ANALYSIS OF MULTIPHASE FLUID FLOW MOTION IN SANDSTONE MODELS

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 7, Issue 5, September October 2016, pp , Article ID: IJMET_07_05_012 Available online at Journal Impact Factor (2016): (Calculated by GISI) ISSN Print: and ISSN Online: IAEME Publication ANALYSIS OF MULTIPHASE FLUID FLOW MOTION IN SANDSTONE MODELS Pramod Kumar Pant Department of Mathematics, Bhagwant University, Ajmer, Rajasthan, India, Dr. Ajay Kumar Gupta Associate Professor, Bhagwant Institute of Technology, Muzaffar Nagar, India. Dr. Mohammad Miyan Corresponding Author/Associate Professor, Department of Mathematics, Shia P. G. College, University of Lucknow, India. ABSTRACT The Rocks are traditionally seen as crystallite aggregates; in general the mechanical properties and treatment of the composition never includes solid non crystalline components. Such glasses like materials are difficult to detect in small quantities by standard techniques i. e., thin-section polarimetry, x-ray diffraction but if they exist at the critical locations i.e., grain contacts then they would affect the behavior of the rocks considerably. The neutron scattering measurements on a solid sample of the Fontainebleau sandstone have shown the clear evidence for the presence of an unexpected glass like component. The atomic pair distribution function (PDF) analysis shows the significant local structural deviations from the pure quartz. These deviations appear as an excess of 5-10% of nearest neighbor (NN) Si-O and O-O bonds that is consistent with a 5-10% volume fraction of the vitreous silica. These measurements will give some significant information about the still unexplained causes for the peculiar dynamics and the mechanics of the sedimentary rocks. Key words: Fluid flow, Multiphase, Sandstone. Cite this Article: Pramod Kumar Pant, Dr. Ajay Kumar Gupta and Dr. Mohammad Miyan, Analysis of Multiphase Fluid Flow Motion in Sandstone Models. International Journal of Mechanical Engineering and Technology, 7(5), 2016, pp INTRODUCTION The method of numerical solution of the Navier-Stokes equation or stokes equation for the multiphase flow motion was much famous among the researchers. Here, we are computing fluid flow through porous media by using the Lattice Boltzmann method. In the Lattice Boltzmann method, the particles are allowed to move and collide on the lattice. The rules that govern the collisions are designed in such a way that the time average motion of particles is consistent with Navier-Stokes equations. The Lattice-Boltzman method 97 editor@iaeme.com

2 Pramod Kumar Pant, Dr. Ajay Kumar Gupta and Dr. Mohammad Miyan has multiple advantages as its space efficient and time computations are straight forward to parallelize, it handles complex boundaries without any difficulty and it directly links macroscopic and microscopic phenomena. The process of visualization helps us to develop the conceptual framework for understanding the complex physical processes. In particular with the fluid flow, visual comparisons with the experiment are very important for the verification and validation of the models. The visualizations were created by using the types of some scientific computational visualization techniques. The two-dimensional images were generated by taking the cross-section of three-dimensional model and mapping fluid density to intensity and color. In the three-dimensional images, the fluids are depicted with the iso-surfaces and the volume visualization techniques by using intensity, color, and transparency to show the fluid density. The fluid movement is expressed with the help of assembling visualizations at the series of the time steps into animations. The three-dimensional images also use iso-surfaces to delineate the structure of medium like as the sandstone, or the surface of the tube structure within that the fluid flow is being modeled. The fluid flow in porous media is of very important role in several environmental and technological processes like in the recovery of hydrocarbons from oil reservoirs etc. For characterizing the fluid flow in the porous media, the experimental analysis is also used. To simulate flow at pore scale, it is viable to use simplified representations of porous media, like as physical micro-models that can be constructed in the form of the pseudo-2d networks. The lattice-boltzmann method has shown big potential for solving flow in the complex geometries and it has been successfully used in the study of the fluid flow in porous media at pore scale. The most of consolidated sandstones stand out due to their acoustic and mechanical properties. The grain bonds and volume near the contact where stresses are high due to some small cross-section are responsible for peculiar nonlinear properties. The structure of the sandstones appears mainly to the quartz. The pure quartz, however, does not display the non-usual properties i.e., observed in the consolidated sandstones and the detailed picture of short and also long range structure is critical to the deeper understanding of the mechanical behavior of sandstone and many other sedimentary rocks. 2. EXPERIMENTAL ANALYSIS In the present paper, there is discussion about the experiment performed by Alostolmi, U. et al. [1]. They simulated the lattice-boltzmann method and the finite-difference method, in which permeability of the tomographic image of the sample of the Fontainebleau sandstone and of the three different constructions of the sample based on its correlation behavior and properties. The linear size of the sample was 3 mm approximately and the image resolution was 7.5 µm. The accuracies of the two methods are discussed and the results are also compared with the help of the experiment. The Fontainebleau sandstone is the reference standard sample because of its exceptional chemical, micro-structural and crystallographic simplicity. The inter-granular porosity of the sandstone ranges approx. from 0.03 to 0.3. A computed microtomographic image was taken of the micro-plug from a larger original core of porosity ϕ = and permeability k = 1.3 D (1 D = µm 2 ). Figure 2.1 (Visualization of Fluid Flow in Porous Media) 98 editor@iaeme.com

