Sanchez-Vila, X.; Guadagnini, A.; Fernàndez-Garcia, D. Conditional probability density functions of concentrations for mixing-controlled reactive transport in heterogeneous aquifers. (English) Zbl 1213.76208 Math. Geosci. 41, No. 3, 323-351 (2009). Summary: This paper presents an approach conducive to an evaluation of the probability density function (pdf) of spatio-temporal distributions of concentrations of reactive solutes (and associated reaction rates) evolving in a randomly heterogeneous aquifer. Most existing approaches to solute transport in heterogeneous media focus on providing expressions for space-time moments of concentrations. In general, only low order moments (unconditional or conditional mean and covariance) are computed. In some cases, this allows for obtaining a confidence interval associated with predictions of local concentrations. Common applications, such as risk assessment and vulnerability practices, require the assessment of extreme (low or high) concentration values. We start from the well-known approach of deconstructing the reactive transport problem into the analysis of a conservative transport process followed by speciation to (a) provide a partial differential equation (PDE) for the (conditional) pdf of conservative aqueous species, and (b) derive expressions for the pdf of reactive species and the associated reaction rate. When transport at the local scale is described by an Advection Dispersion Equation (ADE), the equation satisfied by the pdf of conservative species is non-local in space and time. It is similar to an ADE and includes an additional source term. The latter involves the contribution of dilution effects that counteract dispersive fluxes. In general, the PDE we provide must be solved numerically, in a Monte Carlo framework. In some cases, an approximation can be obtained through suitable localization of the governing equation. We illustrate the methodology to depict key features of transport in randomly stratified media in the absence of transverse dispersion effects. In this case, all the pdfs can be explicitly obtained, and their evolution with space and time is discussed. Cited in 2 Documents MSC: 76S05 Flows in porous media; filtration; seepage 86A05 Hydrology, hydrography, oceanography Keywords:reactive transport; heterogeneous porous media Software:PhreeqcRM; PHREEQC PDFBibTeX XMLCite \textit{X. Sanchez-Vila} et al., Math. Geosci. 41, No. 3, 323--351 (2009; Zbl 1213.76208) Full Text: DOI Link References: [1] Andricevic R, Cvetkovic V (1998) Relative dispersion for solute flux in aquifers. J Fluid Mech 361:145–174 · Zbl 0910.76083 [2] Bellin A, Tonina D (2007) Probability density function of non-reactive solute concentration in heterogeneous porous formations. J Contam Hydrol 94(1–2):109–125 [3] Bellin A, Rubin Y, Rinaldo A (1994) Eulerian–Lagrangian approach for modeling of flow and transport in heterogeneous geological formations. Water Resour Res 30(11):2913–2924 [4] Botter GE, Daly A, Porporato I, Rodriguez-Iturbe I, Rinaldo A (2008) Probabilistic dynamics of soil nitrate: coupling of ecohydrological and biogeochemical processes. Water Resour Res 44:W03416. doi: 10.129/2007WR006108 [5] Caroni E, Fiorotto V (2005) Analysis of concentration as sampled in natural aquifers. Transp Porous Media 59(1):19–45 [6] Chen HD, Chen SY, Kraichnan RH (1989) Probability distribution of a stochastically advected scalar field. Phys Rev Lett 63(24):2657–2660 [7] Cirpka OA, Valocchi AJ (2007) Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state. Adv Water Resour 30(6–7):1668–1679 [8] Cirpka OA, Olsson Å, Ju Q, Rahman MA, Grathwohl P (2006) Determination of transverse dispersion coefficients from reactive plume lengths. Ground Water 44(2):212–221 [9] Cirpka OA, Schwede RL, Luo J, Dentz M (2008) Concentration statistics for mixing-controlled reactive transport in random heterogeneous media. J Contam Hydrol 98(1–2):61–74 [10] Clement TP, Sun Y, Hooker BS, Petersen JN (1998) Modeling multispecies reactive transport in ground water. Ground Water Monit Rem 18(2):79–92 [11] Cvetkovic VD, Shapiro AM, Dagan