A simplified technique for measuring diffusion coefficients

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1 WATER RESOURCES RESEARCH, VOL. 37, NO. 5, PAGES , MAY 2001 A simplified technique for measuring diffusion coefficients in rock blocks Motomu Ibaraki Department of Geological Sciences, The Ohio State University, Columbus, Ohio Abstract. This study demonstrates a simple laboratory method to measure diffusion coefficients of reactive and nonreactive solutes in rock blocks. A demonstrative diffusion measurement device consists of a small plastic cell with magnetic stirrers, the porous rock block to be measured, and nonreactive tracer solution. Concentrations of the tracer (as specific conductance) are measured in the cell continuously with data recorded using a PC. The specific conductance versus time data are fitted to an analytical solution that is numerically inverted to estimate the diffusion coefficient. Mathematical development is demonstrated, and a sensitivity analysis shows that diffusion coefficient as low as 10-9 cm2/s can be measured over a test period of 12 days. 1. Introduction Matrix diffusion can play an important role in the migration of contaminants discretely fractured porous media [Sudicky and McLaren, 1992; Ibaraki and Sudicky, 1995]. This process can effectively retard the migration of contaminant in fractures otherwise fast groundwater flow promotes rapid migration. In order to evaluate such a matrix diffusion process quantitatively it is important to measure the diffusion coefficient of solute in porous media accurately. Wafer diffusion experiments are commonly used to measure diffusion coefficients in lithified porous media [e.g., Hubert and Calvet, 1979; Bradbury and Green, 1985]. The experiment involves monitoring the transport of a tracer from one reservoir to another through a thin slab of the medium. In practice, this type of experiment is quite tedious and prone to difficulties in preparing both intact wafers and experimental apparatus. Another commonly used technique is the so-called "half-cell" technique [e.g., Krom and Berner, 1980]. Variations on this technique include the reservoir method in which a porous medium cell and another cell containing a solution column are joined together [van Rees et al., 1991]. Still another approach was demonstrated by Novakowski and van der Kamp [1996]. In their method, radial diffusion from or into a cylindrical reservoir in a core-sized sample was considered. Diffusion coefficients of porous media are calculated on the basis of concentration changes in the cylindrical reservoir. The purpose of this paper is to demonstrate a simple laboratory method to calculate diffusion coefficients in porous matrix. The device consists of a small plastic cell with magnetic stirrers, porous rock block to be measured, and solute solution. conductance or concentration of the solution is measured over time. For the experiments reported here, I measured the specific conductance continuously with data recorded using a PC, although my analytical solution also includes periodic sampling for chemical analysis. The data are interpreted by fitting to an analytical solution describing the diffusion of a tracer into the rock block. Section 2.1 outlines the mathematical development of the analytical solution Mathematical Development Solute diffusion into a porous rock block from the surrounding tracer solution is described by two coupled equations: one for the porous matrix and one for the reservoir. It is assumed that the porous matrix