As a tool for addressing problems of scale, we consider an evolving approach known as the thermodynamically constrained averaging theory (TCAT), which has broad applicability to hydrology. We consider the case of modeling of two-fluid-phase flow in porous media, and we focus on issues of scale as they relate to various measures of pressure, capillary pressure, and state equations needed to produce solvable models. We apply TCAT to perform physics-based data assimilation to understand how the internal behavior influences the macroscale state of two-fluid porous medium systems. A microfluidic experimental method and a lattice Boltzmann simulation method are used to examine a key deficiency associated with standard approaches. In a hydrologic process such as evaporation, the water content will ultimately be reduced below the irreducible wetting-phase saturation determined from experiments. This is problematic since the derived closure relationships cannot predict the associated capillary pressures for these states. We demonstrate that the irreducible wetting-phase saturation is an artifact of the experimental design, caused by the fact that the boundary pressure difference does not approximate the true capillary pressure. Using averaging methods, we compute the true capillary pressure for fluid configurations at and below the irreducible wetting-phase saturation. Results of our analysis include a state function for the capillary pressure expressed as a function of fluid saturation and interfacial area.

Hydrologic systems are typically investigated using some combination of
experimental, computational, and theoretical approaches. Each of these
classes of approaches has played a central role in advancing knowledge. The
years spanning the career of Eric F. Wood have witnessed a remarkable
development in the ability to study experimentally the elements that comprise
the hydrologic universe. The subsurface is a porous medium system that
receives experimental attention designed to identify the small-scale fluid
distributions within the solid matrix, intermediate-scale behavior through
laboratory study, and also the response of an aquifer to imposed forces

Complementing the advancing ability of experimental study is the development
of simulation tools for various aspects of hydrologic systems that make use
of advanced computer technology

A third element of advancing modeling of water resources systems is the
development of theory that accounts for physical processes. On one hand,
forming theoretical advances for mechanistic models based upon conservation
equations can be viewed as the standard challenges of accounting more
completely for conserved quantities and of developing closure relations for
dissipative processes. However, the need to pose closure relations at scales
that are consistent with those at which the problems have been formulated
creates a need for a variety of constitutive proposals. Furthermore,
consistency of models requires that equation formulations be consistent
across scales such that variables developed at a smaller scale can inform the
equations employed at a larger scale. Overall, these considerations lead to
identifying scale and scaling behavior in both time and space as important
challenges in posing models

In an era of unprecedented data generation, opportunities to use multiscale
averaging theory to develop physics-based data assimilation strategies
have never been more evident. The challenge of performing meaningful
theoretical, experimental, and computational analyses is constrained by the
need to ensure that the length and timescales of quantities arising in each
approach can be related. The scales of experimental data, variables appearing
in equations, and computed quantities must be the same if they are to be
compared in any meaningful way. As a prerequisite for this to happen, data
generated by any of the methods must be consistent across the range of scales
considered

While the desire for consistencies across scales and approaches is
conceptually simple to understand, it has proven to be a difficult practical
objective to meet. The change in scale of conservation and balance equations
can be accomplished rather easily. The problem with applying these equations
lies in the aforementioned need to average some intensive variables, the
requirement that closure conditions be proposed at the larger scale, and the
need to account for the dynamics of new quantities that arise in the change
of scale. Without accounting for all of these items properly, models are
doomed to fail. An essential element in ensuring success is the averaging of
thermodynamic relations to the larger scale

Challenges in assuring consistency across scales have also been confronted in the modeling of porous medium systems. Special challenges have been encountered for two-fluid-phase flow, where upscaling leads to the introduction of quantities such as specific interfacial area, which is the area where two phases meet normalized by the volume of the region, and specific common curve length, which is length of a curve where three phases meet normalized by the volume of the region. Modeling of multiscale porous medium systems must also employ thermodynamics that is scale consistent and included naturally as a part of the process. Because of the inability to overcome these challenges, most efforts to model multiscale, multiphase porous medium systems do not have thermodynamic constraints and full-scale consistency that would be expected in mature models. The thermodynamically constrained averaging theory (TCAT) approach is relatively refined and provides means to model systems that are inherently multiscale in nature and also to link disparate length scales, while representing the essential physics naturally and hierarchically with varying levels of sophistication. However, realizing these scale-consistent attributes requires new approaches, new equations of state, novel parameterizations, and, as with any new model, evaluation and validation.