$The light and dark colors indicate high and low velocity respectively) The most common crystallographic techniques i.e., x-ray diffraction, can give the long range average structure of material due to Bragg scattering.$

3 Analysis of Multiphase Fluid Flow Motion in Sandstone Models Figure 2.2 The tomographic image of Fontainebleau sandstone Figure 2.3 (The figure shows the fluid flow in the pore space simulated with the lattice-boltzmann method. The light and dark colors indicate high and low velocity respectively) The most common crystallographic techniques i.e., x-ray diffraction, can give the long range average structure of material due to Bragg scattering. The deviations from this long range average structure show themselves in diffuse scattering present in scattering experiment. An easiest way to study the short, medium and also long range order in the materials is Pair Distribution Function (PDF) method. The PDF method has originate in the study of the materials without long range order like glasses and liquids. More recently, the systems studied using the PDF method has been extended to disordered crystalline and also nano-crystalline objects [2]. The pair distribution function, G(r), describes the probability of the two atoms that is separated by a distance r. The function G(r) is obtained through the Fourier Transform of total diffraction pattern as given by the equation (2.1). = 2 1sin 2.1 Where Q is the momentum transfer function (Q = 4π sin (θ)/λ) and S (Q) is normalized structure function derived from the experimental diffraction intensity [2]. Now, we use neutron PDF analysis for investigating the short and long range structure of the intact sample of the Fontainebleau sandstone and show some valuable, unexplained deviations from the pure crystalline structure indicating the presence of the glassy phase. The solid Fontainebleau sandstone sample was cored from the large rock quarried near Paris, France. The composition was obtained through X-ray diffraction from the powdered piece of the same rock. This sandstone sample contains more than 99% quartz, with trace amounts of hematite, calcite and possibly feldspar. The sample grains are roughly micron in the size. The porosity of sample is 24 ± 0.5% and individual grains are bonded at the several points like contacts. A cylindrical piece of the sandstone, approximately 50 mm high with a diameter 9 mm was sealed in the vanadium can for experiment. The Neutron powder diffraction data were collected at the ambient conditions on the Neutron Powder 99 editor@iaeme.com

4 Pramod Kumar Pant, Dr. Ajay Kumar Gupta and Dr. Mohammad Miyan Diffractometer (NPDF) [9] at Manuel Lujan Neutron Scattering Center of Los Alamos National Laboratory. The instrument was upgraded and optimized for PDF studies of the disordered materials. The total time for data collection of this sample was 24 hours that resulted in an unprecedented level of quality of collected data. The PDF was obtained from diffraction data by using the program PDFgetN [8]. The cut off in the value of Q was given at 40Å 1 at which point the signal to noise ratio became non-favorable. This high cut off results in the high real space resolution in PDF. The experimental data were corrected for incident neutron spectrum, background scattering, absorption, and multiple scattering, and then normalized by the incident flux and the total sample scattering cross section to give S (Q). S (Q) was subsequently Fourier transformed to give G(r) by using equation (3.1). 3. RESULTS AND DISCUSSION 3.1. Rietveld Refinements The Rietveld refinement analysis was performed with a GSAS package [6]. The data was modelled with the crystalline quartz phase. The lattice parameters and atomic positions and also isotropic thermal parameters were refined. The refinement result for 90 degree bank is shown by figure 3.1. The model indexes all peaks and fits data well, indicates that there is no other long range crystalline phases present in sample. The result is consistent with the X-ray diffraction result. It has observed some modulations in the background visible in difference curve shown in figure 3.1. The resulting Si-O and O-O bond lengths are given in table-3.1. The neutron diffraction pattern does not contain an evidence for trace amount of other minerals suggested by the X-ray analysis. Figure 3.1.Rietveld results for Fontainebleau sandstone editor@iaeme.com

