NUMERICAL MODEL FOR NON-DARCY FLOW THROUGH COARSE POROUS MEDIA USING THE MOVING PARTICLE SIMULATION METHOD

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1 THERMAL SCIENCE: Year 2018, Vol. 22, No. 5, pp NUMERICAL MODEL FOR NON-DARCY FLOW THROUGH COARSE POROUS MEDIA USING THE MOVING PARTICLE SIMULATION METHOD Introducton by Tomok IZUMI a* and Junya MIZUTA b a Graduate School of Agrculture, Ehme Unversty, Ehme, Japan b Fukuyama Cty Offce, Hroshma, Japan Orgnal scentfc paper A numercal model for non-darcy flow, whch occurs when water moves through coarse porous meda under hgh Reynolds number, s developed. The governng equaton for ncompressble vscous flow through porous meda s composed of a contnuty equaton and a momentum equaton, whch s the Naver-Stokes equaton wth an addtonal non-lnear resstance term based on Forchhemer s law. For the dscretzaton scheme, movng partcle smulaton method s employed. In order to assess the model valdty, seepage experments n dfferent knds of coarse porous meda are mplemented, and then reproducblty of the numercal results s examned. From the results, t s found that the computatonal flow veloctes at mddle part of porous meda are n good agreement wth expermental ones whle veloctes at outflow are overestmated. Key words: non-darcy flow, Forchhemer s law, partcle method, numercal smulaton, seepage experment, coarse porous meda It s mportant to properly manage groundwater resources for sustanable water use n small ranfall area where water supply deps on groundwater at a large rate. Although groundwater flow or water movement through porous meda s generally descrbed by Darcy s law, t has been ndcated that flow under hgh Reynolds numbers does not always satsfy the law [1]. Ths flow s known as non-darcy flow, and t occurs n groundwater flow through coarse gravel rverbeds [2] and n ppe flow on heterogeneous mountan slopes [3]. Among them, studes on groundwater flow through coarse gravel rverbeds n fluval fan area are useful for sustanable groundwater use snce t can be consdered that groundwater s recharged through nteracton between rver water and groundwater. Thus, t s mportant to develop a numercal model for non-darcy flow n order to analyze groundwater flow and/or to estmate groundwater recharge because feld observatons of groundwater are relatvely dffcult compared wth those of surface water such as rver water. Forchhemer equaton and Izbash equaton have been proposed as moton equaton for non-darcy flow, and the model parameters ncluded n the equatons have also been dscussed. For example, Sdropoulou et al. [4] compared and evaluated expressons of Forchhemer coeffcents based on theoretcal approaches and expermental analyss. Son et al. [5] determned Izbash coeffcents based on expermental results and dscussed the factors nflu- * Correspondng author; e-mal: t_zum@agr.ehme-u.ac.jp

2 1956 THERMAL SCIENCE: Year 2018, Vol. 22, No. 5, pp encng the coeffcents. Studes on numercal model for non-darcy flow are found n the feld of coastal engneerng. Huang et al. [6] proposed a 2-D numercal model to smulate the nteracton between a soltary wave and submerged porous breakwater. Shao [7] presented an ncompressble smoothed partcle hydrodynamcs (ISPH) flow model to smulate wave nteractons wth a porous medum. Akbar and Namn [8] presented an ISPH flow model whch can solve two porous and pure flud flows smultaneously to smulate wave nteracton wth porous structures. Akbar [9] ntroduced modfed movng partcle method n porous meda (MMPP) for smulatng a flow nteracton wth porous structures. In the feld of thermal scence, Hamdan [10] performed a numercal nvestgaton on the flow and convectve heat transfer characterstcs for sothermal parallel plate channel flled wth porous meda usng the fnte dfference method. Motsa and Anmasaun [11] developed a spectral collocaton method to solve a problem of unsteady heat and mass transfer past a sem-nfnte vertcal plate wth dffuson-thermo and thermophoress effects n the presence of sucton. Sayehvand et al. [12] numercally nvestgated forced