1D Verification Examples

Transcription

1 1 Introduction 1D Verification Examples Software verification involves comparing the numerical solution with an analytical solution. The objective of this example is to compare the results from CTRAN/W analyses to well-established closedform solutions for four different transport scenarios. The transport scenarios include: Case 1: no adsorption and no decay Case 2: adsorption included Case 3: decay included Case 4: adsorption and decay included. Equations for the closed-form solutions are solved using a Microsoft Excel spreadsheet. The solutions require the use of a complimentary error function (erfc). This function is available in MS Excel 2007 as part of the Analysis ToolPak Add-in, which can be installed under the Excel Options Add-Ins dialogue box. 2 Feature Highlights GeoStudio feature highlights include: 1. Comparing CTRAN/W to closed-form solutions for the advection-dispersion equation; 2. Including adsorption and decay processes; and, 3. Comparison of backward difference and central difference iteration schemes. 3 Closed-Form Solutions Closed-form analytical solutions to the advection-dispersion equation are available in the literature for one-dimensional problems involving steady-state seepage flow. The advection-dispersion equation is written as: 2 C C C S ndl nv 2 x = n + ρd nλc nλs x x t t [1] C = concentration of solute in liquid phase, t = time, D L = longitudinal hydrodynamic dispersion, ρ d = dry density, n = porosity or volumetric water content (θ), S = amount of solute sorbed per unit weight of soil, and λ = coefficient of decay. CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 1 of 12

2 The first and second terms on the left side of Equation 1 represent the dispersive and advective transport of the solute, respectively. All of the terms on the right side account for changes in mass or concentration that may occur with time due to mass flux (term 1), sorption processes (term 2), and radioactive decay of mass in solution (term 3) and mass sorbed onto the solid (term 4). 3.1 Case 1: No Adsorption and No Decay Ogata (1970) provided an analytical solution to the advection-dispersion equation for a homogeneous, isotropic, and saturated porous geological medium. The concentration at any lateral distance for a given elapsed time can be determined: C o x vt vx x + vt C = erfc exp erfc Dt D 2 Dt [2] C = concentration, C o = specified concentration in the source boundary, D = hydrodynamic dispersion coefficient, v = average linear velocity, t = elapsed time, x = distance from the source boundary, and erfc = complementary error function. The solution assumes that the concentration at a distance of x = 0 is maintained at C o for all time (i.e. C(0,t) = C o ), the initial concentration every in the flow domain is zero (i.e. C(x,0) = 0), and the flow domain is infinitely long with a concentration of zero at the far boundary (i.e. C(,t) = 0). It should be noted that the complimentary error function is related to the error function (erf) by the following Erfc(x) = 1 erf(x) and that Erf(0) = 0; erf( ) = 1; erf( x) = erf(x) 3.2 Case 2: Adsorption Included Adsorption is the physical process by which a solute adheres to a solid surface. The relationship between the mass of solute sorbed onto the solid (S) and the concentration of the solute (C) can take a variety of linear and nonlinear forms (Figure 1). The slope of the linear sorption isotherm is often referred to as the distribution coefficient (K d ). The amount of solute sorbed per dry unit weight of solid is given by S = K C [3] d CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 2 of 12

3 Linear Isotherm Non Linear Isotherm S (g/g) Concentration (g/m3) Figure 1 - Linear and nonlinear sorption isotherms If the equation for the linear sorption isotherm is substituted into the advection-dispersion equation [1] and ignoring the decay terms, the governing partial differential equation is ( KC) 2 C C C d ndl nv 2 x = n + ρ d x x t t Ignoring dispersive transport and dividing by the porosity (n) yields the following [4] C ρd C vx = 1+ Kd x n t [5] ρ + n 1 d K d is referred to as the retardation factor (R). For this specific case, the sorption process reduces the velocity by a factor of R (i.e., v x /R). Accordingly, the arrival time of the C/C o = 0.5 front for a transport problem involving advection-only is reduced by the factor R. The solution to Equation 4 for the same boundary conditions discussed above is given by (Bear, 1972) ( v ) ( v vx ) C x t x t o C erfc R + = exp erfc R Dt D 2 Dt R R [6] CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 3 of 12

