High Temperature Chemical Processes, Stadiou str., Platani, Patras, Greece

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1 SCA /6 FRO A ULTI-REGION ODEL TO A ULTI-FLOW PATH ODEL: INTERPRETATION OF FLOW TESTS ON HETEROGENEOUS CORES Christos A.Aggelopoulos 1,2 and Christos D. Tsairoglou 1 1 Foundation or Research and Technology Hellas- Institute o Cheical Engineering and High Teperature Cheical Processes, Stadiou str., Platani, Patras, Greece 2 Departent o Physics, University o Patras, Greece This paper was prepared or presentation at the International Syposiu o the Society o Core Analysts held in Calgary, Canada, Septeber 2007 ABSTRACT Iiscible and iscible displaceent tests perored on long undisturbed soil coluns are eployed to quantiy the soil heterogeneity at the acro- and icro- scale, respectively. The transient responses o the solute concentration (breathrough curve) over three cross-sections are inverted by using a ulti-region odel to estiate the longitudinal dispersivity along with the variance o the icroscopic pereability distribution (icro-heterogeneity). The transient responses o the total pressure drop across the soil colun, and luid saturation averaged over ive successive segents are inverted by using a ulti-low path odel to estiate the variance o the acroscopic pereability distribution (acro-heterogeneity) along with the tortuosity, and paraeters quantiying the capillary pressure and relative pereability curves. INTRODUCTION Iiscible and iscible displaceent experients are coonly used to deterine the two-phase low and hydrodynaic dispersion coeicients o soils, respectively. The capillary pressure and relative pereability curves are estiated by inverting the transient responses o pressure drop and axial luid saturation proile, whereas the hydrodynaic dispersion coeicients can be estiated by inverting the transient responses o solute concentration breathrough curves (Tsairoglou et al., 2005). The goal o the present wor is to develop coputational tools that will enable us to interpret the results o low tests, not only by estiating the corresponding ultiphase eective transport coeicients but also by quantiying the heterogeneity o soils at ultiple scales. EXPERIENTAL SETUP An experiental apparatus was constructed to easure the electrical conductance o disturbed and undisturbed sandy soils during iiscible and iscible displaceent experients. The apparatus consists o the saple holder, an in house constructed ultipoint conductivity eter, a HPLC pup, and a data-acquisition syste installed in the host coputer (Aggelopoulos and Tsairoglou, 2005).

2 SCA /6 A core holder was constructed to peror low tests on undisturbed soil coluns. The undisturbed soil saple (length L =32.7 c, diaeter D =4.75 c) is transerred careully ro the sapler into a rubber sleeve which is equipped with rod electrodes (Fig.1) and is closed by placing two coposite caps on its ends. The entire syste is placed inside an adjustable length cell body with the overburden pressure exerted on the sleeve to be slightly higher than the axiu pore pressure so that soil expansion and luid leaages are avoided. Two undisturbed soil coluns (S3, S4) were collected ro dierent depths o the unsaturated zone o an abandoned ilitary airport, situated in North Poland and containated by jet uel. Properties o soil coluns: (S3) porosityφ =0.25, pereability 0 =50 D, oration actor F=10.1; (S4) φ =0.25, 0 =142 D and F=8.3. Beore testing, each soil colun was cleaned careully by treating it with ethanol, ethanol and toluene, and vice versa to ensure the reoval o any traces o jet uel. INVERSE ODELING OF ISCIBLE DISPLACEENT TESTS BSE iages o pore casts (Fig.2a) and Hg injection tests (Fig.2b) have revealed that the pore structure o the investigated soils is characterized by a broad range o pore length scales spanning several orders o agnitude. In order to describe solute transport through such icro-scale heterogeneities (Fig.2a) a 2-paraeter ulti-region odel was used. A two-paraeter ulti-region odel o iscible displaceent The pore space is regarded as an asseblage o intercounicating parallel regions characterized by an area-based distribution o pereabilities ( ; ) σ, where the diensionless icro-scale pereability is deined as = 0, 0 is the easured total pereability, and σ is the standard deviation o the icro-scale pereability distribution unction, and quantiies icro-scale heterogeneities (Fig.2). Solute dispersion in each region is described by the analytic solution o the 1-D advection-dispersion equation in hoogeneous porous edia (Tsairoglou et al., 2005). The intercounication between the various regions ensures that luid pressure does not change over a cross-section and no convective solute transport occurs because o lateral pressure gradient. In addition, any lateral diusive lux is also ignored. Under the aboveentioned assuptions, the cross-sectional averaged solute concentration is given by C = 0 C ( ) d C where the local solute concentration is given by the analytic solution o the advection-dispersion equation. The local low velocity and dispersion coeicient o each region vary according to the relations p u = u D = D + α u a (2) p0 L e L p (1)