convecton heat transfer from two tandem crcular cylnders embedded n a porous medum usng the fnte volume method. In ther works, Forchhemer equaton s used as moton equaton. Studes on numercal model for non-darcy flow cannot be found n the feld of groundwater engneerng. In ths study, a numercal model for non-darcy flow s thus proposed as a powerful tool for sustanable groundwater use n the area where water supply deps on groundwater at a large rate. Accordng to prevous studes, the governng equaton s composed of a contnuty equaton and a momentum equaton, whch s the Naver-Stokes equaton to whch a non-lnear resstance based on Forchhemer s law nstead of Darcy s law s added. For the dscretzaton, however, movng partcle smulaton (MPS) method [13], one of the Lagrangan partcle approach, s employed nstead of the Euleran grd approach such as the fnte dfference and fnte volume method. Ths s because dscretzaton of convectve dervatve term, whch occurs undesrous numercal oscllaton, s not requred orgnally. MPS method was orgnally proposed by Koshzuka et al. [14] and was composed of partcle nteracton models localzed by usng a weght functon. Koshzuka et al. [14] nvestgated the accuracy of MPS method n test calculatons; a parabolc profle of the flow velocty was obtaned n a Poseulle flow. However, numercal stablty of the ncompressblty model was senstve to a correcton parameter used n the method. Therefore, Koshzuka and Oka [13] modfed MPS method of [14] n terms of the weght functon and modellng of ncompressblty. ISPH method [15] s also one of the Lagrangan partcle approach. The dfference between MPS and ISPH method s found n weght functon (or kernel). Whle ISPH method requres dfferentablty for the weght functon, MPS method does not. Thus, MPS method s easy to handle for modellng. In order to assess the model valdty, snce the numercal accuracy of MPS method was shown as mentoned above, seepage experments n dfferent knds of coarse porous meda are mplemented, and then reproducblty of the numercal results for the expermental results s examned. Numercal model for non-darcy flow Numercal model for non-darcy flow proposed n ths study, governng equaton, dscretzaton method computaton algorthm, and boundary condtons are explaned. Governng equaton The governng equaton for ncompressble vscous flow through porous meda s composed of a contnuty equaton and a momentum equaton, whch s the Naver-Stokes equaton to whch a non-lnear resstance based on Forchhemer s law s added:

3 THERMAL SCIENCE: Year 2018, Vol. 22, No. 5, pp wth Cr Du n Dt w 1 0 t (1) 2 p E u agu bgu u g (2) w C 1 n w r 1 (3) nw w E (4) n w where u s the mean velocty vector, C r the nerta coeffcent, n w the porosty, t the tme, w the water densty, p the pressure, E the effectve knematc vscosty, a and b T are the coeffcents of Forchhemer s law, respectvely, g {0,0, g} the gravtatonal acceleraton vector, a non-dmensonal coeffcent whch accounts for the extra amount of momentum that s needed to accelerate the same volume of water n a porous medum [16], and w the knematc vscosty. The frst term on the rght hand sde of eq. (2) denotes the pressure force caused by the pressure gradent, the second term denotes the vscous force, the thrd and fourth terms denote the resstance force by the porous meda, and the last term denotes the gravtatonal force. Snce eq. (2) corresponds to the momentum equaton based on the Darcy s law when the coeffcent of b = 0, the coeffcent a s descrbed as 1/k s where k s s the hydraulc conductvty. The coeffcent b s dealt wth a fttng parameter n ths study because t deps on the propertes of the porous medum only. Dscretzaton method The MPS method [13] s used for the dscretzaton scheme n ths study. Ths method dscretzes dfferental operators such as gradents or the Laplacan, based on the nteracton model between partcles wthn an nteracton area determned by effectve radus r e, fg. 1. In the nteracton model, the gradent and Laplacan models are defned: D ( )w( ) (5) 0 j r 2 j r rj r n 0 j rj r Fgure 1. Effectve radus and nteracton area n the nteracton model 2D ( )w( ) (6) 2 0 j rj r n 0 j where s a scalar varable, D 0 s the number of space dmensons, n 0 the partcle number densty (whch should be kept constant to satsfy the ncompressble condton), r the poston vector, w(r) the weght functon (whch descrbes the weght of nteracton between two partcles separated by dstance r), the model parameter, and and j are partcle numbers, respectvely. The weght functon, model parameter, and the partcle number densty are defned:

