Using PHREEQC to Simulate Solute Transport in Fractured Bedrock by David S. Lipson,2,JohnE.McCray 3,andGeoffreyD.Thyne 4 Abstract The geochemical computer model PHREEQC can simulate solute transport in fractured bedrock aquifers that can be conceptualized as dual-porosity flow systems subject to one-dimensional advective-dispersive transport in the bedrock fractures and diffusive transport in the bedrock matrix. This article demonstrates how the physical characteristics of such flow systems can be parameterized for use in PHREEQC, it provides a method for minimizing numerical dispersion in PHREEQC simulations, and it compares PHREEQC simulations with results of an analytical solution. The simulations assumed a dual-porosity conceptual model involving advective-reactivedispersive transport in the mobile zone (bedrock fracture) and diffusive-reactive transport in the immobile zone (bedrock matrix). The results from the PHREEQC dual-porosity transport model that uses a finite-difference approach showed excellent agreement compared with an analytical solution. Introduction There are many mathematical models that provide solutions to one-dimensional (D), two-dimensional (2D), and three-dimensional ground water flow and transport problems in fractured bedrock (e.g., Bear et al. 993; Therrien and Sudicky 996; Xu and Pruess 200). However, these fracture flow models are limited in terms of simulating geochemical processes such as ion exchange, surface complexation, fixed-volume gas-phase, and kinetically controlled reactions. In contrast, there are several public-domain geochemical models that offer well-developed chemical formulations that can be used to model Corresponding author: Department of Geology and Geological Engineering, Colorado School of Mines, Golden, CO 8040; (303) 23-95; fax (303) 23-957; David.Lipson@arcadis-us.com 2 ARCADIS BBL, 442 Denver West Parkway, Golden, CO 8040. 3 Hydrologic Science and Engineering Program, Division of Environmental Science and Engineering, Colorado School of Mines, Golden, CO 8040. 4 Hydrologic Science and Engineering Program, Department of Geology and Geological Engineering, Colorado School of Mines, Golden, CO 8040. Received May 2006, accepted January 2007. Copyright ª 2007 The Author(s) Journal compilation ª 2007 National Ground Water Association. doi: 0./j.745-6584.2007.0038.x 468 Vol. 45, No. 4 GROUND WATER July August 2007 (pages 468 472) transport in addition to geochemical reactions. The geochemical computer model PHREEQC can be used to model solute transport in fractured bedrock aquifers because it simulates solute water-rock chemical reactions involving natural or polluted ground water, and it also includes a dual-porosity transport model that can simulate solute diffusion in the bedrock matrix. For example, Gaus et al. (2002) successfully calibrated PHREEQC models to evaluate potential water quality changes associated with aquifer storage and recovery scenarios in four major bedrock aquifers in the United Kingdom, namely the Chalk, Lincolnshire Limestone, Sherwood Sandstone, and Lower Greensand aquifers. To accomplish this, Gaus et al. (2002) adjusted the PHREEQC dual-porosity transport parameters to fit calibration data, but they did not relate the dual-porosity transport model to physical characteristics of the bedrock aquifer systems. Furthermore, their fitted PHREEQC input parameters resulted in numerical dispersion. The objectives of this article are to demonstrate how fractured bedrock physical characteristics can be parameterized for use in PHREEQC, address the numerical dispersion issue, describe conditions under which dual-porosity numerical models can be used to simulate solute transport in fractured bedrock aquifers, and compare PHREEQC simulations with results of an analytical solution.