3 SCA /6 where u p0 is the ean pore velocity and D e the eective diussivity. For acrodispersion, the exponent a = 1, and the ulti-region odel is quantiied by two paraeters: α,. The Bayesian estiator o ATHENA Visual Studio 10 was used to L σ estiate the longitudinal dispersivity, α L, and σ by itting the predictions o the ultiregion odel siultaneously to solute breathrough curves easured at three cross- sections o soils S3 & S4 (Table 1). The estiated dispersivity (Table 1) is coparable to corresponding values reported in literature (Roychoudhury, 2001); α L is a easure o the length-scale o soil acro-heterogeneities, and changes wealy with low velocity (Table 1). The easured breathrough curves are reproduced satisactorily by the ulti-region odel (Fig.3a,b). It is worth entioning the physical inoration ebedded into the paraeter values o the ulti-region odel. On the one hand, the dispersivity is indicative o the length-scale o acro-heterogeneities. On the other hand, the standard deviation o pereability distribution is indicative o the icro-heterogeneities associated with the variability o the pore length scales (Fig.2). In other words, iscible displaceent experients on undisturbed soils ay be treated as a diagnostic tool o the soil heterogeneity and provide quantitative inoration about it. INVERSE ODELING OF TWO-PHASE FLOW EXPERIENTS During rate-controlled iiscible displaceent experients on both soil coluns S3 and S4, oil breathrough occurred very ast, and the variation o water saturation in the lower segents o the soil colun was quite sall, and very diicult to be detected precisely. Oil oves along preerential low paths and ost o the water is bypassed, resulting in a high value o irreducible wetting phase saturation. In order to account or such characteristics o the 3-D low ield the ulti-low path odel was adopted. ulti-low path odel The broad range o pore length scales probed in undisturbed soils (Fig.2), induces the creation o highly pereable (critical) pathways which transer ost o the low and control the pereability and electrical conductivity o the soil (Tsairoglou et al., 2006). These low pathways exhibit the sallest capillary resistance to the invasion o the nonwetting luid (oil) and hence they also act as preerential oil low paths during drainage. The pore space is represented by a syste o low paths characterized by a distribution o acro-scale pereability, ( ; ) Σ. The standard deviation, Σ, is a easure o the soil acro-heterogeneity, as it indicates the degree o uniority aong the various low paths. Each low path contains a sub-set o the regions entioned earlier (ultiregion odel), and its pereability is governed by the pereability distribution over this sub-set. The low paths are neither straight nor perpendicular to the ain low direction, and their ean length L p is deined by a coon tortuosity actor, λ = L L p, where L is the soil core length. The oil/water displaceent in each low path is regarded as

4 SCA /6 rontal and is described by the irreducible wetting phase saturation, S wi, and end oil 0 relative pereability = ( S = S ) ro ro w wi. In order to copute the capillary pressure o displaceent in each low path, a 2-paraeter Leverett type equation o the or c δ ( ) ( cγ θ ) P cos (3) = ow was used. Using ass and oentu balances, oil/water displaceent was described by a syste o integral & dierential equations with dependent variables the total pressure drop, P t ( τ ) and the positions, ξ ( ) o interaces (ronts) along low paths. This approach is analogous to earlier ones developed or siulating the two-phase low in a disordered syste o parallel capillary tubes (Bartley and Ruth, 1999; Dong et al., 2006). Using Athena Visual Studio 10, the easured transient responses o the water saturation proile and total pressure drop were itted with the ulti-low path odel in order to estiate all pertinent paraeters. The syste o equations is solved iteratively by using the Sipson ethod to integrate the pressure drop over all low paths and a 4 th order Runge-Kutta to calculate the axial position o interaces as a unction o tie. It is evident that the acro-heterogeneity (Table 2) is uch weaer than icro-heterogeneity (Table 1). The transversely averaged luid saturation calculated at the current axial positions o the interaces (ronts), ξ ( ) capillary pressures P c ( ), Eq.(3), to copute P c ( S o ) pereabilities as unctions o the current positions o the ronts in low paths, ( ) were calculated by using the relationships ro ax, was plotted versus the corresponding, ξ 0 ( ξ ) = ( ) d ( ξ ) = ( ) ro ( ξ ) rw ( ), in. Respectively, the relative d ξ, The low levels o oil saturation easured over the lower segents are coparable to the resolution o the technique and or this reason a discrepancy is observed between experient and theoretical prediction (Fig.4). The capillary pressure and relative pereability curves estiated by the ulti-low path odel are shown in Fig.5. REFERENCES 1. Aggelopoulos, C., and C.D. Tsairoglou, Siultaneous deterination o the two-phase low and hydrodynaic dispersion coeicients o soils ro transient iiscible and icible displaceent experients, Proceeding o the 2005 Annual Syposiu o the Society o Core Analysts, Paper SCA , Toronto, Canada, August Bartley, J.T. and D.W. Ruth, Relative pereability analysis o tube bundle odels, Transp. Porous edia (1999) 36, Dong,., F.A.L. Dullien, L. Dai, and D. Li, Iiscible displaceent in the interacting capillary bundle odel. Part II. Applications o odel and coparison o interacting and non-interacting capillary bundle odel, Transp. Porous edia (2006) 63, (4)

