand columns (10 × 10 × 20 cm 3 ). Different packing structures were generated using two different sand types. One purpose of the study was to test the influence of packing structures on the movement of water. The second purpose was to assess the quality of predictions for the outflow curves in both columns made with an upscaled model. The heterogeneous structures of the columns can be considered as two opposing extremes. The first column was packed with a random arrangement of two sand types that is not stochastically homogeneous and where a cluster running through the column exists for both materials. The second column was packed with a periodic pattern of coarse-sand inclusions in a fine-sand background and has a clearly defined unit cell. The depth-averaged (2D) spatial distribution of the water content in the columns was monitored during the whole multi-step outflow experiment using neutron radiography. The 3D water content was measured at the steady states by neutron tomography. The experimental results are compared with the model predictions of an upscaled model derived with the homogenization theory. The parameters for the upscaled model are calculated from the hydraulic parameters of the two sand types. These hydraulic parameters were first identified in independent measurements on samples of the two individual sand types, separately. Additionally, the hydraulic parameters of both sands were identified by fitting a numerical model to the measured outflow curves. The different column structures showed a significant effect on water retention and the effective retention function, as water was trapped in the coarse-sand inclusions of the periodic structure. We included this trapping effect in the effective retention function of the upscaled model with an apparent air entry pressure. Contrary to the retention, the different packing structures had no large effect on the dynamic behavior of the outflow. The effective conductivity of the columns is therefore not significantly influenced by the structure. The upscaled models predicted the movement of the averaged water content in the two columns well. This confirms the applicability of upscaled models even if the underlying requirements are not strictly met.

We investigate the effect of dynamic boundary conditions on solute transport in unsaturated, heterogeneous, bimodal porous media. Solute transport is studied with two-dimensional numerical flow and transport models for scenarios where either (i) solely infiltration or (ii) more realistic dynamic (infiltration–evaporation) boundary conditions are imposed at the soil surface. Travel times of solute are affected by duration and intensity of infiltration and evaporation events even when cycle-averaged inflow rates of the scenarios are identical. Three main transport mechanisms could be identified based on a criterion for the infiltration rate that is related to the hydraulic conductivity curves of the media. If, based on this criterion, infiltration rates are low, the transport paths for upward and downward transport do not differ significantly, and the breakthrough curves of solute are similar to the one obtained under stationary infiltration. If infiltration rates are moderate, travel paths deviate between upward and downward flow, leading to a trapping of solute and strong tailing of the breakthrough curves. If infiltration and evaporation rates are very high, lateral advective–diffusive transport can lead to very efficient and fast downward transport. Thus, solute breakthrough depends strongly on lateral flow paths enforced by the boundary conditions at the soil surface. If heterogeneity of the materials is not strong and the structure is tortuous, dynamic boundary conditions mainly lead to increased macrodispersion. We test simplified upscaled transport models based on stationary flow rates to estimate breakthrough curves and demonstrate how the transport mechanisms are captured in the model parameters.

International audience; The spatial distribution of fluid phases and the geometry of fluid-fluid interfaces resulting from immiscible displacement in fractures cast decisive influence on a range of macroscopic flow parameters. Most importantly, these are the relative permeabilities of the fluids as well as the macroscopic irreducible saturations. They also influence parameters for component (solute) transport, as it usually occurs through one of the fluid phase only. Here, we present a numerical investigation on the critical role of aperture variation and spatial correlation on fluid trapping and the morphology of fluid phase distributions in a geological fracture. We consider drainage in the capillary dominated regime. The correlation scale, that is, the scale over which the two facing fracture walls are matched, varies among the investigated geometries between L/256 and L (self-affine fields), L being the domain/fracture length. The aperture variability is quantified by the coefficient of variation (δ), ranging among the various geometries from 0.05 to 0.25. We use an invasion percolation based model which has been shown to properly reproduce displacement patterns observed in previous experiments. We present a quantitative analysis of the size distribution of trapped fluid clusters. We show that when the in-plane curvature is considered, the amount of trapped fluid mass first increases with increasing correlation scale Lc and then decreases as Lc further increases from some intermediate scale towards the domain length scale L. The in-plane curvature contributes to smoothening the invasion front and to dampening the entrapment of fluid clusters of a certain size range that depends on the combination of random aperture standard deviation and spatial correlation.

