Saturation Overshoot Research Articles

Modeling 2D gravity-driven flow in unsaturated porous media for different infiltration rates

Abstract. The gravity-driven flow in an unsaturated porous medium remains one of the most important unsolved problems in multiphase flow. Sometimes a diffusion-like flow with a uniform wetting front, known as stable flow, is observed, but, at other times, the flow is unstable with distinct preferential pathways. The formation of an unstable wetting front in a porous medium depends on many factors, including the type of porous medium, the initial saturation, and the applied infiltration rate. As the infiltration rate increases, the wetting front first transitions from stable to unstable at low infiltration rates and then from unstable to stable at high infiltration rates. We propose a governing equation and its discretized form, the semi-continuum model, to describe this significant non-monotonic transition. We show that the semi-continuum model is able to capture the influx dependence together with the correct finger width and spacing. Moreover, we demonstrate that the instability of the wetting front is closely related to the saturation overshoot in one dimension. Finally, we show that the flow can still be preferential even when the porous medium is completely wetted.

Semi-continuum modeling of unsaturated porous media flow to explain Bauters' paradox

Abstract. In the gravity-driven free infiltration of a wetting liquid into a homogeneous unsaturated porous medium, the flow pattern is known to depend significantly on the initial saturation. Point source infiltration of a liquid into an initially dry porous medium produces a single finger with an oversaturated tip and an undersaturated tail. In an initially wet medium, a diffusion-like plume is produced with a monotonic saturation profile. We present a semi-continuum model, based on a proper scaling of the retention curve, which is discrete in space and continuous in time. We show that the semi-continuum model is able to describe this transition and to capture the experimentally observed dependence of the saturation overshoot and the finger velocity on the initial saturation.

Astrochronology of the Paleocene-Eocene Thermal Maximum on the Atlantic Coastal Plain

The chronology of the Paleocene-Eocene Thermal Maximum (PETM, ~56 Ma) remains disputed, hampering complete understanding of the possible trigger mechanisms of this event. Here we present an astrochronology for the PETM carbon isotope excursion from Howards Tract, Maryland a paleoshelf environment, on the mid-Atlantic Coastal Plain. Statistical evaluation of variations in calcium content and magnetic susceptibility indicates astronomical forcing was involved and the PETM onset lasted about 6 kyr. The astrochronology and Earth system modeling suggest that the PETM onset occurred at an extreme in precession during a maximum in eccentricity, thus favoring high temperatures, indicating that astronomical forcing could have played a role in triggering the event. Ca content data on the paleo-shelf, along with other marine records, support the notion that a carbonate saturation overshoot followed global ocean acidification during the PETM.

Modeling Gravity‐Driven Unstable Flow in Subcritical Water‐Repellent Soils With a Time‐Dependent Contact Angle

AbstractUnstable flow in homogeneous dry soils, including saturation overshoot, is associated with a nonzero soil‐water contact angle (CA); 1D and 2D models for such flow, based on the moving‐boundary concept, were recently developed, solved, and verified for a constant high CA. However, in many natural soils rendered water‐repellent by natural organic matter, the CA decreases with time to a value that enables water infiltration. Thus, a mathematical model that includes the effect of time‐dependent CA on water‐content distribution and flow in the soil profile is developed in this study. This model, which also uses the moving‐boundary approach, simulates the effect of time‐dependent CA on unstable infiltration patterns. Comparison with a constant CA sheds light on the time‐dependent CA's influence on the aforementioned parameters. The 1D simulations indicate that a higher rate of CA decrease induces a higher wetting‐front velocity and shorter saturation‐overshoot length than a constant CA. However, due to flux imbalance at the wetting front for specific decreasing CA rates, the wetting‐front velocity first increases, and then decreases to an equilibrium value. The 2D simulations show that a time‐dependent CA significantly reduces water‐content accumulation at the finger tip. Moreover, a faster rate of decreasing CA results in a broader and longer plume shape, the latter being more pronounced. Effects of incoming flux at the soil surface and initial time‐dependent CA are also detailed for 1D and 2D flow. This theoretical study demonstrates that a time‐dependent CA significantly influences the formation of saturation overshoot and further impacts unstable flow generation.

