Upscaling of two-phase flow processes in heterogeneous porous media: determination of constitutive relationships

Transcription

1 28 Calibration and Reliability in Groundwater Modelling: A Few Steps Closer to Reality (Proceedings of ModclCARH'2002 (Prague. Czech Republic. June 2002). IAHS Publ. no Upscaling of two-phase flow processes in heterogeneous porous media: determination of constitutive relationships RAINER HELMIG University of Stuttgart, Institute of Hydraulic Engineering, Pfaffenwaldring 61, D Stuttgart, Germany helmig(5),i ws.uni-stuftgart.de CHRISTOPHERUS BRAUN Technologieberatung Grundwasser und Umwelt GmbH (TGU), Maria Trost 3, D Koblenz, Germany SABINE MAM IIFY University of Stuttgart, Institute of Hydraulic Engineering, Pfaffenwaldring 61, D Stuttgart, Germany Abstract For the numerical simulation of groundwater contamination by e.g. DNAPL, effective parameters representing the macroscale are needed to ensure reasonable computation times. When we use effective parameters on the macroscale, they should reproduce the influence of local-scale heterogeneities on the multiphase flow processes. Our upscaling approach attempts to incorporate the influence of local-scale heterogeneities on the multiphase flow behaviour at the macroscale in the constitutive relationships, taking into account that, in addition to gravity, capillary forces, which are most noticeable at the interface of heterogeneities, dominate the flow processes. The resultant relative permeability-saturation relationship for the macroscale shows a saturation dependent anisotropy. Moreover residual saturations of the nonwetting phase and hysteresis effects are introduced at the macroscale although they do not have a counterpart at the local scale. Keywords effective parameters; hysteresis; relative permeability-saturation relationship; saturation dependent anisotropy; upscaling INTRODUCTION When subsurface contamination by dense non-aqueous phase liquids (DNAPL) is to be assessed, e.g. with regard to the application of remediation techniques, numerical simulations are often employed. Ideally, effective parameters for the macroscale are required which make a discrete reproduction of local-scale heterogeneities futile. Local-scale heterogeneities can have an important impact on multiphase flow processes in the subsurface. Zones of lower permeability and higher entry pressure than the surrounding material cause a pooling of DNAPL until the critical saturation and thus the entry pressure of the lower permeability region is overcome. The pooling causes a lateral spreading of DNAPL. The residual saturation of the DNAPL is also increased. These effects should be captured by the effective parameters for the macroscale. We achieve a representation of these processes in the constitutive

2 Upscaling of two-phase flow processes in heterogeneous porous media 29 relationships by means of an upscaling procedure from the local to the macroscale which takes into account the structures of the local-scale heterogeneities. At the local-scale, hysteresis and residual saturations are not considered. Possible dynamic effects or functional dependencies of the constitutive relationships, e.g. on the interfacial area, are also neglected. Furthermore, the porous medium is assumed to be isotropic on the local scale. Thus, the intrinsic permeability and the relative permeabilitysaturation relationship are scalar functions. The constitutive relationships on the local scale are parameterized after the well-known approach of Brooks & Corey/Burdine. Description of the upscaling approach In our upscaling approach, one crucial assumption is an equilibrium of (capillary) forces, achieved by reducing the number of variables. The application of one capillary pressure at the boundaries gives a unique saturation distribution. To compute the saturation distribution for a heterogeneous system, made up of different REV, a sitepercolation model (e.g. Yortsos et al, 1993; Kueper & Girgrah, 1994) can be applied, for example. The weighted arithmetic mean of the saturation distribution yields one value of the averaged, macroscopic capillary pressure-saturation relationship. The procedure is repeated with different capillary pressures until the macroscopic P C S U. relationship is obtained with sufficient accuracy. With the saturation distribution, it is also possible to compute a distribution of the relative permeabilities and thus the conductivities. On the macroscale, the conductivities are averaged either arithmetically or harmonically, depending on the structure and the flow direction, with a renormalization method (Williams, 1992; King, 1996). With the renormalization, the size of the REV of the considered medium is also identified. The conductivities thus derived are then set into a relation with the saturated conductivity. As a result, we get a macroscopic relative permeability saturation relationship. One advantage of the approach is its easy applicability. Furthermore, its physical motivation facilitates a physical interpretation of the upscaled parameters. In the following, the approach will be applied to a geostatistical example and to a bench-scale laboratory experiment. Geostatistical example With the help of a geostatistical example we will show how hysteresis effects can arise at the macroscale although, on the local scale, hysteretic behaviour is neglected in the constitutive relationships. A permeability field (log(&o,rc/) = , standard deviation s = , anisotropy factor a = 10) is generated with rf2d (random field 2d; van Lent, 1992; Dykaar & Kitanidis, 1992). Then the Brooks-Corey parameter P c/, the entry pressure, is scaled (with the help of the Leverett function): with the reference values for Âro^/as the average permeability and P ( i.ref= 550 Pa (see Fig. 1). The Brooks-Corey parameter X is constant (k = 2.0) over the whole domain. (1)