5 SCA /6 4. Roychoudhury, A.N., Dispersion in unconsolidated aquatic sedients, Estuar. Coast. Shel. S. (2001) 53, Tsairoglou, C.D.,.A. Theodoropoulou, V. Karoutsos, and D. Papanicolaou, Deterination o the eective transport coeicients o pore networs ro transient iiscible and iscible displaceent experients, Water Resources Research, (2005) 41, W Tsairoglou, C.D.,.A. Ioannidis, E. Airtharaj, O. Vizia, A new approach or the characterization o the pore structure o dual porosity rocs, Proceeding o the 2006 Annual Syposiu o the Society o Core Analysts, Paper SCA , Trondhei, Norway, Sept ACKNOWLEDGEENTS This wor was perored under Global Change and Ecosystes contract nuber SSPI- CT STRESOIL ( ) supported by the European Coission. Table 1. Estiated paraeter values o the ulti-region odel u Soil colun S3 Soil colun S4 p0 (/s) α L (c) σ α L (c) σ 1.50e e e e e e Table 2. Estiated paraeter values o the ulti-low path odel Paraeter Soil S3 Soil S4 Σ λ c 1.2x x10-4 δ S wi ro Figure 1. Rubber sleeve with soil colun and builtin rod electrodes. V Hg (c 3 /g) Saple: C-15 Hg intrusion Hg retraction 0.02 Porosity, φ=0.235 Pore volue, V p = c 3 /g E Pore diaeter, D (µ) Figure 2. BSE iage (agniication=300x) o the 2-D cross-section o saple C15 originating ro the sae zone with that o soil colun S4. Hg intrusion / retraction curves o saple C15

6 SCA / C =(C-C i )/(C 0 -C i ) u p0 =1.50x10-6 /s a L =2.96 c σ = eas. pred. x=25.65 c eas. pred. x=19.45 c eas. pred. x=7.05 c x10 5 2x10 5 3x10 5 4x10 5 5x10 5 C =(C-C i )/(C 0 -C i ) u p0 =3.76x10-6 /s a L =5.96 c σ = eas. pred. x=25.65 c eas. pred. x=19.45 c eas. pred. x=7.05 c x10 5 2x10 5 3x10 5 4x10 5 t (s) Figure 3. Experientally easured vs theoretically predicted (ulti-region odel) solute concentration breathrough curves at various ean pore velocities or soil S3, soil S4. ean oil saturation, <S oj > Observed Predicted ξ 1 ξ 2 ξ j Diensionless tie, τ P = P/(ρ w gl) t(s) τ=tu 0 /(φl) easured Predicted Figure 4. Nuerically predicted vs experientally easured axial saturation proiles, pressure drop across the soil colun Soil S4 history atching Hg intrusion (C15) P c (Pa) 10 rw, ro 0.2 Soil S4 1 ro 0.1 1E-5 1E-4 1E rw 0.1 1E-6 1E-5 1E-4 1E S o Figure 5. Estiated capillary pressure curve vs oil/water capillary pressure curve resulting ro Hg intrusion data. Estiated relative pereability curves. S o