4 It should be noted that the governing partial differential equation is formulated in CTRAN using the more general form shown in Equation 1. Accordingly, the formulation can accommodate unsaturated soil and non-linear adsorption functions. The retardation factor is therefore given by S R = θ + ρ d C [7] θ = volumetric water content, and S C = slope of the sorption isotherm. 3.3 Case 3: Decay Included Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting radiation. This process will reduce the concentration of radionuclides in both the dissolved and adsorbed phases. The coefficient of decay (λ) in equation [1] is given by: ln 2 λ = [8] t 1/2 t 1/2 = half-life of the radionuclide. Bear (1972, 1979) provided the following analytical solution subject to the same boundary conditions discussed above Co vx C = exp exp x erfc exp x erfc + 2 2D 2 Dt 2 Dt 2 2 x t ( v) + 4λD x+ t ( v) + 4λD ( β ) ( β ) [9] v x λ β = + 2D D 2 [10] 3.4 Case 4: Adsorption and Decay Included If radioactive decay and adsorption are included, the analytical solution becomes (Bear, 1972, 1979) ( ) ( ) 2 2 x t v 4λD x t v 4λD Co vx R R R R C = exp exp( βx) erfc + exp( βx) erfc 2 2D 2 Dt 2 Dt R R CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 4 of 12

5 v λr β = + 2D D 2 4 Boundary Conditions and Material Properties Figure 2 presents the geometry and mesh of the model domain. The model is comprised of a onedimensional column that is 3 m in length and 0.2 m in height. The mesh consists of 80 elements and 162 nodes. Although the dimensions are arbitrary, the column length was chosen to ensure that the far-field boundary condition has no effect on the results. This condition is in keeping with the analytical solutions, which assume that the C = 0 boundary is located at infinite distance. 1D Advection-Dispersion Steady-State Seepage Figure 2 Model geometry and mesh A screen capture of the KeyIn Analyses dialogue box is presented in Figure 3. A steady-state seepage analysis forms the parent analysis for each transport model. In the seepage analysis, a unit flux of m/sec was applied to the left boundary and the right boundary was assigned a hydraulic head of 1 m. The soil was assigned a hydraulic conductivity and saturated volumetric water content of m/sec and 0.5, respectively. Accordingly, the average linear velocity (v) in the flow domain is m/sec (i.e. v = q/n). Figure 3 Model structure for the 1D verification example CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 5 of 12

6 For each transport model, the left and right boundaries were set to constant concentrations of 1.0 and 0.0 g/m 3, respectively. Table 1 presents the material properties used in the analyses. The coefficient of diffusion (D) and dispersivity were set to zero and 0.1 m for all cases, yielding a hydrodynamic dispersion (αv) of m 2 /sec. A distribution coefficient K d of m 3 /g was used for the cases with adsorption, producing a retardation factor R equal to 4.0. Note that the activation concentration is set to 0 g/m 3 for each material under the KeyIn Materials dialogue box. Each model was run for an elapsed time of 6000 seconds using 60 time steps. Table 1 Material properties for transport analyses Case Disp. (m) K d (m 3 /g) Dry Density (g/m 3 ) Decay Half-life (sec) Case 1: No Adsorp./No Decay 0.1 Case 2: Adsorp. Included Case 3: Decay Included Case 4: Adsorp. & Decay Results and Discussion 5.1 Case 1: No Adsorption and No Decay Figure 4 presents results for the Case 1 analyses (no adsorption or decay) at elapsed times of 2000, 4000, and 6000 seconds. CTRAN/W was solved using the backward difference time integration scheme. The CTRAN/W results compare very well to the analytical solution; however, the analytical solution is slightly steeper than CTRAN/W. In other words, the CTRAN/W solution is slightly more spread out or smeared compared to the closed-form solution. This phenomenon is due to numerical dispersion, which is inherent in the finite element solution of the transport equation. A better match can be achieved when CTRAN/W is solved using the central difference time integration scheme (Figure 5). In general, the central difference technique provides a better solution than using backward difference for most transport problems. However, the central difference technique is susceptible to numerical oscillation, which can cause the computed concentrations to be larger or smaller than the specified maximum or minimum concentrations. Figure 6 shows a more extreme case of both numerical dispersion and oscillation. Figure 1Numerical dispersion and oscillation can only be minimized, not eliminated. Techniques for minimizing numerical dispersion and oscillation are presented in the CTRAN/W documentation. CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 6 of 12