A two dimensional semi-continuum model to explain wetting front instability in porous media

Modelling fluid flow in an unsaturated porous medium is a complex problem with many practical applications. There is enough experimental and theoretical evidence that the standard continuum mechanics based modelling approach is unable to capture many important features of porous media flow. In this paper, a two-dimensional semi-continuum model is presented that combines ideas from continuum mechanics with invasion percolation models. The medium is divided into blocks of finite size that retain the nature of a porous medium. Each block is characterized by its porosity, permeability, and a retention curve. The saturation and pressure of the fluids are assumed to be uniform throughout each block. It is demonstrated that the resulting semi-continuum model is able to reproduce (1) gravity induced preferential flow with a spatially rich system of rivulets (fingers) characterized by saturation overshoot, (2) diffusion-like flow with a monotonic saturation profile, (3) the transition between the two. The model helps to explain the formation of the preferential pathways and their persistence and structure (the core and fringe of the fingers), the effect of the initial saturation of the matrix, and the saturation overshoot phenomenon.

Correction to: “Measurement of saturation overshoot under grass cover” by Miloslav Šír et al.

The original version of the article was published in Biologia 75 (6): …–… (2020), https://doi.org/10.2478/s11756-020-00477-z. Unfortunately, Eq. (1) is missing in the Methods section of original version of this article. Here we display this equation.

The Moving‐Boundary Approach for Modeling 2‐D Gravity‐Driven Stable and Unstable Flow in Partially Wettable Soils

AbstractThe moving‐boundary approach, which has been successfully used to model stable and unstable 1‐D flow in initially dry soils of various contact angles (Brindt & Wallach, 2017 https://doi.org/10.1002/2016WR019252), was extended here for 2‐D flow. The wetting front is the plume perimeter that is partly formed by the capillary driving force, the remaining part by the combined capillary and gravity driving forces. The moving‐boundary approach overcomes the limitation of the Richards equation for describing gravity‐driven unstable flow with nonmonotonic water‐content distribution. According to this approach, the 2‐D flow domain is divided into two subdomains with a sharp change in fluid saturation between them—the wetting front (moving boundary). The 2‐D Richards equation was solved for the subdomain behind the wetting front for a given flux boundary condition at the soil surface, while the location of the other boundary, for which a no‐flux condition is imposed, was part of the solution. The moving‐boundary solution was used after verification to demonstrate the synergistic effect of contact angle and incoming flux on flow stability and its associated plume shapes. The contact angle that hinders spontaneous invasion of the dry pores decreases the water‐entry capillary pressure, ψwe, while the flux‐dependent dynamic water‐entry value, ψwed, is even lower, both inducing water accumulation behind the wetting front (saturation overshoot). This innovative physically based model for the 2‐D unsaturated flow problem for an initially dry soil of zero and nonzero contact angle using the moving‐boundary approach fulfills several criteria raised by researchers to adequately describe gravity‐driven unstable flow.

Measurement of saturation overshoot under grass cover

The paper deals with non-destructive measurement of water infiltration into homogenous soil (Haplic Chernozem) covered by grass. The experimental site was located in Most pri Bratislave (Slovakia) in the Pannonian Basin at a height of 130 m above sea level. Radioactive tracer technique was used to determine the water content of the soil profile during water infiltration. A mixture of the radioactive iodine 131I (chemical form Na131I) and the NaI solution was used as a tracer. The tracer distribution in the soil was measured by the Geiger-Muller detector connected to the nuclear analyser. Gravity-driven fingers were detected in the soil under the grass cover as a consequence of the wetting front instability. The instability manifested itself as a non-monotonic moisture profile caused by saturation overshoot and a varying discharge from the soil profile in time.