4 1958 THERMAL SCIENCE: Year 2018, Vol. 22, No. 5, pp re 1 ( r re ) w( r) r 0 ( r re ) 2 rj r w( rj r ) j j w( r r ) n w( r r ) j j j (7) (8) (9) Computaton algorthm A sem-mplct tme marchng scheme s used for the computaton. Fgure 2 shows the computaton algorthm. In each tme step k, the dffuson and source terms are explctly calculated, and then the temporal veloctes u * and temporal postons r * are obtaned: n t u u ( u agu bgu u g) (10) * k w 2 k k k k E Cr * k * r r u t Next, the Posson equaton for pressure (descrbed below) s teratvely solved usng the ncomplete Cholesky conjugate gradent (ICCG) method, after whch the tme pressure values p k+1 are obtaned: * 0 2 k 1 Cr w n n p 2 0 n ( ) w t n (11) (12) Fnally, the next tme veloctes u k+1 and postons r k+1 are obtaned, after the temporal values have been corrected accordng to the pressure gradent. Boundary condtons A free surface boundary s determned based on partcle number densty. When the partcle number densty satsfes: n 0 n (13) Fgure 2. Computaton algorthm each partcle s regarded as a free surface, where s a parameter below 1.0. Under the condton, pressure s set to zero n the pressure calculaton. A wall boundary s represented as fxed partcles. Accordng to r e, more than three layers of fxed partcles are used to represent the wall n order to calculate the partcle densty number accurately. Velocty s set at zero for all fxed partcles. Pressures are calculated only at the fxed partcles n the nnermost layer. The fxed partcles n the other layers are only requred to keep the partcle number densty around n 0 wthout calculatng pressure. An nflow boundary s represented as movng partcles. Veloctes and pressures are gven to the partcles at the boundary.

5 THERMAL SCIENCE: Year 2018, Vol. 22, No. 5, pp Valdaton Valdty of the numercal model proposed n ths study s nvestgated n terms of the reproducblty for expermental results snce the numercal accuracy of MPS method was shown n the prevous studes. In ths secton, seepage experments usng dfferent knds of porous meda are frstly explaned, condtons of smulaton by MPS method are then descrbed, and computaton results are fnally dscussed. Table 1. Physcal propertes of porous meda Experment of flow through porous meda Seepage experments usng three knds of porous meda are mplemented. Physcal propertes of the meda are obtaned by laboratory tests and summarzed n tab. 1. The porosty, mean partcle dameter, and hydraulc conductvty for the porous meda are measured by water evaporaton method, seve analyss, and constant head permeablty test, respectvely. The schematc of the experment s shown n fg. 3. Each porous medum s set at the sze of m (heght length wdth). Water s poured nto the rght hand sde of sol layer to keep water depth h 2 constant. Then outflow from the left hand sde of sol layer and water depth at the nflow and outflow s (h 1 and h 2 ) are measured. For each porous medum, at most sx knds of hydraulc gradent are set. Smulaton condtons Numercal smulatons of 2-D seepage flow n the cases correspondng to the seepage experments mentoned above are mplemented. Fgure 4 shows an example of the ntal state of the smulaton doman for the MPS method, usng computatonal partcles wth 0.01 m n dameter. The rght hand sde s set as the nflow boundary and the nflow partcles are generated to keep water depth h 2 constant. The smulatons are contnued untl achevng steady-state wth tme step of seconds. Parameters used n the smulatons are summarzed n tab. 2. Results and dscusson Gravel 1 Gravel 2 Glass beads Porosty Mean partcle dameter [m] Hydraulc conductvty [ms 1 ] Fgure 3. Schematc of the experment Fgures 5-7 show the relatons between the velocty wth hydraulc gradent n order to compare computatonal wth expermental flow velocty at the mddle part and outflow n each porous medum. Fgures 5-7 also show the plots based on Darcy s law n terms of expermental flow velocty at the outflow, whch are specfed as u D. The relaton representng lnearty such as u D means that flow can be regarded as Darcy flow, and the relaton re-