is isotropic and homogeneous, that the reservoir is well mixed, and that isothermal conditions prevail. For the porous block the diffusion process is described by oc at o[o'c R b- x + + Oz2] ac] =0, (1) c is the solute concentration, t is time, D is the diffusion coefficient, and R is the retardation factor. The following ini- tial and boundary conditions are imposed on the equation': c(x, y, z, 0) = 0, (2a) c(o, y, z, t) = c(lx,y,z, t) = Cr(t), (2b) c(x, O, z, t) = c(x, Ly, z, t) = Cr(t), (2c) c(x, y, O, t) -- c(x, y, Lz, t) - Cr(t), (2d) Cr(t ) is the concentration in the solution reservoir at any 2. Method time t. Lx, Ly, and Lz are the lengths of the porouslab in the The test involves monitoring the rate of loss of a tracer from x, y, and z directions, respectively. a reservoir as solute diffuses into a saturated rock block. The In the reservoir, samples of volume Vs can be taken N times block is placed in a small plastic cell with magnetic stirrers at times ti from the reservoir for chemical analysis. However, (Figure 1). This experiment begins when the water-saturated immediately upon the removal of a sample, a tracer-free liquid porous block is placed in the cell filled with the tracer. Specific of volume Vs is added to the reservoir to eliminate any volume Copyright 2001 by the American Geophysical Union. change in the reservoir. Thus the reservoir concentration is Paper number 2001WR periodically diluted. Hence the solute concentration at any /01/2001WR time t in the reservoir is governed by 1519

2 IBARAKI: TECHNICAL NOTE Conductivity Probe Porous Block It is now defined that (x, y, z, p) = 'r(p) + V(X, y, Z, p). (7) Hence (5) can be written as P 'r+pv-- D I - 02v + + b-p] = 0. (8) Accordingly, the boundary conditions (6a)-(6c) can be written as v(o, y, z, p) = v(lx, y, z, p) = 0, v(x, 0, z, p) = v(x, Ly, z, p) = 0, v(x, y, O, p) = v(x, y, Lz, p) = 0. (9a) (9b) (9c) Magnetic Stirrer Figure 1. Schematic diagram of an experimental setup. Applying the finite Fourier transform inx, y, and z to (8) yields œxœz = nml,r3 [1- (--1)hi[1 (--1)m][1 - (--1)l]p 'r q-p OC r Ot fo z x o Oc z=0 x=o fo Lz fo Lx Oc dzdy + + dy dx q- rr ½r (t -- ti) = 0, 0 and Vr are the effective porosity and the volume of solution reservoir, respectively, and ( ) is the Dirac delta function. The three integral terms in (3) represent solute flux entering into the porous slab from the solution reservoir. These terms provide for coupling of the governing equations for the porous matrix (1) and the reservoir (3). The sampling procedure involving sampling and chemical analysis is appropriate when diffusion coefficients for organic chemicals are measured. There are no direct-reading elements that can be used for this purpose. In the illustrative experiment, results presented in this paper involve potassium chlorine solution with specific conductance measured continuously using a probe. The initial condition for the reservoir is i=1 y=o (3) Cr(O) = CO, (4) Co is the initial concentration in the solution reservoir. The Laplace transform is applied to (1) for c with the initial condition (2a). This step leads to D[02 ' ß ] P ' - b- + b- + 0 z 2] = 0, (5) p is the Laplace transform variable, which is defined as (x, y, z, p) = c(x, y, z, t) exp (-pt) dt. The boundary conditions (2b)-(2d) will be also transformed: (0, y, z, p) = 6(mx, y, z, p) = 'r(p), '(X, O, Z, p) = '(X, my, z, p): 'r(p), 6(x, y, O, p) -- 6(x, y, mz, p) =. 