Numerical resolution of a pseudo-parabolic Buckley-Leverett model with gravity and dynamic capillary pressure in heterogeneous porous media

We discuss a numerical approach for the simulation of a hydrodynamic model for immiscible incompressible two-phase flow in heterogeneous porous media to deal with effects of dynamic capillary pressure and gravity. The governing system of equations consists of a nonlinear pseudo-parabolic Buckley-Leverett saturation transport model coupled with an elliptic pressure-velocity problem. Upon certain manipulation of the equations, we rewrite the transport model as a nonlinear elliptic reaction-diffusion problem along with a simpler time-dependent relation to the dynamic capillary pressure. Hybridized mixed finite elements and domain decomposition procedures are used for the spatial discretization of the equations. Proper locally conservative finite volume techniques are used for a balancing discretization of the first-order hyperbolic flux and the dispersive behavior inherent of the pseudo-parabolic model. After discretizing in time with an implicit method, the resulting nonlinear algebraic equations is solved with a sequential time-marching approach and the arising systems of linear equations are solved efficiently with an algebraic multigrid method. Thus, we formulate and test numerically a fully coupled and implicit formulation of the transport problem of pseudo-parabolic nature. We find that the new procedure is accurate, robust, efficient and easy for implementation. We simulated the two-dimensional transport of the full nonlinear modified Buckley-Leverett model with a dynamic capillary pressure model in a variety of flow regimes along with heterogeneous permeability fields and domination of gravity. Our numerical experiments demonstrated the viability of the proposed formulation based on computational simulations of typical flow situations and saturation overshoot.

Traveling wave solutions for the Richards equation with hysteresis

Abstract We investigate the one-dimensional non-equilibrium Richards equation with play-type hysteresis. It is known that regularized versions of this equation permit traveling wave solutions that show oscillations and, in particular, the physically relevant effect of a saturation overshoot. We investigate here the non-regularized hysteresis operator and combine it with a positive $\tau $-term. Our result is that the model has monotone traveling wave solutions. These traveling waves describe the behavior of fronts in a bounded domain. In a two-dimensional interpretation, the result characterizes the speed of fingers in non-homogeneous solutions.

A semi-continuum model of saturation overshoot in one dimensional unsaturated porous media flow

A semi-continuum model for fluid flow in saturated-unsaturated porous medium in one spatial dimension is presented. The model is based on well-established physics, measurable parameters and material characteristics. The porous material is characterized by porosity, intrinsic permeability, main wetting and draining branches of the retention curve, and the saturation dependence of the relative permeability. The fluid is characterized by its density and dynamic viscosity. The only physics involved is the mass balance of fluid in porous media together with the Darcy-Buckingham Law for fluid flow in unsaturated porous media. The model is a cellular automaton based on the Macro Modified Invasion Percolation concept of dividing the porous medium into blocks which are not infinitesimal and are assumed to retain the characteristics of a porous medium. The cellular automaton repeats three successive rules: saturation update in each block, pressure update in each block, and flux update between neighboring blocks. The model tracks the evolution of the relative saturation, the fluid capillary pressure, and the fluid flux. The model is shown to reproduce qualitatively and quantitatively all features of one dimensional saturation overshoot behavior reported in the literature.

Experimental and Numerical Studies of Saturation Overshoot during Infiltration into a Dry Soil

Core Ideas Experiments measured saturation overshoot during infiltration of water into dry soil. Two models based on Richards' equation with a dynamic capillarity term simulated the data. The IFA model is an alternative approach to simulate saturation overshoot. The IFA model and dynamic capillarity equation were combined to simulate experiments. Downward infiltration of water into almost dry soil, when there is no ponding at the soil surface, often occurs in the form of fingers, with saturation overshoot at the finger tips. While this is well known, there is still uncertainty about the exact saturation pattern within fingers. We performed a series of one‐dimensional water infiltration experiments into a dry soil to study the non‐monotonicity of the saturation. We observed that saturation showed a non‐monotonic behavior as a function of time. The overshoot was somewhat plateau shaped at relatively low flow rates but was quite sharp at higher flow rates. Two mathematical models, referred to as the extended standard (ESD) model and the interfacial area (IFA) model, were used to simulate the experimental results. Both models were based on extended forms of the Richards equation by including a dynamic capillary term. In the ESD model, standard equations for hysteresis were used. In the IFA model, the specific interfacial area was introduced to simulate hysteresis. Parameter values for both models were obtained from preliminary experiments or using empirical formulas. Only one parameter, the dynamic capillarity coefficient τ, was optimized to model saturation overshoot. While the ESD model did not reproduce the form of saturation overshoot for any combination of parameter values, the IFA model could provide good agreement with the data. To our knowledge, this is the first time where a combination of the IFA model and the dynamic capillarity equation has been used to simulate a set of experiments.