6 1960 THERMAL SCIENCE: Year 2018, Vol. 22, No. 5, pp presentng non-lnearty, on the other hand, means that flow can be regarded as non- Darcy flow. Also, the plots of expermental flow velocty have the error bar of 20% for the expermental data. Addtonally, tab. 3 shows the errors between computatonal and expermental flow velocty at the mddle part and outflow n each porous medum. Note that superscrpt o and m denote outflow and mddle, respectvely. The coeffcents b n gravel 1, gravel 2, and glass beads whch are to be fttng parameters n ths study are 60 m 1, 60 m 1, and 110 m 1, respectvely. From fgs. 5-7, t can be seen that all relatons between the expermental flow velocty and hydraulc gradent are nonlnear and thus flows through porous meda are non-darcy flow. It s also found that the computatonal flow veloctes at the mddle part agree wth expermental ones. Except for the plots at hgher hydraulc gradent, computatonal flow veloctes are plotted wthn 20% of expermental ones. From tab. 3, errors are n the range of m/s. However, all of the computatonal flow veloctes at the outflow are overestmated. Errors are n the range of m/s and t seems that errors become large as porosty and/or mean partcle dameter ncrease. To nvestgate the dscrepancy n velocty, the coeffcents n Forchhemer s law are dscussed. Snce the coeffcent b s dealt wth a fttng parameter n computaton, the coeff- cent a s focused on. Table 4 shows the coeffcents of computatons and experments. In computaton, a s defned as a recprocal number of hydraulc conductvty. The coeffcents of experment are obtaned from the fttng curve for expermental results shown n fgs From tab. 4, t s found that values n computaton are approxmately ten tmes larger than those n experment. Thus t s consdered that modellng of coeffcent a nfluences on the accuracy of computatonal re- Fgure 4. Intal state of the smulaton doman for the MPS method Table 2. Parameters used n the smulatons Unt Value Knematc vscosty ( w ) m 2 s Densty of water partcles ( w ) kgm 3 1,000 Partcle dameter (l 0 ) m 0.01 Effectve radus (r e ) m 2.1 l Fgure 5. Comparson of computatonal wth expermental velocty n gravel 1 Fgure 6. Comparson of computatonal wth expermental velocty n gravel 2

7 THERMAL SCIENCE: Year 2018, Vol. 22, No. 5, pp sults. There are some prevous studes on determnaton of the coeffcents a and b n Forchhemer s law, e. g. [4]. Therefore t s necessary to exam the prevous models for the coeffcents a and b n Forchhemer s law and to revse descrpton of a and b n order to mprove the reproducblty of the numercal model developed. Concluson A numercal model for non-darcy flow through coarse porous meda s developed. The governng equaton s a contnuty equaton and momentum equaton whch s Naver-Stokes equaton to whch a non-lnear resstance based on Forchhemer s law nstead of Darcy s law s added. MPS method s used for the dscretzaton scheme. In order to exam- Table 3. Errors between computatonal and expermental flow velocty (= u com u exp )* at the mddle and outflow Acknowledgment Ths study s supported by JSPS 17K Fgure 7. Comparson of computatonal wth expermental velocty n glass beads Hydraulc (1) (2) (3) (4) (5) (6) gradent Outflow Mddle Outflow Mddle Outflow Mddle Outflow Mddle Outflow Mddle Outflow Mddle Gravel Gravel Glass beads * n [ms 1 ] Table 4. Comparson of coeffcent a n Forchhemer s law a Computaton Experment Gravel Gravel Glass beads ne the model valdty, seepage experments for three knds of porous meda are mplemented. From the comparson of computatonal and expermental results, t s