Upscaling of the Richards equation is important for large-scale modeling of water flow in the unsaturated zone. We derive an upscaled model for Richards' equation in a layered porous medium. Homogenization theory is applied to derive the upscaled equations for slow flow processes. Two flow regimes are compared. The first flow regime is quantified by small Bond numbers, meaning forces due to pressure gradients are dominant on the small scale. The second flow regime is quantified by large Bond numbers, meaning that forces due to pressure gradients and gravity contribute equally on the small scale, while the large scale is dominated by gravity forces. The case of intermediate Bond numbers is also addressed. We compare the effective relative permeability function and the effective capillary pressure head–saturation function for both flow regimes. In the case of small Bond numbers the effective curves can be derived based on capillary equilibrium. The procedure to calculate effective curves is then quite simple. In the case of large Bond numbers the procedure is more complex. However, comparing effective curves of different test cases showed that the difference between the effective curves for different Bond numbers is not that large for moderate parameter contrasts and when the gradients of the capillary–saturation curve do not become too small. The effective parameter functions calculated with a capillary equilibrium assumption are therefore often a good estimate.

The spatial correlation structure of heterogeneous parameters plays an important role in upscaling of many flow and transport processes in porous media and fractures. We analyze upscaling of a slow drainage process in a horizontal open rough walled fracture. The applicability of upscaled continuum models for this kind of flow process is discussed. We consider two different fracture aperture distributions with different spatial correlation properties. One is described by a slowly decaying, the other one by a Gaussian autocovariance. The flow process is modelled using an invasion percolation model, which takes the in plane curvature of the fluid cluster into account. Results are compared to laboratory experiments in artificial plexiglas fractures. Different averaging procedures are tested in order to assess the applicability of continuum models. Independently of the correlation structure of the fracture aperture field, spatial averages do not converge to a continuum formulation. Ensemble averages do converge, however, they only give probabilistic information and are not applicable to model drainage in a fracture.

We derive a large scale mixing parameter for a displacement process of one fluid by another immiscible one in a two-dimensional heterogeneous porous medium. The mixing of the displacing fluid saturation due to the heterogeneities of the permeabilities is captured by a dispersive flux term in the large scale homogeneous flow equation. By making use of the stochastic approach we develop a definition of the dispersion coefficient and apply a Eulerian perturbation theory to determine explicit results to second order in the fluctuations of the total velocity. We apply this method to a uniform flow configuration as well as to a radial one. The dispersion coefficient is found to depend on the mean total velocity and can therefore be time varying. The results are compared to numerical multi-realization calculations. We found that the use of single phase flow stochastics cannot capture all phenomena observed in the numerical simulations.

Prediction of infiltration in porous media is challenged by finger formation and unstable displacement of the wetting front. We present a systematic experimental study on the effect of packings and infiltration rates on unstable flow into initially dry porous media. We conducted small two- and three-dimensional experiments where water contents were measured with neutron radiography and larger two-dimensional experiments, which we evaluated by obtaining commonly used finger properties such as width and velocity from image analysis. Our results from experiments in macroscopically homogeneous packings were tested for correspondence with theoretical finger property predictions and are in good agreement. For the smaller experiments, water content profiles including “overshoots” at the finger tips as well as finger properties depending on the packing matched well results reported in the literature. For all homogeneous experiments, we found a strong dependence of finger properties on porosity. Homogeneous experiments served as reference cases for comparison with larger two-dimensional experiments where heterogeneity was induced by block-shaped inclusions. Unstable flow propagation in heterogeneous experiments was documented by inclusion fill rates in addition to finger properties. We found a multifaceted finger propagation behavior that depends on the impingement position of instabilities on an inclusion. Thereby, we are able to show that the location of the impingement on an inclusion is crucial to the filling rate of the inclusion and the propagation velocity of the finger. Results from the conducted experiments present a data set for testing of extended Richards’ models.