The overall goal of this work is to examine the impact of phase connectivity
on scale consistency for two-fluid-phase porous medium systems. From the
mathematical standpoint, the microscale and macroscale must provide a
consistent view of the physics. In our approach, macroscale variables (such
as phase pressures and capillary pressure) are explicitly defined in terms of
microscale quantities to ensure that physical consistency is achieved. The
resultant rigorous connection between the microscale and the macroscale can
be exploited to understand and characterize how phase connectivity influences
key macroscale quantities. In other words, we ensure consistency between
information at small and large scales by using precise mathematics to change
the scale of variables; and we also ensure that variables denoted as
pertaining to theory, experiment, or simulation are defined such that they
refer to quantities defined at the same scale and are directly comparable.
The specific objectives of this work are

to formulate explicitly related microscale and macroscale descriptions of state variables important for traditional and evolving descriptions of capillary pressure

to determine state variables for capillary pressure using complementary experimental and computational approaches

to compare a traditional state equation approximation approach with a carefully formulated approach based in multiscale TCAT theory

to demonstrate the limitations of traditional state equation approaches for macroscale capillary pressure and

to examine the uniqueness of alternative state equation formulations for capillary pressure.

Two spatial scales are of primary interest for the porous medium problems of
focus herein: the microscale, which is often referred to as the pore scale;
and the macroscale, which is often referred to as the porous medium continuum
scale. At the microscale, the geometry of all phase distributions are fully
resolved in space and in time, which makes it possible to locate interfaces
where two phases meet and common curves where three phases meet. The
equations governing the conservation of mass, momentum, and energy, the
balance of entropy, and equilibrium thermodynamic relations are well
established at the microscale. Microscale experimental work and modeling are
active areas of research because of their relevance to understanding
operative processes in complex porous medium systems that were previously
impossible to observe. The macroscale is a scale for which a point is
associated with some averaged properties of an averaging region comprising
all phases, interfaces, and common curves present in the system. Notions such
as volume fraction and specific interfacial area arise when a system is
represented at the macroscale in terms of averaged measures of the state of
the system. These additional measures are quantities that must be determined
in the model solution process. Because of historical limitations on both
computational and observational data, the macroscale has been the traditional
scale at which models of natural porous media systems have been formulated
and solved. Closure relations at this scale are needed to yield well-posed
models. Traditionally, these closure relations have been posited empirically
and parameter estimation has been accomplished based upon relatively simple
laboratory experiments. In general, traditional macroscale models, while the
dominant class of model, suffer from several limitations related to the way
in which such models are formulated and closed

As efforts to model and link hydrologic elements in models advance, the
ability to address scales effectively will become essential. For porous
media, methods such as averaging, mixture theory, percolation theory, and
homogenization have been employed to transform governing system equations
from smaller to larger length scales

Averaging procedures have been used for analysis of porous media for
approximately 50 years

The problems associated with trying to model multiple fluid phases in porous media include (1) difficulties in properly accounting for interface properties, (2) lack of definition of macroscale intensive thermodynamic variables, (3) failure to account for system kinematics, and (4) challenges representing other important physical phenomena explicitly, such as contact angles and common curve behavior. These four difficulties sometimes impact the system description in combination.

Multiple-fluid-phase porous media differ from a single-fluid-phase porous
medium system by the presence of the interface between the fluids. This
interface is different from a fluid–solid interface because of its dynamics.
The total amount of solid surface is roughly constant, or is slowly varying,
for most natural solid materials. The fluid–fluid-specific interfacial area
changes in response to flow in the system and redistribution of phases. The
timescale of this change is between that of the pore diameter divided by
flow velocity and that of pore diameter divided by solid phase movement.
These specific interfacial areas are important for their extent, surface
tension, and curvature. They are the location where capillary forces are
present. Thus, a physically consistent model must account for mass, momentum,
and energy conservation at the interfaces; a model concerned only with phase
behavior cannot represent capillary pressure in a mechanistically
high-fidelity fashion

Intensive variables that are introduced at the macroscale without
consideration of microscale precursor values are also poorly defined. For
example, a range of procedures for averaging microscale temperature can be
employed that will lead to different macroscale values unless the microscale
temperature is constant over the averaging region. Thus, mere speculation
that a macroscale value exists fails to identify how or if this value is
related to unique microscale variables and most certainly does not relate the
macroscale variable to microscale quantities. The absence of a theoretical
relation makes it impossible to reliably relate microscale measurements to
larger-scale representations

The importance of kinematics is recognized, at least implicitly, in modeling
many systems at reduced dimensionality or when averaging over a region the
system occupies. For example, in the derivation of vertically integrated
shallow water flow equations, a kinematic condition on the top surface is
imposed based on the condition that no fluid crosses that surface