G (1992) A solute flux approach in transport in heterogeneous formations: 2. Uncertainty analysis. Water Resour Res 28(5):1377–1388 [12] Dagan G (1989) Flow and transport in porous formations. Springer, New York · Zbl 0673.76026 [13] Dagan G (1994) Upscaling of dispersion coefficients in transport through heterogeneous porous formations. In: Computational methods in water resources X. Kluwer Academic, Norwell [14] Dentz M, Kinzelbach H, Attinger S, Kinzelbach W (2000) Temporal behavior of a solute cloud in a heterogeneous porous medium 2. Spatially extended injection. Water Resour Res 36(12):3605–3614 · Zbl 0954.76087 [15] De Simoni M, Carrera J, Sanchez-Vila X, Guadagnini A (2005) A procedure for the solution of multicomponent reactive transport problems. Water Resour Res 41:W11410. doi: 10.1029/2005WR004056 [16] De Simoni M, Sanchez-Vila X, Carrera J, Saaltink MW (2007) A mixing ratios-based formulation for multicomponent reactive transport. Water Resour Res 43:W07419. doi: 10.1029/2006WR005256 [17] Fernandez-Garcia D, Sanchez-Vila X, Guadagnini A (2008) Reaction rates and effective parameters in stratified aquifers. Adv Water Res 31(10):1364–1376 [18] Fiori A (2001) The relative dispersion and mixing of passive solutes in transport in geologic media. Transp Porous Media 42(1–2):69–83 [19] Fiori A, Dagan G (2000) Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications. J Contam Hydrol 45(1–2):139–163 [20] Fiorotto V, Caroni E (2002) Solute concentration statistics in heterogeneous aquifers for finite Peclet values. Transp Porous Media 48(3):331–351 [21] Friedly JC, Rubin J (1992) Solute transport with multiple equilibrium-controlled or kinetically controlled chemical reactions. Water Resour Res 28(6):1935–1953 [22] Gelhar LW, Axness CL (1983) Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resour Res 19(1):161–180 [23] Girimaji SS (1991) Assumed beta-pdf model for turbulent mixing–validation and extension to multiple scalar mixing. Combust Sci Technol 78(4–6):177–196 [24] Guadagnini A, Sanchez-Vila X, Riva M, De Simoni M (2003) Mean travel time of conservative solutes in randomly heterogeneous unbounded domains under mean uniform flow. Water Resour Res 39(3):1050. doi: 10.1029/2002WR001811 [25] Guadagnini A, Sanchez-Vila X, Saaltink MW, Bussini M, Berkowitz B (2008) Application of a mixing-ratios based formulation to model mixing-driven dissolution laboratory experiments. Adv Water Resour. doi: 10.1016/j.advwaters.2008.07.005 [26] Ham PAS, Schotting RJ, Prommer H, Davis GB (2004) Effects of hydrodynamic dispersion on plume lengths for instantaneous bimolecular reactions. Adv Water Resour 27(8):803–813 [27] Hess KM, Wolf SH, Celia MA (1992) Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts, 3. Hydraulic conductivity variability and calculated macrodispersivities. Water Resour Res 28(8):2011–2017 [28] Hu BX, Deng F-W, Cushman JH (1995) Nonlocal reactive transport with physical and chemical heterogeneity: linear nonequilibrium sorption with random Kd. Water Resour Res 31(9):2239–2252 [29] Kapoor V, Gelhar LW (1994) Transport in three-dimensionally heterogeneous aquifers. 1. Dynamics of concentration fluctuations. Water Resour Res 30(6):1775–1788 [30] Kapoor V, Kitanidis PK (1998) Concentration fluctuations and dilution in aquifers. Water Resour Res 34(5):1181–1193 [31] Kitanidis PK (1994) The concept of the dilution index. Water Resour Res 30(7):2011–2026 [32] Lawrence AE, Sanchez-Vila X, Rubin Y (2002) Conditional moments of the breakthrough curves of kinetically sorbing solute in heterogeneous porous media using multirate mass transfer models for sorption and desorption. Water Resour Res 38(11), Article Number 1248 [33] Lichtner PC (1996) Continuous formulation of multicomponent-multiphase reactive transport. In: Lichtner PC, Steefel CI, Oeklers EH (eds) Reactive transport in porous media. Reviews in mineralogy, vol 34. Miner Soc Amer, Washington, pp 1–81 [34] Lichtner PC, Tartakovsky DM (2003) Stochastic analysis of effective rate constant for heterogeneous reactions. Stoch Env Res Risk Assess 17(6):419–429 · Zbl 