The paper presents numerical simulations of water infiltration in unsaturated porous media containing coarse-textured inclusions embedded in fine-textured background material. The calculations are performed using the two-phase model for water and air flow and a simplified model known as the Richards equation. It is shown that the Richards equation cannot correctly describe flow in the presence of heterogeneities. However, its performance can be improved by introducing appropriately defined effective capillary and permeability functions, representing largescale behaviour of the heterogeneous medium.

[1] We consider flow in a porous medium containing coarse-textured inclusions with a low value of air entry pressure, embedded in a fine-textured background material having high entry pressure. During imbibition some air remains trapped in the inclusions, while during drainage the inclusions become drained only after the capillary entry pressure exceeds the pressure of the background material. These effects can only be reproduced by a two-phase flow model, and not by the Richards' equation. However, if an upscaled form of the Richards' equation with appropriately modified capillary and permeability functions is used, the results are in a reasonable agreement with the two-phase flow model.

A major part of the world's remaining oil reserves is located in fractured carbonate reservoirs, which are dual-porosity (fracture-matrix) or multi-porosity (fracture-vug-matrix) in nature. Fractured reservoirs suffer from poor recovery, high water cut, and generally low performance. They are modelled using a dual-porosity approach, which assumes that the high-permeability fractures are mobile and low-permeability matrix is immobile. A single transfer function models the rate at which hydrocarbons migrate from the matrix into the fractures. As shown in many numerical, laboratory, and field experiments, a wide range of transfer rates occurs between the immobile matrix and mobile fractures. These arise, for example, from the different size of matrix blocks (yielding a distribution of shape factors), different porosity types, or the inhomogeneous distribution of saturations in the matrix blocks. Accurate models are hence needed that capture all the transfer rates between immobile matrix and mobile fracture domains, particularly to predict late-time recovery more reliably when the water cut is already high. In this work we propose a novel multi-rate mass transfer model for two-phase flow, which accounts for viscous dominated flow in the fracture domain and capillary flow in the matrix domain. It extends the classical (i.e., single-rate) dual-porosity model in that it allows us to simulate the wide range of transfer rates occurring in naturally fractured multi-porosity rocks. Using numerical simulations of water-flooding in naturally fractured rock masses at the grid-block scale we demonstrate that our multi-rate mass-transfer model matches the observed recovery curves more accurately compared to the classical dual-porosity model. We further discuss how tracer tests can be used to calibrate our multi-rate dual-porosity model before the water-flood commences and how our model could be employed in commercial reservoir simulation workflows.

5 [1] We study the influence of buoyancy and spatial heterogeneity on the spreading of the 6 saturation front of a displacing fluid during injection into a porous medium saturated 7 with another, immiscible fluid. To do so we use a stochastic modeling framework. We 8 derive an effective large‐scale flow equation for the saturation of the displacing fluid that 9 is characterized by six nonlocal flux terms, four that resemble dispersive type terms and 10 two that have the appearance of advection terms. From the effective large‐scale flow 11 equation we derive measures for the spreading of the saturation front. A series of 12 full two‐phase numerical solutions are conducted to complement the analytical 13 developments. We find that the interplay between density and heterogeneity leads to an 14 enhancement of the front spreading on one hand and to a renormalization of the evolution 15 of the mean front position compared with an equivalent homogeneous medium. The 16 quantification of these phenomena plays an important role in several applications, 17 including, for example, carbon sequestration and enhanced oil recovery.

Intrinsic heterogeneities influence the multi-phase flow behavior of a dense non-aqueous phase liquids (DNAPL) infiltrating into a natural soil. Typically, we cannot resolve the scale of these heterogeneities so that upscaling techniques are required. The choice of the appropriate upscaling method depends on the averaging scale, since the relative importance of capillary and gravity forces change with scale. We present an easy and quick upscaling approach for cases in which the flow on the length-scale of heterogeneities is dominated by capillary forces.

