Ijmet 06 10_018

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 6, Issue 10, Oct 2015, pp. 161-170, Article ID: IJMET_06_10_018
Available online at
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=6&IType=10
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
UPSCALING OF MULTIPHASE FLOW
PROCESSES IN HETEROGENEOUS
POROUS MEDIA
Pramod Kumar Pant
Department of Mathematics, Bhagwant University, Ajmer, Rajasthan, India
Dr. Mohammad Miyan
Associate Professor (Supervisor), Department of Mathematics,
Shia P. G. College, University of Lucknow, India
ABSTRACT
The present article gives a mathematical model describing flow of two
fluid phases in a heterogeneous porous medium. The medium contains
disconnected inclusions embedded in the background material. This
background material is characterized by higher value of the non-wetting
phase entry pressure than the inclusions that causes non-standard behavior of
medium at the macroscopic scale. During which the displacement of the non-
wetting fluid by the wetting fluid, some portions of the non-wetting fluid
become trapped in the inclusions. Secondly, if the medium is initially saturated
with the wetting phase then it starts to drain only after the capillary pressure
exceeds the entry pressure of the background material. These effects cannot be
shown by standard upscaling approaches based on the assumption of local
equilibrium of the capillary pressure. So, we can propose a relevant
modification of the upscaled model obtained by asymptotic homogenization.
The modification relates the type of flow equations and the calculation of the
effective hydraulic functions. The heterogeneities of the porous media are
typically well represented in the global fine-scale solutions. In particular, the
connectivity of the media is properly embedded into the global fine-scale
solution. Thus, for the porous media with channelized features, where there
are high or low permeability regions have long-range connectivity. Hence this
type of approach is expected to work better.
Keywords- Heterogeneous, Multiphase, Porous, Upscaling
Cite this Article: Pramod Kumar Pant and Dr. Mohammad Miyan. Breaking
Upscaling of Multiphase Flow Processes In Heterogeneous Porous Media,
International Journal of Mechanical Engineering and Technology, 6(10),
2015, pp. 161-170.
http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=6&IType=10

Pramod Kumar Pant and Dr. Mohammad Miyan
1. INTRODUCTION
As analyzed by Gamal A. A. et al. [1], there are large varieties of natural and artificial
porous materials encountered in practice, such as: soil, sandstone, limestone,
ceramics, foam, rubber, bread, lungs, and kidneys. Aquifers by which water is
pumped and filters for purifying water, reservoirs that yield oil or gas, packed and
fluidized beds in the chemical engineering and the root zone in agricultural industry
may serve as additional examples of porous media domains. In these the common to
all of these examples is the observation that part of the domain is occupied by a
persistent solid phase, known as the solid matrix. The remaining part is known as the
void space, which may be occupied either by a single phase fluid or a number of fluid
phases. In the case of number of fluid phases, each phase occupies a distinct separate
portion of the void space. So that a porous material may be regarded as a material, in
which the solid portion is continuously distributed throughout the complete volume to
form a loosely connected matrix and voids i.e., pores inside the solid matrix which is
filled with fluids. For a connected porous medium the porosity is defined as the
fraction of the total volume of the medium that is occupied by void space. Therefore,
the remaining is the fraction that is occupied by the solid. For a disconnected porous
media i.e., some of the pore space is separated from the remainder, an effective
porosity that is the ratio of the connected void to the complete volume, has to be
defined. A system of identical spherical particles of small radius and equal sizes
affords a simple model of a porous body. These particles may be arranged in different
ways [1].
The multiphase flow through porous media is characterized by complex geometry
and by intimate contact between the solid matrix and the fluid. The extent of this
contact depends on the characteristic features of the porous media, like as porosity.