Theory The PHREEQC user s manual (Parkhurst and Appelo 999) provides example problems that verify the dualporosity transport formulation in a D column filled with spherical clay beads, but an explanation regarding how to formulate the model in terms of fractured bedrock characteristics has yet to be put forth. Upon inspection, it is apparent that the finite-difference dual-porosity solute transport model in PHREEQC provides an approximate solution to the differential equations for dual-porosity solute transport in fractured bedrock described elsewhere (e.g., Sudicky and Frind 982). The PHREEQC dual-porosity transport formulation involves a single row of mobile cells adjacent to an array of immobile cells, which can be conceptualized as bedrock fractures and bedrock matrix, respectively (Figures and 2). D ground water flow and solute transport occur in the bedrock fractures (mobile zone) via advection, dispersion, diffusion, and solute water-matrix interactions. Only diffusive and reactive solute transport occurs in the bedrock matrix (immobile zone), where ground water is assumed to be stagnant. Mixing factors are used to approximate the diffusive transport of solutes between the bedrock fractures and the rock matrix, and within the bedrock matrix. The formulation involves discretizing the rock matrix into a 2D orthogonal array of immobile cells with mixing factors defined explicitly for each cell. The mixing factor is defined as (Parkhurst and Appelo 999): mixf ij ¼ D ita ij f bc h ij V j ðþ where D i is the solute diffusion coefficient, A ij is the surface area between adjacent cells, f bc is a boundary condition correction factor, h ij is the distance between midpoints of adjacent cells, and V j is volume of the central cell. The diffusion coefficient D i may account for solute diffusion parallel to flow (D x ), perpendicular to flow (D z ), or across fracture walls (D f ). The boundary condition correction factor f bc accounts for the volumetric ratio of adjacent cells and may be set to.0 if adjacent cells have equal volumes (Parkhurst and Appelo 999). At the fracture-matrix interface, where adjacent cell volumes may be unequal, f bc may be specified explicitly based on the geometry of the problem, or it may be set to 2.0 if the concentration in the fracture is constant (Appelo and Postma 993). To maintain the boundary condition assumption at the fracture matrix interface (f bc ), fracture and matrix grid spacing values (x and z, respectively) must be selected so that the amount of solute potentially exiting the fracture due to diffusion does not exceed the amount of solute entering the fracture due to advection and dispersion. To control this condition, we introduce a new dual-porosity dimensionless quantity (w): w ¼ v D f b z / im x ð2þ where v is the average linear ground water velocity in the fractures defined as x/t, where t is time step, b is fracture aperture, and / im is porosity of the bedrock matrix. From Equation 2, it is seen that w is the ratio of advective to diffusive solute flux in the fracture. w values greater than.0 indicate that advective flux into the fractures exceeds the diffusive flux out of the fractures, and w values less than.0 indicate that diffusive flux exceeds advective flux. If a constant concentration boundary condition is assumed at the fracture origin (i.e., f bc ¼ 2.0), then the model time step and grid spacing parameters should be selected to produce w values that exceed 0 to ensure that advective flux in the fracture exceeds diffusive flux by at least order of magnitude. Total dispersion (a total ) in the finite-difference transport model is as follows: a total ¼ a L a num ð3þ where a L is physical longitudinal dispersivity in the fracture. Minimizing numerical dispersion (a num ) is paramount in the finite-difference transport model because it can lead to instability and inaccuracy. Numerical dispersion in finite-difference transport models is estimated by (Appelo and Postma 993): a num ¼ x 2 2 vt 2R ð4þ Figure. Example problem showing dual-porosity solute transport conceptual model (not to scale). where t is time step and R is solute retardation factor. In most transport models, the solute retardation factor R is assumed to be constant in time and space, and numerical dispersion can be eliminated by use of Equation 4 and appropriate selection of grid spacing and time step. However, in dual-porosity fractured bedrock systems, solute retardation increases with time and distance and approaches a steady-state value (Lipson et al. 2005). From Equation 4, it follows that a num also increases with time and distance and eventually approaches a steady-state value due to the transient nature of R. Thus, numerical dispersion cannot be completely eliminated in the PHREEQC dual-porosity model or other finite-difference dual-porosity models. To minimize the influence of D.S. Lipson et al. GROUND WATER 45, no. 4: 468 472 469