3 30 Rainer Helmig et al. Pc [Pa] ^ X Fig. 1 Generated heterogeneity field (64 2 cells) (Braun, 2000) water saturation [-] Fig. 2 Macroscopic P C ~S. for the heterogeneity field (Braun, 2000). The macroscopic P C S W relationship for this heterogeneity field is computed as described above. Although on the local scale no hysteresis effects were assumed in the P C S W relationship for the different REV, process-dependent relationship evolve at the macroscale (see Fig. 2). For low capillary pressures, the nonwetting phase is trapped in the domain, leading to a macroscopic residual saturation of the nonwetting phase. The trapping occurs when high permeability regions with low entry pressure still show nonwetting phase saturations at low capillary pressures whereas surrounding regions with lower permeabilities and higher entry pressure are already fully saturated with water and thus do not provide a mobility for the DNAPL to flow out of the high permeability regions. The relative permeability-saturation relationship for the drainage and the imbibition cycle both show a saturation-dependent anisotropy (see Fig. 3 and 4).

4 Upscaling of two-phase flow processes in heterogeneous porous media S water saturation S [-] Fig. 3 k-s relationships for imbibition of the heterogeneity field (Braun, 2000) water saturation S [-] Fig. 4 kr-s relationships for the drainage cycle of the heterogeneity field (Braun, 2000). The relative permeabilities of the nonwetting phase do not differ from drainage to imbibition because hysteresis effects would arise at small capillary pressures when the saturation of the nonwetting phase is small. But the nonwetting phase only becomes mobile at saturations higher than S = This is a threshold value, below which there is no continuous path for the nonwetting phase in the domain. It corresponds to a residual saturation. The threshold value depends on various parameters, e.g. the dimension of the percolation model or the kind of spatial discretization (Stauffer, 1985). The relative permeabilities of the drainage and imbibition cycle of the wetting phase show a similar behaviour for saturations of up to S w ~ 0.9. Then, the macroscopic residual saturation of the nonwetting phase is reached, i.e. the saturation of the wetting phase cannot be increased further. At the residual saturation of the nonwetting phase, the relative permeability of the wetting phase becomes 1. Thus, a hysteresis effect on the macroscale arises not only for the P c -S w relationship but also for the k n -S w relationship. Verification and application example The upscaling approach was verified with a numerical experiment with a periodically layered medium where DNAPL infiltrated from the top (Braun et al, 1998). Here we

5 32 Rainer Helmig et al. Table 1 Properties of the sands for the VEGAS experiment (Allan, 1998). Fine sand Medium sand Coarse sand Permeability k 0 (m 2 ) 6.38 x lfr x lfr x lfr 10 Entry pressure P rf (Pa) Form factor X (-) Residual saturation S wr (-) Residual saturation S m. (-) gravel source zone constant DNAPL flux fine lenses coarse lenses 120 cm pressure tensiometers very fine sand Fig. 5 Set-up of infiltration experiment: distribution of the lenses (Allan, 1998). will demonstrate the application to an infiltration experiment (Fig. 5) conducted in the VEGAS facility, University of Stuttgart (Allan et al, 1998). Table 1 shows the properties of the sands used. During the experiment, DNAPL (trichlorethylene, TCE, density p = 1460 kg nr 3, dynamic viscosity u = 0.57 x 10" J Pa s) infiltrated for 2970 s with a constant flux of 29 ml min" 1 along 0.02 m of the top. Using this experiment, we compare two different upscaling approaches. First, we apply a simple geometric average of the parameters, taking only the volume fractions of the different sands into account. We then derive the averaged constitutive relationships for a geometric arrangement of the sands by means of our upscaling approach. For the simple geometric average, the resulting constitutive relationships resemble those of the medium sand because it has the highest volume fraction (see Fig. 6 and Fig. 7). The relative permeability-saturation relationships are isotropic. Figure 8 shows the results of the application of these constitutive relationships in a numerical simulation of the experiment (isolines) compared to the experimental results (hatched areas) after one hour. It is obvious that the averaging approach of the weighted geometric mean does not suffice to reproduce the experimental results correctly. The infiltration of the DNAPL in the vertical direction is overestimated whereas the dispersion in the horizontal direction is underestimated. Consequently, in addition to the information about the volume fractions, the geometric arrangement of the materials in the tank (lenses) is taken into account by applying

6 Upscaling of two-phase flow processes in heterogeneous porous media water saturation S [-] Fig. 6 Relative permeability-saturation relationship for the case of the geometric average (Braun, 2000) \ I rd Q. ' ' G water saturation S w [-] Fig. 7 Capillary pressure-saturation relationship for the case of the geometric average (Braun, 2000) m x TCE, observed: Fig. 8 Comparison of the saturation distribution of the DNAPL after one hour: numerical simulation with effective parameters derived by a weighted geometric mean (isolines) and experimental results (hatched areas) (Braun, 2000).

