UPSCALING OF MULTIPHASE FLOW PROCESSES IN HETEROGENEOUS POROUS MEDIA

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 6, Issue 10, Oct 2015, pp , Article ID: IJMET_06_10_018 Available online at ISSN Print: and ISSN Online: 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 nonwetting 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 editor@iaeme.com

2 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 quasihomogeneous 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] editor@iaeme.com

3 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 editor@iaeme.com

4 Pramod Kumar Pant and Dr. Mohammad Miyan 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 editor@iaeme.com

5 Upscaling of Multiphase Flow Processes In Heterogeneous Porous Media 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 editor@iaeme.com

6 Pramod Kumar Pant and Dr. Mohammad Miyan 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 fluidphase 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 pointwise 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 editor@iaeme.com

7 Upscaling of Multiphase Flow Processes In Heterogeneous 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 REVscale flow processes. 3. To summarize conventional continuum-scale mathematical models editor@iaeme.com

8 Pramod Kumar Pant and Dr. Mohammad Miyan 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. REFERENCES [1] A. A. Gamal and Piotr Furmański. (1997). Problems of modeling flow and heat transfer in porous media, Biuletyn Instytutu Techniki Cieplnej Politechniki Warszawskiej, [2] A. S. Mayer and C. T. Miller (1993). An experimental investigation of pore-scale distributions of nonaqueous phase liquids at residual saturation. Transport in Porous Media, 10(1): editor@iaeme.com

9 Upscaling of Multiphase Flow Processes In Heterogeneous Porous Media [3] A. S. Mayer and C. T. Miller (1996). The influence of mass transfer characteristics and porous media heterogeneity on nonaqueous phase dissolution. Water Resources Research, 32(6): [4] B. H. Kueper, W. Abbott and G. Farquhar (1989). Experimental observations of multiphase flow in heterogeneous porous media. Journal of Contaminant Hydrology, 5: [5] B. H. Kueper and E. O. Frind (1991). Two-phase flow in heterogeneous porous media, 1. Model development. Water Resources Research, 27(6): [6] B. H. Kueper and E. O. Frind (1991). Two-phase flow in heterogeneous porous media, 2. Model application. Water Resources Research, 27(6): [7] C. E. Brennen (2005). Fundamentals of Multiphase Flows. California Institute of Technology Pasadena, California, Cambridge University Press, [8] D. T. Hristopulos and G. Christakos (1997). An analysis of hydraulic conductivity upscaling. Non-linear Analysis, 30(8): [9] Farad Kamyabi Trondheim. (2014). Multiphase Flow in Porous Media. TPG4920- Petroleum Engineering, Master Thesis, Norway, 1. [10] J. C. Parker and R. J. Lenhard (1987). A model for hysteretic constitutive relations governing multiphase flow 1. Saturation-pressure relations. Water Resources Research, 23(12): [11] M. Hilpert, J. F. McBride and C. T. Miller (2001). Investigation of the residualfunicular non wetting-phase-saturation relation. Advances in Water Resources, 24(2): , Mini-Workshop: Numerical Upscaling, Theory and Applications [12] M. Miyan and P. K. Pant (2015). Analysis of Multiphase Flow in Porous Media for Slightly Compressible Flow and Rock, J. Math. Comput. Sci. 5, No. 3, pp [13] M. Miyan and P. K. Pant (2015). Flow and Diffusion Equations for Fluid Flow in Porous Rocks for the Multiphase Flow Phenomena, American Journal of Engineering Research (AJER), Vol. 4, Issue-7, pp [14] National Research Council. (1984). Groundwater contamination: Overview and recommendations. In National Research Council, editor, Groundwater Contamination, National Academy Press, Washington, DC, [15] National Research Council. (2000). Natural Attenuation for Groundwater Remediation, National Academy Press, Washington, DC. [16] P. Bastian and Heidelberg. (1999). Numerical Computation of Multiphase Flows in Porous Media, pp.11. [17] P. M. Adler (1992). Porous Media: Geometry and Transports. Butterworth- Heinemann, Boston. [18] R. Helmig (1997). Multiphase flow and transport processes in the subsurface - a contribution to the modeling of hydrosystems, Springer Verlag. [19] R. Helmig (2005). Upscaling of two-phase flow processes in highly heterogeneous porous media including interfaces on different scales. Report No. 20/2005, Mini-Workshop: Numerical Upscaling: Theory and Applications, [20] R. Helmig and R. Huber (1998). Comparison of Galerkin-type discretization techniques for two phase flow in heterogeneous porous media, Advances in Water Resources, 21(8): editor@iaeme.com

10 Pramod Kumar Pant and Dr. Mohammad Miyan [21] R. J. Lenhard and J. C. Parker (1987). Measurement and prediction of saturationpressure relationships in three-phase porous media systems. Journal of Contaminant Hydrology, 1: [22] R. J. Lenhard, J. C. Parker and J. J. Kaluarachchi (1989) A model for hysteretic constitutive relations governing multiphase flow 3. Refinements and numerical simulations. Water Resources Research, 25(7): [23] S. Chen and G. D. Doolen (1998). Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 30: [24] Dr P.Ravinder Reddy, Dr K.Srihari, Dr S. Raji Reddy. Combined Heat and Mass Transfer In MHD Three-Dimensional Porous Flow With Periodic Permeability & Heat Absorption, International Journal of Mechanical Engineering and Technology, 3(2), 2012, pp [25] Ajay Kumar Kapardar, Dr. R. P. Sharma, Experimental Investigation of Solar Air Heater Using Porous Medium, International Journal of Mechanical Engineering and Technology, 3(2), 2012, pp [26] Ajay Kumar Kapardar, Dr. R. P. Sharma, Numerical and CFD Based Analysis of Porous Media Solar Air Heater, International Journal of Mechanical Engineering and Technology, 3(2), 2012, pp editor@iaeme.com