The microscopic nature of the flow is much complicated and random. The analysis of
the heat transfer inside the tortuous void passages in these medium by taking into
account the interaction between different mechanisms of transport does not appear to
be possible analytically. So that the solutions founded by applying the classical
models of fluid mechanics to both the fluid and solid phases are misleading and have
no interest in practice. This is also due to the lack of information concerning the
microscopic configuration of the interface boundaries and the fluid paths as it moves
inside the porous media. Hence a continuum model has to be defined in order to
simulate mathematically the transport phenomena existing in these media. A quasi-
homogeneous continuum model is defined for the porous media for which the phases
are assumed to behave as a continuum which fills up the entire domain i.e., each
phase occupies its own continuum. The spaces occupied by these overlapping
continue is referred to as the macroscopic space. This continuum model of the porous
media has some advantages. Firstly, it does not need the exact configuration of the
interface boundaries to be specified for acquiring the knowledge of which is an
invisible task. Secondly, it describes processes occurring in porous media in terms of
differentiable quantities, then enabling the solution of many problems by employing
methods of mathematical analysis. These advantages are at the expense of the loss of
detailed information related to the microscopic configuration of the interface
boundaries and the actual variation of quantities within every phase. Now the
macroscopic effects of these factors are still retained in the form of coefficients,
whose structure and relationship to the statistical properties of the void space
configuration could be analyzed and determined [1].

Upscaling of Multiphase Flow Processes In Heterogeneous Porous Media
The porous media modeling demands thorough explanation of rock and fluid
properties. According to Farad Kamyabi Trondheim [9], the tortuous structure of
porous media naturally gives to complicated fluid transport through the pores. As
there is no interaction between fluids, single phase flow is comparatively easy to
visualize. In this kind of system, flow efficiency is a function of permeability which is
a property of rock and independent of the fluid saturating it. Single phase fluid flow
through a porous medium is well described by Darcy’s law, and the primary elements
of the subject have been well understood for many decades. Multiphase flow through
porous media is important for a various applications such as carbon dioxide
sequestration, and enhanced oil recovery. These often involve the displacement of a
non-wetting invading fluid from a porous medium by a wetting fluid, a physical
phenomenon known as imbibitions. Modeling of multiphase flow, on the other hand is
still an enormous technical challenge. For capturing the best model of multiphase
flow, true analysis of fluid interactions such as capillary pressure and relative
permeability is inevitable. By considering these parameters, the complexity of
numerical calculation in reservoir simulation process will increase. In many cases,
these two parameters will create instability in numerical simulation. The numerical
analysis of multiphase flow is much interesting, and there exists a growing body of
literature for addressing and analyzing the subject. The modeling of such physical
flow process mainly requires solving the mass and momentum conservation equations
associated with equations of capillary pressure, saturation and relative permeability
[9]. P. Bastian has given the definitions related to homogeneous and heterogeneous
behavior. The porous medium is said to be homogeneous with respect to a
macroscopic quantity if that parameter has the same value throughout the domain,
otherwise it is said to be heterogeneous. Macroscopic tensor quantities can also vary
with direction, in that case the porous medium is called anisotropic with respect to
that quantity, and otherwise it is called isotropic. The corresponding macroscopic
quantity called permeability will be anisotropic [16].
By a point of view of practical engineering one of the major design difficulties in
dealing with multiphase flow is that the mass, momentum and energy transfer rates
and processes that can be quite sensitive to the geometric distribution or topology of
the components within the flow. For example, the geometry may strongly affect the
interfacial area available for mass, momentum or energy exchange between different
phases. So that the flow within each phase or component will clearly depends on that
geometric distribution. Then we recognize that there is a complicated two-way
coupling between the flow in each of the phases or components and the geometry of
the flow as well as the rates of change of that geometry. Now the complexity of this
two-way coupling presents a major challenge in the study of multiphase flows and
there is much that remains to be done before even a superficial understanding is
achieved. The appropriate starting point is a phenomenological description of the
geometric distributions or flow patterns that are observed in common multiphase
flows [7].