7 T ; 34 Rainer Helmig et al. U if coarse sand lortso [height 2 cm) J 1 1 ] fine sand loose (height 2 cm) crj - 0_ ' ' Q_ - fd s~ to (/) CD Q '5_ nj o I - _!,!,!,!,! i!,!,!, j, f water saturation S w [-] Fig. 9 Capillary pressure-saturation relationship for the case of the upscaling approach applied to a medium with lenses (Braun, 2000). water saturation S w [-] Fig. 10 Relative permeability-saturation relationship for the case of the upscaling approach applied to a medium with lenses (Braun, 2000). our upscaling approach (see Fig. 9). The relative permeability-saturation relation now shows an anisotropy for both phases (see Fig. 10). The anisotropy can be related to characteristic points in the capillary pressuresaturation relations of the different sands. As long as the capillary pressure is higher than the entry pressure of the coarse sand and lower than the entry pressure of the medium sand, the DNAPL can only infiltrate into the coarse sand. However, as the lenses are not continuous, this does not effect the mobility of the DNAPL. With decreasing water saturation, the capillary pressure increases until the entry pressure of the medium sand is exceeded. Only then does the DNAPL start to be mobile. Taken that the fluid is no longer mobile at a relative permeability lower than 1 x l(h, the mobilization of the DNAPL in the horizontal direction sets in at saturations of

8 Upscaling of two-phase flow processes in heterogeneous porous media 35 = and in the vertical direction at saturations of S, m, = These saturations can be seen as residual saturations of the nonwetting fluid. These residual saturations have to be exceeded before the nonwetting fluid becomes mobile and spreads in the respective direction. Moreover, the mobility of the DNAPL in the vertical direction is inhibited by the fine sand lens until its entry pressure is overcome as well. Thus, a larger DNAPL saturation has to be reached for the DNAPL to reach high mobilities in the vertical direction compared to the horizontal direction. The comparison of the numerical simulation with the effective parameters for the REV with the lenses to the experimental results (Fig. 11) shows a good agreement between the simulation and the experiment. The simulation reproduces the spreading in the horizontal direction. The isolines encompass the main part of the infiltrated area and do reproduce the characteristics of the mobilization process correctly m z X TCE, observed: W///W/W/M Fig. 11 Comparison of the saturation distribution of the DNAPL after one hour: numerical simulation with effective parameters derived with the upscaling approach for a medium with lenses (isolines) (Braun, 2000). CONCLUSION During the upscaling process for heterogeneous media, new effects may evolve on the macroscale which do not occur on the local scale. First, a saturation-dependent anisotropy of the relative permeability-saturation relationship can be observed. The twophase flow behaviour 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 nonwetting phase are an important parameter for assessing the success of remediation processes. Moreover, hysteresis effects can be observed on the macroscale, 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 macroscale. The incorporation of the geometry of these structures in the upscaling process enhances the quality of the effective parameters.

9 36 Rainer Helmig et al. REFERENCES Allan, J., Ewing, J., Braun, J. & Helmig, R. (1998) Scale effects in multiphase How modelling. In: Remediation of Chlorinated and Récalcitrant Compounds (ed. by G. B. Wickramanayake & R. E. Hinchee) (First Int. Conf., May 1998, Monterey, California), Battelle Press, Columbus, Ohio, USA. Braun, C. (2000) Ein Upscaling-Verfahren fur Mehrphascnstromungen in potosen Medien (An upscaling approach for multiphase flow in porous media). Scientific report 103, Dissertation, Institute of Hydraulic Engineering, University of Stuttgart, Stuttgart, Germany. Braun, C, Hassanizadeh, S. M. & Helmig, R. (1998) Two-pliase flow in stratified porous media effective parameters and constitutive relationships. In: Computational Methods in Water Resources XII (ed. by V. N. Burganos, G. P. Karatzas & A. C. Payatakes), Vol. I, Comp. Mech. Publ., Southampton, UK and Boston, USA. Dykaar, B. B. & Kitanidis, P. K. (1992) Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach. 1 Method. Water Resour. Res. 28(4), King, P. R. (1996) Upscaling permeability: Error analysis for renormalisalion. Transp. Porous Media 23, Kuepcr, B. H. & Girgrah, B. (1994) Effective large-scale parameters for two-phase flow in heterogeneous porous media. In: Transport and Reactive Processes in Aquifers (ed. by T. H. Dracos & F. StaufFer) (Proc. IAHR Symp., April 1994, Ziirich, Switzerland), Balkema, Rotterdam, The Netherlands. Van Lent, T. (1992) Numerical spectral methods applied to flow in highly heterogeneous aquifers. PhD Thesis, Stanford University, California, USA. Stauffer, D. ( 1985) Introduction to Percolation Theory. Taylor & Francis, London, UK. Williams, J. K. (1989) Simple renormalisalion schemes for calculating effective properties of heterogeneous reservoirs. In: Mathematics of Oil Recovery (ed. by P. R. King) (First Int. Conf., July 1989, Cambridge, UK), Clarendon Press, Oxford, UK. Yortsos, Y. C, Satik, C., Bacri, J. C. & Salin, D. (1993) Large-scale percolation theorv of drainage. Transp. Porous Media 10,