7 Case 1: Backward Difference 2000 seconds 4000 seconds 6000 seconds CTRAN 2000 seconds CTRAN 4000 seconds CTRAN 6000 seconds Figure 4 Case 1 (no adsorption or decay) results: CTRAN/W solved using backward difference time integration scheme CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 7 of 12

8 Case 1: Central Difference 2000 seconds 4000 seconds 6000 seconds CTRAN 2000 seconds CTRAN 4000 seconds CTRAN 6000 seconds Figure 5 Case 1 results: CTRAN/W solved using central difference time integration scheme CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 8 of 12

9 seconds 0.8 Backward Difference Central Difference Figure 6 Example of numerical dispersion and oscillation 5.2 Case 2: Adsorption Included Figure 7 presents results for the Case 2 analyses, in which adsorption (R = 4.0) is included. CTRAN/W was solved using the central difference time integration scheme. Note that the scale of the x-axis ranges from 0 to 1 m. Again, CTRAN/W is in close agreement with the analytical solution. The adsorption process slows the arrival front as anticipated. For example, the position of C/C o = 0.5 front at a time of 4000 seconds is about 0.27 m, compared to 0.90 m without adsorption (i.e., approximately one-quarter; v*t/r). The small discrepancy is due to the inclusion of dispersivity in this example. CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 9 of 12

10 Case 2: Central Difference 2000 seconds 4000 seconds 6000 seconds CTRAN 2000 sec CTRAN 4000 sec CTRAN 6000 sec Figure 7 Case 2 results (adsorption included) 5.3 Case 3: Decay Included Figure 8 presents the results for Case 3 (decay included) at an elapsed time of 4000 seconds. The results from Case 1 are also included for comparison. CTRAN/W compares very well with closed-form analytical solution. The largest effect of decay occurs near the source boundary the concentration is the highest. As the concentration approaches zero, the decay component has less effect as there is little mass to decay. The affects of decay are more apparent when a slug of contaminant is introduced into the system. This was modeled by creating a region 0.1 m in length on the left side of the model domain. The region was assigned the same material used for Case 3, but with an activation concentration of 1.0 g/m 3. Accordingly, the initial mass in the system is calculated as 0.1 m 0.2 m 1 m g/m 3 = 0.01 g (i.e. length height unit width porosity Concentration). The column length was extended to 6 m and the model was run for a time of 18,000 seconds. Figure 9 presents the concentration verses distance profiles for three elapsed times. The amount of mass in the model domain (i.e. area under the curve) decreases with time due to decay. This can be checked by hand-calculation via the relationship M=M o e -λt, M o is the initial mass and λ is the coefficient of decay. For example, the calculated mass remaining in the system after an elapsed time of 6900 seconds should be about g using a λ = s -1. CTRAN/W reports the same value for the Total System Mass under View Mass Accumulation. CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 10 of 12

11 Case 3: Central Difference Case 1: 4000 Seconds seconds CTRAN 4000 sec Figure 8 Case 3 results (decay included) sec 6900 sec sec Figure 9 Effects decay on the transport of a contaminant slug CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 11 of 12

12 5.4 Case 4: Adsorption and Decay Included A comparison between CTRAN/W and the analytical solution for Case 4 is presented in Figure 10, along with the results from Case 1. There is excellent agreement between the CTRAN/W solution and the analytical solution. In fact, the two solutions are almost identical for all three elapsed times. Furthermore, a comparison between Case 1 and 4 demonstrates that CTRAN/W correctly captures the effect of both adsorption and decay. Mass is lost due to radioactive decay and the contaminant front is slowed due to adsorption Case 4: Central Difference 2000 seconds 4000 seconds 6000 seconds CTRAN 2000 sec CTRAN 4000 sec CTRAN 6000 sec Case 1: 2000 sec Case 2: 4000 sec Case 3: 6000 sec Figure 10 Case 4 results (adsorption and decay) compared to Case 1 (no adsorption/no decay) 6 Concluding Remarks In this example, results from CTRAN/W are compared to the closed-form solution of the advectiondispersion equation for four different transport scenarios. The results demonstrate that CTRAN/W is capable of modeling geochemical processes such as adsorption and decay. In general, the results from CTRAN/W provide a better-match to the analytical solution using the central difference time integration scheme. CTRAN Example File: 1D Verification Examples (pdf)(gsz) Page 12 of 12