5 Analysis of Multiphase Fluid Flow Motion in Sandstone Models Table 3.1 The Average Bond Lengths for Various Refinement Models Model Si-O (Å) O-O (Å) Scale The number in parentheses gives standard deviation for the final digit of the bond lengths. Rietveld 1.611(3) 2.63(2) - 3 to 20Å PDF 1.611(3) 2.64(2) 1.119(4) 1 to 3Å PDF 1.605(3) 2.62(2) 1.002(2) 3.2. PDF Analysis For real space PDF analysis, a crystalline quartz model with atomic positions taken from Rietveld refinement was fit to data. The refinements were completed by using program PDFFIT [8], [9]. In the beginning, standard fitting was employed: thermal parameters, lattice parameters, scaling factors, and peak sharpening due to correlated motion were refined over the range of 1 < r < 20Å. The correlated motion is the tendency of near neighbor atoms to move in phase thereby sharpening the near neighbor PDF peaks. The agreement was not good as compared to the sample of pure quartz. That indicates some local deviations from average structure. For investigating the nature of this disorder, various refinements of PDF over different ranges in r were undertaken. The structural model is valid for the range in r that has been refined. It has refined the PDF using a crystalline quartz model with the range 1 < r < 20Å, shown in figure 3.2.The two observations can be made from inspecting figure 3.2. The first was for atom-atom distances with r > 3Å is excellent. The second was for the first two PDF peaks in the range of 1 < r < 3Å is too poor. The first two PDF peaks in model are somewhat broader compared to the experimental PDF. That is originated by the difficulty of obtaining the suitable value for correlated motion parameter that requires a refinement over the large range in r. However, it is apparent that integrated area under the PDF peaks is larger in data than that in model PDF. The integral is directly proportional to the coordination number [10] in material that means there is excess Si-O and O-O nearest neighbor atom pairs that are not a part of the crystalline quartz phase. We confirmed it by refining PDF over two ranges from 1 < r < 3Å and 3 < r < 20Å. In both cases it has obtained a good agreement and the resulting Si-O and O-O bond lengths and also the resulting scale factors for PDF refinements are given in table 3.1. Now it has found a significantly different scale factor for both refinements that actually should be unity and independent of fitting range in the single phase system. Really, the low r range refinement results in the scale factor of one that means in this range we account for all elements contributing to scattering. For refinement using the range 3 < r < 20Å the calculated PDF needs to be multiplied with the larger scale to match experimental values. All this points the existence of excess O-O and Si-O bonds that accounts for 5 10% of material. By considering the size of the grains in Fontainebleau sandstone and the amount of the excess phase, it is clear that additional material cannot be ascribed to disorder in the few atomic layers at grain surface editor@iaeme.com

6 Pramod Kumar Pant, Dr. Ajay Kumar Gupta and Dr. Mohammad Miyan Figure 3.2 The PDF of the Fontainebleau sandstone with 1 to 20Å fit model. The data are marked by the crosses, the model by circles, and the difference curve by a line on the separate (lower) axis Evidence about an Amorphous Phase In the speculation about the source of the extra O-O and Si-O bond contributions, other SiO2 diffraction patterns were analyzed. The PDF for amorphous SiO2 collected on NPDF is given by figure 3.3 by dotted circles. It should be noted that sample in this case was fused silica tube measured as the background for next experiment. The statistics are not nearly on same level as the sandstone data. However, it is immediately apparent that the short range structure is much similar to the Fontainebleau sandstone data shown by crosses in figure 3.3. That is of course to be expecting, since local structure consists of SiO4 tetrahedra in the both cases. It is noticed that there are only two sharp structural peaks in spectrum of amorphous silica and they present in figure 3.3. Consequently, the Fontainebleau PDF data would be understood as the two phase system consisting of crystalline phase and an amorphous silica phase. Figure 3.3 The nearest neighbor peaks for the amorphous SiO 2. The sandstone data are marked with crosses while amorphous SiO 2 data are marked with dotted circles editor@iaeme.com

7 Analysis of Multiphase Fluid Flow Motion in Sandstone Models For the further investigation of this idea, we refined a two phase model using PDFFIT. The both phases were structurally derived from single phase model result. Now, the amorphous phase was contributing correlations up to rmax = 3Å. Structural parameters and also the scale factor were refined for every phase. The refinement result is given in figure 3.4. It can be seen a much improved overall agreement. Now, the low r region is in need of the further improvement. It is likely that the correlated motion correction used is very simple for rigid units like as the SiO4 tetrahedra. It can be found that the resulting phase fractions are somewhat dependent on details of the modelling of the correlated motion. So, in all cases resulting in the comparable agreement with the data we find the phase fractions of amorphous phase of 5 10% consistent with estimate given previously. Figure 3.4 Two phase modeling for Fontainebleau sandstone. The data are displayed with crosses and the model is given by circles. The difference curve appears below the two data sets on a separate axis. 4. DISCUSSION The results found in the present paper show that there is a non-crystalline contribution structurally consistent with a volume fraction of 5 10% of amorphous silica in Fontainebleau sandstone. It has been observed that the crystalline quartz cans amorphize under stress [7]. The point like contacts between grains observed in Fontainebleau sandstone amplifies the stress due to small area for transmitting force, so that there is a likelihood that some significant volume in region around the contact may have been permanently amorphized by the pressure during sedimentation and bond formation. The stress gradient is too high in this near-contact volume that may produce strain broadening of the diffracted peaks in the applied stress experiments like [3]. If, however, the region is amorphous, there will be no contribution to Bragg peaks, despite large stresses and gradients. It can be analyzed the previous data carefully for these effects at 5 10% volume level. The difference in mechanical and annealing and fracture properties of the amorphous and crystalline silica in these high stress regions may be significant elements in explanation of many macroscopic behaviors in the consolidated sandstones. Finally it is worth pointing out that the present example clearly illustrates the power of PDF technique in characterizing the materials with mixed amorphous and crystalline phases editor@iaeme.com