numerical dispersion, we draw on the steady-state, matrix diffusion derived plume attenuation factor b described in Lipson et al. (2005): b ¼ / im =/ m ð5þ where / m is fracture porosity. Substituting b for R in Equation 4 can help select the time step and grid spacing and minimize numerical dispersion. Use of this strategy is successfully demonstrated subsequently. Comparison with Analytical Solution The conceptual model is illustrated in Figure and involves a 3-m-long bedrock block with matrix porosity (/ im ) of 0.2 and parallel 400-lm-aperture planar fractures spaced 0.5 m apart (Figure and Table ). This geometry results in a fracture porosity (/ m ) of 0.0008. Given the symmetry of this fracture system, this problem was simulated in PHREEQC as a half-aperture of 200 lm with fracture half-spacing of 0.25 m. The PHREEQC finite-difference configuration is shown in Figure 2, and transport parameters are listed in Table 2. As shown, the fracture was simulated as a single row of mobile cells with grid spacing parallel to flow (x) of 0.05 m. The bedrock matrix adjacent to the fracture was simulated as an array of five layers of stagnant cells with grid spacing perpendicular to flow (z) of 0.05 m (Table 2). In this example, the plume attenuation factor b is 250, suggesting that numerical dispersion may be significant. To account for numerical dispersion, x and t were selected so that a total in PHREEQC was consistent with a total in the analytical model (cf. Table and Table 2). As shown, a total in both models was 0.008 m. Furthermore, parameters in this example were selected so Table Physical Quantities and Analytical Model Parameters for the Example Problem Parameters Value Units Source concentration (c 0 ) Initial concentration (c, c9) 0 Fracture ground.74 3 0 25 m/s water velocity (v) Fracture dispersivity (a L ) 0.008 m Fracture aperture (b) 0.0004 m Fracture spacing (S f ) 0.5 m Bedrock matrix 0.2 porosity (/ im ) Solute diffusion 4.4 3 0 20 m 2 /s coefficient (D9) Solute retardation factor (R) Solute decay coefficient (k, k9) NA /s Note: indicates dimensionless units; NA ¼ not applicable. Figure 2. Section of finite-difference configuration (not to scale). that w exceeded 00, indicating that the fracture matrix boundary condition assumption is valid. Mixing factors for the stagnant cells are shown in Table 3. As shown, solute diffusion coefficients were assumed constant in all directions. Different diffusion coefficients could potentially be used to account for the presence of heterogeneous materials and anisotropic diffusion. These mixing factors were calculated using Equation and were not derived as fitting parameters. Figures 3 and 4 show that the PHREEQC simulations were in excellent agreement with the analytical solution provided by Sudicky and Frind (982). Correlation Table 2 PHREEQC Dual-Porosity Transport Model Parameters Parameters Value Units Transport parameters Source concentration (c 0 ) Initial concentration (c, c9) 0 Cells 200 Shifts Variable Flow direction Forward Time step (t) 846 s Boundary condition Flux-flux Diffusion coefficient mobile zone 0 m 2 /s Length of mobile zone cells (x) 0.05 m Dispersivity in mobile zone (a L ) 0.00 m Number of stagnant layers 5 Length of immobile zone cells (z) 0.05 m Derived/dimensionless parameters Fracture porosity (/ m ) 0.0008 Retardation factor asymptotic (R E ) 250 Numerical dispersivity (a num ) 0.007 m Total dispersivity (a total ) 0.008 m Advective to diffusive flux ratio (C) 3 Based on PHREEQC input information, not required as input. 470 D.S. Lipson et al. GROUND WATER 45, no. 4: 468 472