The mixed success in posing appropriate theoretical models, making use of
relevant data, and harnessing effective computer power to advance
understanding of hydrologic systems is attributable to the inherent
difficulty of each of these scientific activities. For progress to be made in
enhancing understanding, a significant hurdle must be overcome that requires
consistency among these three approaches and within each approach
individually. We have found that by performing complementary microscale
experimental and computational studies, we have formed a basis for being able
to upscale data spatially with insights into the operative timescales for
the system

An important aspect of the issues of concern in this work is related to the various ways in which capillary pressure can be measured and the consequences of using traditional approaches that observe fluid pressures on the boundary of an experimental cell and approximate the capillary pressure based upon the difference between the non-wetting phase pressure and the wetting phase pressure. However, even alternative approaches such as those based upon measurements using microtensiometers cannot resolve the issues of concern identified in this work. The differences among approaches are important, and commonly used approaches are flawed. In the formulation that follows, we show how microscale pressures can be averaged in a variety of ways as well as the relationship of these averaged pressures to the true capillary pressure. We note that averaging of pressures is inherent in the formulation of macroscale models; and indeed measurement devices themselves provide averages over a length scale depending upon the device. The issues related to averaging cannot be avoided.

Averaging of any intensive variable (e.g., pressure, temperature, chemical potential) is problematic because there is no unique averaging procedure that can be employed. This is in contrast to obtaining an upscaled value of mass per volume by integrating the microscale density over a volume to obtain the total mass and then dividing by the volume to get the upscaled density. Pressure, for example, is a force per area or, alternatively, an energy per volume. Averaging pressure over some area in a region as opposed to averaging over the volume of the region can give different values. Thus, it is imperative to identify pressure averages in ways that they arise in equations and in data collection. Correct identification of an averaged pressure and association of that average with a particular process or element of an equation is essential if the physics of a system are going to be described well at the macroscale. For this reason, we carefully define the larger-scale variables that will be used in analyzing the simulated system and describing system physics in this section. We also highlight the importance of identifying capillary pressure as an intrinsic property of an interface rather than as having an identity that is based on properties of juxtaposed phases.

Direct upscaling can be performed based on microscale information, providing
an opportunity to explore aspects of macroscale system behavior that have
previously been overlooked. Underpinning this exploration is the precise
definition of macroscale quantities. TCAT models are derived from
first principles starting from the microscale. At the macroscale, important
quantities such as phase pressures, specific interfacial areas, curvatures,
and other averaged quantities are defined unambiguously based on the
microscale state

Macroscale quantities can be determined explicitly from microscale
information based on averages. In this work, the form for averages is

The volume fractions, specific interfacial areas, and specific common curve
length are each extent measures that can be formulated as

At the macroscale, various averages arise for the fluid pressures. For flow
processes, the relevant quantity is an intrinsic average of the microscale
fluid pressure,

The capillary pressure of the two-fluid-phase system depends on the curvature
of the interface between the fluids. The curvature of the boundary of phase

Since the capillary pressure is defined for the interface between the two
fluids,

In previously published work, we have considered the impact of non-wetting-phase connectivity in detail

The definitions of pressures provided demonstrate that several different
pressures are of interest for two-fluid systems. In general these pressures
will not be equivalent. Thus, care is needed in analyzing the system state and
in proposing relations among pressures. Typically only the pressure defined
by Eq. (

An experimental approach was sought to investigate the distribution of
capillary pressure in a porous medium system. To meet the objectives of this
work, we needed directly to observe capillary pressure at high resolution,
which requires computation of the average curvature of the fluid–fluid
interface as a function of the averaging region. Because we wished to observe
systems at true equilibrium and knew from recent experience that extended
periods of time are necessary to obtain such a state

A depiction of the two-dimensional micromodel that was used in the displacement experiment. The solid phase consists of pore-space-free solid cylinders of varying radii distributed in the horizontal plane represented by black and the regions accessible to fluid flow by white within the porous medium cell.

Experiments involving two-fluid flow through porous media are typically
conducted using a setup similar to the one shown in Fig.

The solid geometry used in our microfluidic experiments was designed to allow for high capillary pressure at the end of primary drainage. At the wetting-fluid-phase reservoir, a layer of evenly spaced homogeneous cylinders was placed such that the gap between cylinders was uniformly small. This allowed for a large pressure difference between the fluid reservoirs, since the non-wetting fluid phase did not penetrate the wetting-fluid-phase reservoir over a wide range of pressure differences.