1053.92050 [35] Liedl R, Valocchi AJ, Dietrich P, Grathwohl P (2005) Finiteness of steady state plumes. Water Resour Res 31(12):W12501. doi: 10.1029/2005WR004000 [36] Luo J, Dentz M, Carrera J, Kitanidis PK (2008) Effective reaction parameters for mixing controlled reactions in heterogeneous media. Water Resour Res 44(2):W02416. doi: 10.1029/2006WR005658 [37] Molins S, Carrera J, Ayora C, Saaltink MW (2004) A formulation for decoupling components in reactive transport problems. Water Resour Res 40(10):W10301. doi: 10.1029/2003WR002970 [38] Neuman SP, Zhang Y-K (1990) A quasi-linear theory of non-Fickian subsurface dispersion, 1. Theoretical analysis with application to isotropic media. Water Resour Res 26(5):887–902 [39] Parkhurst DL, Appelo CAJ (1999) User’s guide to PHREEQC (Version2)–a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations. US Geological Survey water-resources investigations report 99-4259 [40] Phillips OM (1991) Flow and reactions in permeable rocks. Cambridge Univ Press, New York [41] Pope SB (1981) Transport equation for the joint probability density function of velocity and scalars in turbulent flow. Phys Fluids 24(4):588–596 · Zbl 0477.76079 [42] Quinodoz HAM, Valocchi AJ (1993) Stochastic analysis of the transport of kinetically sorbing solutes in aquifers with randomly heterogeneous hydraulic conductivity. Water Resour Res 29(9):3227–3240 [43] Reichle R, Kinzelbach W, Kinzelbach H (1998) Effective parameters in heterogeneous and homogeneous transport models with kinetic sorption. Water Resour Res 34(4):583–594 [44] Riva M, Sanchez-Vila X, Guadagnini A, De Simoni M, Willmann M (2006) Travel time and trajectory moments of conservative solutes in two-dimensional convergent flows. J Contam Hydrol 82:23–43 [45] Riva M, Guadagnini A, Fernandez-Garcia D, Sanchez-Vila X, Ptak T (2008a) Relative importance of geostatistical and transport models in describing heavily tailed breakthrough curves at the Lauswiesen site. J Contam Hydrol 101(1–4):1–13 [46] Riva M, Guadagnini A, Sanchez-Vila X (2008b) Effect of sorption heterogeneity on moments of solute residence time in convergent flows. Comput Geosci, submitted · Zbl 1425.76253 [47] Robinson BA, Viswanathan HS, Valocchi AJ (2000) Efficient numerical techniques for modeling multicomponent ground-water transport based upon simultaneous solution of strongly coupled subsets of chemical components. Adv Water Resour 23(4):307–324 [48] Rubin J (1990) Solute transport with multisegment, equilibrium-controlled reactions: a feed forward simulation method. Water Resour Res 26(9):2029–2055 [49] Saaltink MW, Ayora C, Carrera J (1998) A mathematical formulation for reactive transport that eliminates mineral concentrations. Water Resour Res 34(7):1649–1656 [50] Sanchez-Vila X, Guadagnini A (2005) Travel time and trajectory moments of conservative solutes in three dimensional heterogeneous porous media under mean uniform flow. Adv Water Res 28:429–439 [51] Shvidler M, Karasaki K (2003) Probability density functions for solute transport in random field. Transp Porous Media 50:243–266 [52] Singurindy O, Berkowitz B, Lowell RP (2004) Carbonate dissolution and precipitation in coastal environments: laboratory analysis and theoretical consideration. Water Resour Res 40:W04401. doi: 10.1029/2003WR002651 [53] Steefel CI, MacQuarrie KTB (1996) Approaches to modelling reactive transport. In: Reactive transport in porous media. Reviews in mineralogy, vol 34. Miner Soc Amer, Washington, pp 83–129 [54] Tebes-Stevens C, Valocchi AJ, VanBriesen JM, Rittmann BE (1998) Multicomponent transport with coupled geochemical and microbiological reactions: model description and example simulations. J Hydrol 209(1–4):8–26 [55] Vanderborght J (2001) Concentration variance and spatial covariance in second-order stationary heterogeneous conductivity fields. Water Resour Res 37(7):1893–1912 [56] Yeh GT, Tripathi VS (1991) A model for simulating transport of reactive multispecies components: model development and demonstration. Water Resour Res 27(12):3075–3094 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.