2. MULTIPHASE FLOW PATTERNS
Christopher E. Brennen has said that the particular category of geometric distribution
of the components is said to be a flow pattern or flow regime and many of the names
given to these flow patterns such as annular flow or bubbly flow, are now quite
standard [7]. Generally the flow patterns are recognized by visual inspection, though
other means such as analysis of the spectral content of the unsteady pressures or the
fluctuations in the volume fraction have been devised for those cases in which visual

information is difficult to obtain. For some of the simpler flows, such as those in
vertical or horizontal pipes, various numbers of investigations have been conducted
for determining the dependence of the flow pattern on component volume fluxes, on
volume fraction and on the fluid properties such as density, viscosity and surface
tension. The results of the analysis are often shown in the form of a flow pattern map
that recognizes the flow patterns existing in various parts of a parameter space defined
by the component flow rates. The flow rates used may be the volume fluxes, mass
fluxes, momentum fluxes or other similar quantities depending on the analysis. The
most widely used of these flow pattern maps is that for the horizontal gas or liquid
flow. The generalization of these flow pattern studies and the various empirical laws
extracted from these are a common feature in reviews of multiphase flow. The
boundaries between the various flow patterns in a flow pattern map exist since a
regime becomes unstable as the boundary is approached and growth of this instability
causes transition to other flow pattern. Now the laminar-to-turbulent transition in
single phase flow, these multiphase transitions can be unexpected since they may
depend on otherwise minor features of the flow, like as the roughness of the walls or
the entrance conditions. So, the flow pattern boundaries are not distinctive lines, these
are more poorly defined transition zones. Since, there are other serious difficulties
with the existing analysis on flow pattern maps. In which, one of the basic fluid
mechanical problems is that these maps are dimensional and so that apply only to the
specific pipe sizes and fluids employed by the analyzer. The number of investigators
has attempted to find generalized coordinates that can allow the map to analyze
different fluids and pipes of different sizes. However, such generalizations can only
have limited value since several transitions are represented in most flow pattern maps
and the respected instabilities are governed by different sets of fluid properties. As an
example, one transition might occur at a critical Weber number, whereas other
boundary may be characterized by a particular Reynolds number. So, even for the
simplest duct geometries, there does not exists any universal, dimensionless flow
pattern maps that incorporate the complete parametric dependence of the boundaries
on the characteristics of fluid. For single phase flow it is well established that an
entrance length of 25 to 50 diameters is necessary to establish fully developed
turbulent pipe flow. The suitable entrance lengths for multiphase flow patterns are
less well established and it is possible that some of the experimental analyses are for
temporary flow patterns. Now the implicit supposition is that there exists a unique
flow pattern for given fluids with given flow rates. Consequently, there will be several
possible flow patterns whose existence may depend on the initial conditions,
especially on the manner in which the multiphase flow is generated. Hence, there
remain many challenges and questions associated with a fundamental understanding
of flow patterns in multiphase flow and considerable suitable research is necessary
before reliable design methods become available. So, we will give emphasis on some
of the qualitative features of the boundaries between flow patterns and on the
underlying instabilities that give rise to these transitions [7].
3. UPSCALING METHODOLOGY
The upscaling in the heterogeneous flow was deeply discussed by Rainer Helmig
[19]. According to him the environmental remediation and protection has provided an
especially important motivation for the multiphase research in the course of the last
20 years [15], [18]. The release of non-aqueous phase liquids, both lighter and denser
than water i.e., LNAPLs and DNAPLs into the environment is a problem of particular
importance to researchers and analyzers [10], [12], [13], [14]. Latterly, such work has

focused on the construction of mathematical models which can be used to test and
advance our understanding of complex multiphase systems, that evaluate risks to
human and ecological health both and aid in the design of control and remediation
methods. The one of the foremost problems facing the reliable modeling of
multiphase porous medium systems is the problem of scale. In general, a model is
assembled from a set of conservation equations and constitutive or closure relations.
Analyzer must identify constitutive relations and system-specific parameters those are
appropriate for the spatial and temporal scales of interest. The disparity exists
between the measurement scale in the field or laboratory and the scale of the model
application in the field. Now, neither the measurement nor the field application scales
are commensurate with the scale of theoretical or empirical process analyses [5], [6].