found that the computatonal flow veloctes at mddle part of porous meda are n good agreement wth expermental ones whle veloctes at outflow are overestmated. Improvement of the reproducblty of the numercal model should be nvestgated for future task. Nomenclature a coeffcent of lnear term n Forchhemer s law, [sm 1 ] b coeffcent of non-lnear term n Forchhemer s law, [s 2 m 2 ] C r nerta coeffcent, [ ] D 0 number of space dmenson, [ ] d 50 mean partcle sze, [m] g gravtatonal acceleraton, [ms 2 ] g gravtatonal acceleraton vector, [ms 2 ] h 1 water depth at an outflow of porous meda, [m] h 2 water depth at an nflow of porous meda, [m] k s hydraulc conductvty, [ms 1 ] n 0 partcle number densty to be kept constant, [ ]

8 1962 THERMAL SCIENCE: Year 2018, Vol. 22, No. 5, pp n 0 partcle number densty, [ ] n w porosty, [ ] p pressure, [Nm 2 ] r poston vector, [m] r dstance between two partcles ( rj r ), [m] r e effectve radus, [m] t tme, [s] u mean velocty vector, [ms 1 ] w() weght functon Greek symbols parameter used when determnng a free surface, [ ] coeffcent whch accounts for the extra amount of momentum, [ ] References relaxaton coeffcent n Laplacan model of MPS model, [ ] E effectve knematc vscosty, [m 2 s 1 ] w knematc vscosty of water, [m 2 s 1 ] w water densty, [kgm 3 ] scalar varable Superscrpt k tme step Subscrpts, j partcle number exp expermental com computatonal [1] Bear, J., Dynamcs of Fluds n Porous Meda, Amercan Elsever Publshng Company, Inc., Cambrdge, Massachusetts, USA, 1972 [2] Yamada, H., et al., Measurng Hydraulc Permeablty n a Streamed Usng the Packer Test, Hydrologcal Processes, 19 (2005), 13, pp [3] Ktahara, H., Characterstcs of Ppe Flow n a Sub-surface Sol Layer on a Gentle Slope (II): Hydraulc Propertes of Ppes (n Japanese), Journal of the Japanese Forest Socety, 71 (1989), 8, pp [4] Sdropoulou, M. G., et al., Determnaton of Forchhemer Equaton Coeffcents a and b, Hydrologcal Processes, 21 (2007), 4, pp [5] Son, J. P., et al., An Expermental Evaluaton of non-darcan Flow n Porous Meda, Journal of Hydrology, 38 (1978), 3-4, pp [6] Huang, C.-J., et al., Structural Permeablty Effects on the Interacton of a Soltary Wave and a Submerged Breakwater, Coastal Engneerng, 49 (2003), 1-2, pp [7] Shao, S., Incompressble SPH Flow Model for Wave Interactons wth Porous Meda, Coastal Engneerng, 57 (2010), 3, pp [8] Akba, H., Namn, M. M., Movng Partcle Method for Modelng Wave Interacton wth Porous Structures, Coastal Engneerng, 74 (2013), Apr., pp [9] Akbar, H., Modfed Movng Partcle Method for Modelng Wave Interacton wth Mult Layered Porous Structures, Coastal Engneerng, 89 (2014), July, pp [10] Hamdan, M. O., An Emprcal Correlaton for Isothermal Parallel Plate Channel Completely Flled wth Porous Meda, Thermal Scence, 17 (2013), 4, pp [11] Motsa, S. S., Anmasaun, I. L., A New Numercal Investgaton of Some Thermos-Physcal Propertes on Unsteady MHD non-darcan Flow Past an Impulsvely Started Vertcal Surface, Thermal Scence, 19 (2015), 1, pp. S429-S258 [12] Sayehvand, H.-O., et al., Numercal Analyss of Forced Convecton Heat Transfer from Two Tandem Crcular Cylnders Embedded n a Porous Medum, Thermal Scence, 21 (2017), 5, pp [13] Koshzuka, S., Oka, Y., Movng-Partcle Sem-mplct Method for Fragmentaton of Incompressble Flud, Nuclear Scence and Engneerng, 123 (1996), 3, pp [14] Koshzuka, S., et al., A Partcle Method for Incompressble Vscous Flow wth Fragmentaton, Computatonal Flud Dynamcs J., 4 (1995), 1, pp [15] Shao, S., Lo, E. Y. M., Incompressble SPH Method for Smulatng Newtonan and non-newtonan Flows wth a Free Surface, Advances n Water Resources, 26 (2003), 7, pp [16] van Gent, M. R. A., Porous Flow Through Rubble-mound Materal, Journal of Waterway, Port, Coastal, and Ocean Engneerng, 121 (1995), 3, pp Paper submtted: December 31, 2017 Paper revsed: March 28, 2018 Paper accepted: March 28, Socety of Thermal Engneers of Serba. Publshed by the Vnča Insttute of Nuclear Scences, Belgrade, Serba. Ths s an open access artcle dstrbuted under the CC BY-NC-ND 4.0 terms and condtons.