The Effect of Dynamic Capillarity in Modeling Saturation Overshoot during Infiltration

Core Ideas We provide detailed analysis of various forms of the dynamic capillarity coefficient. We focus on the region of saturation overshoot at larger saturation. The results are evaluated with the aid of experimental observations. Gravity‐driven fingering has been observed during downward infiltration of water into dry sand. Moreover, the water saturation profile within each finger is non‐monotonic, with a saturation overshoot at the finger tip. As reported in the literature, these effects can be simulated by an extended form of the Richards equation, where a dynamic capillarity term is included. The coefficient of proportionality is called the dynamic capillarity coefficient. The dynamic capillarity coefficient may depend on saturation. However, there is no consensus on the form of this dependence. We provide a detailed traveling wave analysis of four distinctly different functional forms of the dynamic capillarity coefficient. In some forms, the coefficient increases with increasing saturation, and in some forms, it decreases. For each form, we have found an explicit expression for the maximum value of saturation in the overshoot region. In current formulations of dynamic capillarity, if the value of the capillarity coefficient is large, the value of saturation in the overshoot region may exceed unity, which is obviously nonphysical. So, we have been able to ensure boundedness of saturation regardless of the value of the dynamic capillarity coefficient by extending the capillary pressure–saturation relationship. Finally, we provide a qualitative comparison of the results of traveling wave analysis with experimental observations.

An Improved Method for Quantifying Liquid Saturation Using Transparent Soil

Transparent porous media combined with digital image processing technology can be used to quantify the saturation of local fluids. In this regard, the calibration relationship between the pixel intensity of the image and the saturation of the wetting liquid is the crucial premise for whether this technique can be promoted and applied, and it is also a bottleneck that limits whether this technique can be rapidly and extensively developed. However, owing to gravity‐driven effects, wetting liquid in a residual state usually exhibits an irregular boundary caused by the phenomenon of saturation overshoot, which makes it very difficult to use the previous overflow method to obtain the data pair of saturation‐pixel intensity accurately. To this end, we have made some improvements to the specific testing method for wetting liquid saturation in the residual state and proposed a local centrifugal sampling method, which considerably improved the accuracy of this method. The results show that the relationship between the pixel intensity and saturation of the transparent soil obtained through this method is generally consistent with the regularity of the previous calibration method but shows better correlation. The results of this study will drive further potential application of transparent soil in the flow of partially saturated porous media.

Oxyhemoglobin Saturation Overshoot Following Obstructive Breathing Events Mitigates Sleep Apnea-Induced Glucose Elevations.

Background: Obstructive sleep apnea (OSA) and nocturnal hypoxia are associated with disturbances in glucose regulation and diabetes. Temporal associations between OSA, oxygenation profiles and glucose have not been well-described. We hypothesized that oxyhemoglobin desaturation during apneic events and subsequent post-apnea saturation overshoot predict nocturnal glucose.Methods: In 30 OSA patients who underwent polysomnography while subjected to CPAP withdrawal, we characterized SPO2 swings by frequency, desaturation depth, and overshoot height relative to baseline. We examined the associations between frequently sampled glucose and SPO2 swings during the preceding 10 min. We developed multi-variable mixed effects linear regression to examine the independent associations between glucose and each level of these SPO2 swings, while controlling for OSA severity.Results: Desaturation depth was not associated with glucose (p > 0.05). In contrast, overshoot was associated with glucose in a dose-dependent manner. Each SPO2 peak that did not rise to within 1% of baseline was associated with incremental glucose elevations of 0.49 mg/dL (p = 0.01), whereas peaks that exceeded baseline by >1% were associated with glucose reductions of 0.46 mg/dL. Overshoot remained an independent predictor of glucose after adjustment for mean SPO2 and OSA severity (p > 0.05).Conclusions: Vigorous SPO2 improvements after apneic events may protect patients against OSA-related glucose elevations.