Both closure relation forms and parameters are subject to change when the system of
concern is heterogeneous in some relevant respect. The figure-2 graphically depicts
the range of spatial scales of concern in a typical porous medium system. This shows
two important aspects of these natural systems: several orders of magnitude in
potentially relevant length scales exist, and heterogeneity occurs across the entire
range of relevant scales. A similar range of temporal scales exists as well, from the
pico-seconds over which a chemical reaction can occur on a molecular length scale to
the decades of concern in restoring sites contaminated with DNAPLs [2], [21], [22].
Figure-1 DNAPLs below the water table
(Figure taken from Friedrich Schwille 1988, Lewis Publishers)
The careful definition of relevant length scales can clarify any investigation of
scale considerations, although such definitions are a matter of choice and modeling
approach [8]. Now, we define the following length scales of concern: the molecular
length scale, which is of the order of the size of a molecule; the micro-scale or the
minimum continuum length scale on which the individual molecular interactions can
be neglected in favor of an ensemble average of molecular collisions; the local scale,
that is the minimum continuum length scale at which the micro-scale analysis of fluid
movement through individual pores can be neglected in favor of averaging the fluid
movement over a representative elementary volume (REV), so that this scale is also
called the REV-scale; the meso-scale, that is a scale on which local scale properties
vary distinctly and markedly; and the mega-scale or field-scale. The measurements or
observations can give representative information across this entire range of scales
which depends on the aspect of the system analyzed and the nature of the instrument
used to take the observations. For that reason, we do not specifically define a
measurement scale [11], [12], [13].
For the minimum continuum length scale, we take the boundaries of the different
grains directly for consideration. For the micro-scale, we look at a various type of

pore throats and pore volumes. In both scales, we average over the properties of the
fluids only like as, density and viscosity. On REV-scale, we average over both fluid-
phase and solid–phase properties. In the figure- 3, there are schematically the
averaged properties e.g., the porosity. On averaging over a representative elementary
volume (REV), we suppose that the averaged property P does not oscillate
significantly. In the figure- 3, this is the case in the range of V’ to V” with V’< V”, so
that any volume V with V’≤ V ≤ V” can be taken as REV. Accordingly, we do not
assume any heterogeneities on the REV-scale. For our model, we assume that the
effects of the sub–REV–scale heterogeneities are taken into considerations by
effective parameters.
The super REV scale heterogeneities have to be taken for consideration by
applying different parameters according to our interest. Both steady transitions and
jumps have to be taken for the parameters. We can show these heterogeneities with
jumps within the spatial parameters to block heterogeneities. For this, we can consider
that the block heterogeneities can be described by sub-domains with defined
interfaces. In the present paper, we do not take heterogeneities on the field scale.
Since the scale of interest in the present paper is ultimately the meso-scale, so one can
usually ignore molecular-scale phenomena, although these effects are embodied in
continuum conservation equations and associated closure relations. However, we can
consider all other important and relevant scales in the analysis of multiphase porous
medium systems. By concept view point, one wishes to explain phenomena at a given
scale by using the minimum information from smaller scales. The process gives rise
to quantities at each scale that may not be useful at smaller scales. Such as, the fluid
pressures are not suitable for individual collisions at the molecular scale, and point-
wise fluid saturations or volume fractions do not necessarily reflect the micro-scale
fluid composition at the point. From the concept view point that satisfying theoretical
approach, analyzer that could fundamentally increase the field’s maturity, he must
give the method for incorporating models on a given scale sparingly into models on
the next larger scale using different things.