Table 3 Mixing Factors for Finite-Difference Calculation of Solute Transport in Fractured Bedrock Parameter Stagnant layer 2, 3, 4 5 Length x (m) 0.05 0.05 0.05 0.05 Height z (m) 0.0002 0.05 0.05 0.05 Volume V j (m 3 ) 4.5 3 0 28 2.3 3 0 26 2.3 3 0 26 2.3 3 0 26 Area 2 A ij (m 2 ) 4.5 3 0 25 4.5 3 0 25 4.5 3 0 25 4.5 3 0 25 Boundary condition 2.0.0.0 correction factor 3 f bc Mixing factors 4 mixf ij 0.0003 0.0005 0.0005 mixf jj 0.98468 0.99954 0.99969 0.99985 mixf jk 0.0532 0.0005 0.0005 Stagnant layer cell volume assumes width equals length and cell porosity is 0.2. 2 Shared surface area between adjacent cells, also based on porosity of 0.2. 3 Boundary condition correction factor for volumetric ratio of adjacent cells; may be set to 2.0 at the fracture matrix interface if solute concentration in the fracture is constant. 4 Computed with j as the central cell, i as the previous cell, and k as the next cell based on methods in Parkhurst and Appelo (999), using diffusion coefficient of 4.4 3 0 20 m 2 /s and time step (t) of 846 s. Value coefficient (R 2 ) values between numerical and analytical model results were 0.99 or greater for all simulations. Most of the parameters required for modeling plume migration in such situations can be derived from the results of site investigations performed at contaminated properties. For example, Lipson et al (2005) simulated the transport of an organic contaminant in fractured arkosic bedrock using parameters estimated from site investigation techniques, including collecting and analyzing bedrock core samples, collecting and analyzing ground water samples, in situ borehole imaging, and in situ hydraulic testing. Conclusions The dual-porosity transport model available in PHREEQC is able to accurately simulate advective, diffusive, and reactive solute transport processes present in fractured bedrock aquifers where ground water flow in the fractures can be assumed to be D and ground water flow in the bedrock is negligible. This article demonstrates that it is possible to formulate PHREEQC input parameters based on physical characteristics of bedrock aquifers and possible to control numerical dispersion through careful selection of model time step size and grid spacing. If these parameters are beyond the constraints discussed previously, then model results could be subject to significant inaccuracy within relatively few iterations. For example, when model time step and grid spacing parameters in the example problem were changed so that w was equal to 3.3, predicted concentrations were approximately 20% lower than the correct model-predicted concentrations (Figure 3). The new dual-porosity dimensionless quantity w can be used to help select grid spacing values to ensure that the fracture-matrix boundary condition assumption is reasonably approximated not only in PHREEQC but also in other finite-difference dual-porosity solute transport codes. It should be noted that these simulations used a D reactive solute transport model assuming a single planar Figure 3. Concentration in the mobile zone after 00 d of transport. Note solute concentrations are underpredicted when w < 0. Figure 4. Concentration in the immobile zone after 00 d of transport. D.S. Lipson et al. GROUND WATER 45, no. 4: 468 472 47
fracture or system of parallel fractures and a steady-state flow field. These conditions may be approximated by experiments or field sites involving relatively wide source zones. References Appelo, C.A.J., and D. Postma. 993. Geochemistry, Groundwater, and Pollution. Rotterdam, Netherlands: A.A. Balekema. Bear, J., C.F. Tsang, and G. de Marsily, ed. 993. Flow and Contaminant Transport in Fractured Rock. San Diego, California: Academic Press. Gaus, I., P. Shand, I.N. Gale, and A.T. Williams. 2002. Geochemical modeling of fluoride concentration changes during aquifer storage and recovery (ASR) in the chalk aquifer in Wessex, England. Quarterly Journal of Engineering Geology & Hydrogeology 35, no. 2: 203 208. Lipson, D.S., B.H. Kueper, and M.J. Gefell. 2005. Matrix diffusion-derived plume attenuation in fractured bedrock. Ground Water 43, no. : 30 39. Parkhurst, D.L., and C.A.J. Appelo. 999. User s guide to PHREEQC (Version 2) A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations. U.S. Department of the Interior. USGS Water-Resources Investigations Report 99-4259. Reston, Virginia: USGS Sudicky, E.A., and E.O. Frind. 982. Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures. Water Resources Research 8, no. 6: 634 642. Therrien, R., and E.A. Sudicky. 996. Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media. Journal of Contaminant Hydrology 23, no. 2: 44. Xu, T., and K. Pruess. 200. Modeling multiphase nonisothermal fluid flow and reactive geochemical transport in variably saturated fractured rocks:. methodology. American Journal of Science 30, no. : 6 33. 472 D.S. Lipson et al. GROUND WATER 45, no. 4: 468 472