The experimental microfluidics setup described in the previous section provides a way to perform traditional two-fluid-flow experiments and observe the internal dynamics of interface kinematics and equilibrium distributions. Microscale-phase configurations can be observed directly, and averaged geometric measures can be obtained from this data. While boundary pressure values are known, the experiment does not provide a way to measure the microscale pressure field. Accurate computer simulation of the experiment can provide this information and can also be used to generate additional fluid configurations that may not be accessible experimentally. In particular, configurations below the irreducible wetting-phase saturation will be considered. The common identification of a saturation as “irreducible” is a misnomer because wetting-phase saturations beneath this value can be achieved through, for example, evaporation or by initializing a saturation below this value in an experimental setup. In this work, simulation is applied in two contexts: (1) to simulate the microscale pressure field based on experimentally observed fluid configurations, and (2) to simulate two-fluid equilibrium configurations based on random initial conditions. Success with the first set of simulations in matching the experiments provides confidence that the results of the second set of computations represent physically reasonable configurations. Here we summarize each of the approaches.

Simulations are performed using a “color” lattice Boltzmann method (LBM).
Our implementation has been described in detail in the literature

The implementation allows us to initialize fluid configurations directly from
experimental images. Segmented images are generated from gray scale camera
data. These images were used to specify the initial position of the phases in
the simulations with high resolution. The micromodel cell was computationally
resolved within a domain that is 20

A set of simulations was also performed based on random initial conditions.
The approach used to generate random fluid configurations and associated
equilibrium states is described in detail by

Phase connectivity presents a critical challenge for the theory and
simulation of two-fluid-phase flow. When all or part of a phase forms a
fully connected pathway through a porous medium, flow can occur without the
movement of interfaces. However, the case where phase sub-regions are not
connected is a source of history-dependent behavior in traditional models.
Traditional models make use of the capillary pressure proposed as
a function of the fluid saturation only,

To calculate

Comparison between the experimentally measured boundary pressure
difference

Pressure transducers located in each of the two-fluid reservoirs were used to
measure experimental boundary pressures for each fluid. The resulting values
of

Phase connectivity has a direct impact on the meaning of the
macroscale experimental measurements:

In the experimental system, an irreducible wetting-phase saturation was
clearly observed as

In light of this result, it is useful to consider alternative means to
generate two-fluid configurations in porous media. For example, suppose a
fluid configuration was encountered with

Comparing capillary pressures measured from random initial conditions with
those measured from experimental initial conditions provides additional
insight. First, the true capillary pressure measurements based on Eq. (

Since the boundary pressure difference

Contour plot showing the relationship

Comparison of the residual errors for the GAM fits that approximate

In this work, we show that the ability to quantitatively analyze the internal structure of two-fluid porous medium systems has a profound impact on macroscale understanding. We considered the behavior of the capillary pressure based on traditional laboratory boundary measurements and compare this to the true average capillary pressure, a state function, determined by directly averaging the curvature of the interface between fluids. We demonstrate that the difference between the phase pressures as measured from the boundary cannot be used to deduce the capillary pressure of the system. In particular, the high capillary pressure measured for irreducible wetting-phase saturation is an artifact of the experimental design. Three important conclusions result.

First, the true capillary pressure measured at traditionally identified irreducible wetting-phase saturation is significantly lower than predicted from boundary pressure measurements. This can be understood based on the underlying phase connectivity. At irreducible wetting-phase saturation, the wetting-phase reservoir pressure no longer reflects the internal pressure of the system since the reservoir does not connect to the remaining wetting phase inside the system.

Second, randomly generated fluid configurations provide a way to access states where the wetting-phase saturation is below the irreducible wetting-phase saturation. By carrying out direct averaging based on these states, the capillary pressure state function can be computed over the full range of possible saturation values, including configurations that are inaccessible from traditional experiments. We note that modified experimental designs could be used to accomplish the same studies.

Third, we show that the equilibrium relationship among capillary
pressure, fluid saturation, and interfacial area is consistent between
randomly initialized configurations used only in computation and
experimentally initialized configurations. Combining the two data sets,
generalized additive models were used to approximate the surface relating

Data associated with this publication have been made publicly available on the
Digital Rock Portal (

All authors participated in the writing of this manuscript. William G. Gray and Cass T. Miller contributed to the introduction, background, and theory, Amanda L. Dye contributed to the microfluidics, and James E. McClure contributed to lattice Boltzmann modeling. All authors contributed to the discussion and conclusions from this work.

The authors declare that they have no conflict of interest.

This work was supported by Army Research Office grant W911NF-14-1-02877, Department of Energy grant DE-SC0002163, and National Science Foundation grant 1619767. An award of computer time was provided by the Department of Energy INCITE program. This research also used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under contract DE-AC05-00OR22725. Edited by: R. Uijlenhoet Reviewed by: two anonymous referees