Stable Propagation of Saturation Overshoots for Two-Phase Flow in Porous Media

Propagation of saturation overshoots for two-phase flow of immiscible and incompressible fluids in porous media is analyzed using different computational methods. In particular, it is investigated under which conditions a given saturation overshoot remains stable while moving through a porous medium. Two standard formulations are employed in this investigation, a fractional flow formulation and a pressure–saturation formulation. Neumann boundary conditions for pressure are shown to emulate flux boundary conditions in homogeneous media. Gravity driven flows with Dirichlet boundary conditions for pressure that model infiltration into heterogeneous media with position-dependent permeability are found to exhibit pronounced saturation overshoots very similar to those seen in experiment.

Numerical investigations of two-phase flow with dynamic capillary pressure in porous media via a moving mesh method

Motivated by observations of saturation overshoot, this paper investigates numerical modeling of two-phase flow in porous media incorporating dynamic capillary pressure. The effects of the dynamic capillary coefficient, the infiltrating flux rate and the initial and boundary values are systematically studied using a traveling wave ansatz and efficient numerical methods. The traveling wave solutions may exhibit monotonic, non-monotonic or plateau-shaped behavior. Special attention is paid to the non-monotonic profiles. The traveling wave results are confirmed by numerically solving the partial differential equation using an accurate adaptive moving mesh solver. Comparisons between the computed solutions using the Brooks–Corey model and the laboratory measurements of saturation overshoot verify the effectiveness of our approach.

Hysteresis in relative permeabilities suffices for propagation of saturation overshoot: A quantitative comparison with experiment.

Traditional Darcy theory for two-phase flow in porous media is shown to predict the propagation of nonmonotone saturation profiles, also known as saturation overshoot. The phenomenon depends sensitively on the constitutive parameters, on initial conditions, and on boundary conditions. Hysteresis in relative permeabilities is needed to observe the effect. Two hysteresis models are discussed and compared. The shape of overshoot solutions can change as a function of time or remain fixed and time independent. Traveling-wave-like overshoot profiles of fixed width exist in experimentally accessible regions of parameter space. They are compared quantitatively against experiment.

The moving‐boundary approach for modeling gravity‐driven stable and unstable flow in soils

AbstractThe Richards equation is unsuccessful at describing gravity‐driven unstable flow with nonmonotonic water content distribution. This shortcoming is resolved in the current study by introducing the moving‐boundary approach. Following this approach, the flow domain is divided into two subdomains with a sharp change in fluid saturation between them (moving boundary). The upper subdomain consists of water and air, whose relationship varies with space and time following the imposed boundary condition at the soil surface calculated by the Richards equation. The lower subdomain consists of an initially dry soil that remains constant. The location of the boundary between the two subdomains is part of the solution, rendering the problem nonlinear. The moving boundary solution was used after verification to demonstrate the effect of contact angle, soil characteristic curves and incoming flux on the dynamic water‐entry pressure of the soil, which depends on the soil's wettability, incoming flux at the soil surface and the wetting front's propagation rate. Lower soil wettability hinders spontaneous invasion of the dry pores and, together with a higher input flux, induces water accumulation behind the wetting front (saturation overshoot). The wetting front starts to propagate once the pressure building up behind it exceeds the dynamic water‐entry pressure. To conclude, the physically based novel moving‐boundary approach for solving stable and gravity‐driven unstable flow in soils was developed and verified. It supports the conjecture that saturation overshoot is a prerequisite for gravity‐driven fingering.

A Numerical Study of Two-Phase Flow Models with Dynamic Capillary Pressure and Hysteresis

Saturation overshoot and pressure overshoot are studied by incorporating dynamic capillary pressure, capillary pressure hysteresis and hysteretic dynamic coefficient with a traditional fractional flow equation in one-dimensional space. Using the method of lines, the discretizations are constructed by applying the Castillo–Grone’s mimetic operators in the space direction and a semi-implicit integrator in the time direction. Convergence tests and conservation properties of the schemes are presented. Computed profiles capture both the saturation overshoot and pressure overshoot phenomena. Comparisons between numerical results and experiments illustrate the effectiveness and different features of the models.