Figure 2 Different scales for flow in porous media

Figure 3 Different schematically scales for flow in porous media (with respect to
figure-2)
For example, the micro-scale models can be developed to describe fluid flow in
individual pores by solving the Navier-Stokes equations [17] or Boltzmann equation
[23] with a suitable domain. These methods can be used to model systems relating of
many pores, even of a size equivalent to REV for a REV-scale porous medium
system. These approaches have been used to develop REV-scale closure relations
based upon micro-scale methods [11]. This kind of connection does not exist across
suitable length scales for all the phenomena taking into consideration for multiphase
porous medium. The suitable questions come about the importance of heterogeneities
for specific processes, the suitable form and parameters of the closure relations for
heterogeneous multiphase porous medium, and effective methods of simulating such
systems. Except the problems of scale, we need suitable efficient multiphase flow and
transport simulators that can represent the dominant flow and transport mechanisms in
heterogeneous multiphase porous medium. The REV-scale modeling problem has
been separated from the more general problem of cascading scales, although the two
problems are formally entwined. The two have been split apart due to urgent need for
responding such problems, even before we understand them fully. The operational
separation of local scale modeling by the more detailed modeling methods has
resulted in many practical models and experimental analyses of complex multiphase
phenomena [4]. The engineering has played an important role in the implementation
this practical response. By the meso-scopic view point, the two basic classes of
multiphase applications have received attention in the literature and deserve for
further consideration.
 The imbibitions of DNAPL into a heterogeneous porous medium [4].
 The removal of a DNAPL originally in the state of residual saturation [3].
The (a) determines the morphology of the DNAPL distribution at residual
saturation. That determines the initial condition of the second problem (b). Whereas
the public is mostly related with remediating DNAPL contaminated soils, many
questions concerning DNAPL imbibitions and removal still hinder our remediation
efforts. Now the overall goal of the work is to advance our understanding of models
for heterogeneous multiphase porous medium across a range of scales [11]. The
specific objectives will be as follows:
1. To evaluate the role of the spatial scale in determining the dominant process for
multiphase flows.
2. To investigate the influence of pore-scale heterogeneities on micro-scale and REV-
scale flow processes.
3. To summarize conventional continuum-scale mathematical models.

4. To evaluate the accuracy and efficiency of a set of spatial and temporal discretization
approaches [20] for solving the multiphase flow and transport phenomena.
5. To compare numerical simulations with experimental observations for heterogeneous
meso-scopic systems of the phenomena.
6. To point out the way toward important future areas of research in this field.
4. RESULTS AND DISCUSSIONS
The traditional approaches for scale up of pressure equations generally include the
calculation of effective media properties. In these approaches the fine scale
information is built into the effective media parameters, and then the problem related
to coarse scale is solved. For more discussion on upscaled modeling in the multiphase
flows, a number of approaches have been introduced where the coupling of small
scale information is taking through a numerical formulation of the different problems
by incorporating the fine features of the problem into base elements. In the present
analysis, there is a development of a new type of approach by using finite by volume
framework. The finite volume methods will be much suitable in these applications
i.e., the flow in the heterogeneous porous media. Now our method for analysis is
similar to the multi scale finite element methods. Hence we can propose a relevant
modification of the upscaled model obtained by asymptotic homogenization. The
modification relates the type of flow equations and the calculation of the effective
hydraulic functions. The heterogeneities of the porous media are typically well
represented in the global fine scale solutions. In particular, the connectivity of the
media is properly embedded into the global fine scale solution. Thus, for the porous
media with channelized features, where there are high or low permeability regions
have long range connectivity. Hence this type of approach is expected to work better
for multiphase flow in heterogeneous porous media.
5. CONCLUSIONS
During the upscaling process for heterogeneous media, new effects may evolve on the
macro scale which does not occur on the local scale. First, a saturation-dependent
anisotropy of the relative permeability saturation relationship can be observed. The
multiphase flow behavior amplifies the anisotropy of the effective conductivity as
compared to single-phase flow. Furthermore, direction dependent macroscopic
residual saturations evolve, at which the phases are immobile. Residual saturations of
the non wetting phase are an important parameter for assessing the success of
remediation processes. Moreover, hysteresis effects can be observed on the macro
scale, though on the local scale no process dependent parameters were applied. The
application of our upscaling procedure proves that the structures of a porous medium
on the local scale, such as layers or lenses, have an important influence on the
effective parameters on the macro scale. The incorporation of the geometry of these
structures in the upscaling process enhances the quality of the effective parameters.
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