8 Pramod Kumar Pant, Dr. Ajay Kumar Gupta and Dr. Mohammad Miyan 5. REFERENCE [1] Aaltosalmi, U. 2005; Fluid flow in porous media with the Lattice-Boltzmann method, University of Jyväskylä, Research report No. 3/2005, Finland, pp [2] Chupas, Peter J., Xiangyun Qiu, Jonathan C. Hanson, Peter L. Lee, Clare P. Grey and Simon J. L. Billinge, 2003; Rapid Acquisition Pair Distribution Function (RA-PDF) Analysis, Journal of Applied Crystallography, [3] Darling, A. C., Mau, B., Blattner, F. R. and Perna, N. T., 2004; Mauve: multiple alignment of conserved genomic sequence with rearrangements, Genome Res.,14(7): DOI: /gr [4] Martys N. S. and Hagedorn, J. G., 2002; Multiscale modeling of fluid transport in heterogeous materials using descrete Boltzmann methods, Materials and Structures, 35, pp [5] Martys N. S. and Douglas, J. F., 2001; Critical Properties and Phase Separation in Lattice Boltzmann Fluid Mixture, Physical Review E, 63, [6] Larson, A. C. and Von Dreele, R. B. 2000; "General Structure Analysis System (GSAS)", Los Alamos National Laboratory Report LAUR [7] Page, K. L. et al., 2004; Local atomic structure of Fontainebleau sandstone: Evidence for an amorphous phase, Geophysical Research Letters, 31(24). DOI: /2004GL [8] Peterson, P. F., Gutmann, M., Proffen, Th. and Billinge, S. J. L. 2000; PDFgetN: a user-friendly program to extract the total scattering structure factor and the pair distribution function from neutron powder diffraction data, J. Appl. Cryst., 33, [9] M. Y. Gokhale and Ignatius Fernandes, Lattice Boltzmann Simulation of Non-Newtonian Fluid Flow in a LID Driven Cavity. International Journal of Mechanical Engineering and Technology (IJMET), 5(8), 2014,pp [10] Proffen, Th., Page, K. L., S. E. McLain et al., (2004); Local atomic structure of Fontainebleau sandstone: Evidence for an amorphous phase?, GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L doi: /2004GL [11] Soper, A. K. and Luzar, A., 1992; A neutron diffraction study of dimethyl sulphoxide water mixtures, J. Chem. Phys., 97, 1320 (1992). [12] Son, Y., Martys, N. S., Hagedorn, J. G. and Migler, K., 2003; Suppression of Capillary Instability of a Polymeric Thread via Parallel Plate Confinement, Macromolecules, 36, pp [13] Simon, J. L., 2004; BillingeThe atomic pair distribution function: past and present, Z. Kristallogr. 219, [14] Miyan, M. 2016; The Fluid Flow Motion through Porous Media for Slightly Compressible Materials, International Journal of Pure and Applied Researches, Vol. 3(1); pp [15] Prof. Dr. Qasim S. Mahdi and Noor AM. Mohammed, Heat Transfer Augmentation of Laminar Nano fluid Flow in Horizontal Tube Inserted with Twisted Tapes. International Journal of Mechanical Engineering and Technology (IJMET), 7(3), 2016,pp editor@iaeme.com

9 Analysis of Multiphase Fluid Flow Motion in Sandstone Models [16] Miyan, M. 2016; The Two Phase Flow Motions and their P. D. F. Representations, International Journal of Pure and Applied Researches, Vol. 1(1); pp [17] Miyan, M. 2016; Lattice Boltzmann Method for PAKKA-Model Structures in Fluid Flow Motion, International Journal of Pure and Applied Researches, Vol. 3(2); pp [18] Pant, P. K., 2016; Analysis of Multiphase Flow under the Ground Water, International Journal of Pure and Applied Researches, Vol. 1(1), pp

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