[1] Large-scale modeling of transient flow in the unsaturated zone is important for the estimation of the water budget and solute transport in the vadose zone. Upscaled flow models need to capture the impact of small-scale heterogeneities, which are not resolved by the model, on large-scale flow. We perform upscaling of the Richards equation in heterogeneous porous media with continuous distributions of the soil hydraulic parameters using homogenization theory and stochastic averaging techniques. We restrict the analysis to flow regimes in which the capillary-equilibrium assumption holds on the small scale. In order to account for effects of capillary entry pressure we apply the Brooks-Corey model for the soil retention and relative permeability curves and consider Leverett scaling for the coupling of intrinsic permeability and entry pressure. For this model we derive and analyze the ensemble-averaged parameter functions for the macroscopic flow equations. The effects of a definite entry pressure vanish with increasing variance of the log intrinsic permeability. We compare the statistically averaged parameter functions to numerically calculated effective functions for parameter fields with different connectivity properties. These results illustrate that soils with well-connected coarse materials differ in the relative permeability from those with well-connected fine materials or those without particular connectedness.

We consider the infiltration of DNAPL into heterogeneous water-saturated porous media. The heterogeneities are modeled as periodic continuously distributed permeability and capillary entry-pressure fields with log normal distribution. For the spatial correlation, we use different models: fields with a higher connectivity but similar two-point correlation function as introduced by [1], where one time the high permeable material is connected and one time the low permeable material is connected. For multi-phase flow, we assume capillary equilibrium, requiring that capillary forces dominate on the small scale. We determine the effective retention curve for the medium using a percolation approach. The relative permeability curves are obtained by solving the single-phase flow problem for the total permeability field at a given capillary pressure. Two different flow configurations are considered in order to obtain limiting cases for the effective parameter curves. Fields exhibiting connection of low absolute permeability values yield a lower effective residual saturation than fields originated from standard random-field generators. Due to the similarity of the covariance functions of the highly and poorly connected fields, the difference in residual saturation would not be obtained by linear stochastic theory. The two-point cluster function as introduced in [2], applied to an indicator field of the total permeability, is used in order to predict residual saturations and thus to make predictions about the effective parameter functions. Finally, we compare the results of numerical multiphase-flow simulations for the heterogeneous and upscaled fields.

In two-phase flow processes in a porous medium where one fluid displaces another immiscible one various patterns of the displacing front and the fluid configuration after breakthrough can be observed. Under certain conditions the displacement process is stable, leading to a regular and well predictable fluid distribution. However, very often the flow conditions lead to unstable displacement and irregular, complex shapes evolve. In this article, the nature and the development of the different shapes will be discussed.

Prediction of fluid phase dynamics in the vadose zone is hampered by the presence of different types of sharp interfaces, questioning the validity of standard theory that is based on assumption of continuity of air and water phases, uniform distribution of fluids in a control volume, local equilibrium of phase contents and pressures, and slow process velocities. Complexity of fluid front morphology and burst-like redistribution processes may be accentuated across material contrasts or at the interface between soil and atmosphere. This special section presents eleven contributions highlighting the role of interfaces on water and air distributions and outlining methods to improve prediction of interfacial displacement.

We study the impact of mass transfer between high and low conductivity zones on flow in unsaturated media. A dual continuum approach that assumes capillary-dominated flow in the slow continuum gives a system of coupled flow equations for the water saturations in the mobile and in the slow domain, whose coupling term is directly related to the evolution of the averaged water content in the slow domain. We show that linearization of the nonlinear diffusion equation that governs capillary flow in the slow continuum captures well the essential features of the time evolution of the averaged water content in the slow domain. This allows one to derive a non-local Richards equation for the water content in the mobile domain that is characterized by a memory kernel that encodes the local mass transfer dynamics as well as the geometry of the slow zones. Comparison of the model predictions to the results of numerical simulations of infiltration in a vertically layered medium shows that the non-local approach describes well nonequilibrium effects due to mass transfer between high and low conductivity zones.