'r(p). (6a) (6b) (6c) Lx w + -r Ly + 7=0, ½(n, y, z, p) = v(x, y,. z, p) sin Lx ] dx, (n, m, z, p) = ½(n, y, z, p) sin dy, (n,m,l,p)= (n,m,z,p) sin Lz] dz. Equation (10) is rewritten as (lo) --[1-- (--1)n][1 (--1)mill-- (-1)l]LxmyLzp 'r(rlml r3) [ D'rr2 ( m t12 x m my 2 / z2) l -1 ß P+ +-r + ß (11) The inverse Fourier transform is applied to (11) to obtain v. After obtaining v, Laplace transformed concentration in the porous domain ( '(x, y, z, p)) can be calculated by using (7): '(X, y, Z, p) = 'r(p) 1 -- { 64p n=0 m=0 /=0 sin Lx Ly sin ((2/+ Lz 1)z-z) - [(2n ß p D'rr 2/(2n + 1) 2 my + 1)(2m + 1)(2/+ 1)] -1 (2m + 1) 2 q- 2 (12)

3 ß ß The Laplace transform is applied to (3) to obtain reservoir concentration in the Laplace domain ( 'r) with the initial condition (4): pe r -- C o x=0 dz dy + dx dz ør o L fo Lx 0 ' o Lx o L 0 ' Vs N y=0 z=0 dy IBARAKI: TECHNICAL NOTE 1521 dx] + rr exp (--pti)cr(ti) = 0. (13) 1=1 The total flux of ' at the interface x -- 0, which is used in (13), is obtained by differentiating (12)' fol Oa l xx=0 dz dy = -- r Lx,r4 ß Z [(2m + 1)(2/+ 1)] -2 n=0 m=0 /=0 256pLrLz I P + D ((2n Lx + 21) 2 (2m Ly + 1) 2 (2/+ L 2 1)2)1-2 ß ß -1'- 2 (14) The Laplace transformed reservoir concentration ( 'r) can be calculated by substituting (14) and the total flux of ' at the interfaces y = 0 and z = 0 into (13): 'r(p) = Co- rr E exp (--pti)cr(ti) i=1 5120D [ mxmcz E m=0 /=0 1 ß 1+ Vr,2T4 Olnml p-l, (15) OZnml = [(2rl + 1) (2m + 1)(2[+ 1)]_2( p +-- / 2)-1 (2n + 1)2 (2m + 1)2 (2/+ 1)2 = ß J nml Lx 2 Ly Lz Equation (15) is the final solution in the Laplace domain, Which shows concentration in the reservoir. This solution contains a combination of three infinite loops, which imposes a significant computational overload. However, one of the loops can be described with a ß function, and accordingly, only the combination of two infinite loops has to be evaluated, i.e., Co 1 Vs N t'r(p) =pf(p) pf(p) Vr exp (--pti)cr(ti), (16) 5120D F(p) = 1 + Vr T 4 mxmylz Knm n=0 m=0 ( pr LxLy(-2D,rLxLy + 2T2 ) Knm T12L zd 2 r pr LxLyh 1-2D,rLxLy + 2T2 4 T Lz -'- - J 4 DLrLx,r x y\ 1 pr21,31,3(-2d,rlxly- T L zd *r 2T2 ) pr LxLy 1-2D,rLxLy- 2T ( T Lz _.- i- J wit 4 DLrLx,r 81 R(4Ly2n Lx2m 2 + 4Lx2m + Lx2),r 2} nc ß {(2m + 1)2(2n + 1)2} -', T = plxlyr + 4D,r Lyn + 4D,r Lyn + D r Ly + 4D,r2Lx2m 2 + 4D,r2Lx2m + D qt 2 Lx, 2 T2 = (-pdlxlylzr D 2 r 2r LyLzn 2r D 2 r 2r :L, ylzrt 2r 2 D2 r2my2m2 z 4D 2 r 2 Lxœ m D 2 r 2 LxLz) / D r LxLzm The reader should note thathe first term in (16) represents the solution for the case no sampling of the reservoir occurs and the second term represents the reduction in concentration due to the dilution with sampling and fluid replace- ment. The concentration value of the reservoir at steady state, i.e., no matrix diffusion, is derived by applying final value theorem to (16). c r(t= ) = limp ' r(p ) p-->0 (17) = Co-- r E Cr(ti) 1 + Because all parameters in (17) except effective porosity 0 can be measured quite easily, I can use this equation to calculate the effective porosity of the porous block. That is, 0 = Co- rr.= Cr(ti) C r(t=o ) 1 RLxLrL z. (18) Final values of the solution in the time domain are obtained by numerical inversion of (16), which describes the tracer concentration in the reservoir in the Laplace domain. Numerical inversion of the transformed solution provides two advantages. First, the difficult step of deriving an analytical inverse is eliminated, and second, the transformed solution is generally easier to evaluate. This numerical inversion approach has been used by others [e.g., Moench and Ogata, 1981; Goltz and Oxley, 1991; Neville et al., 20 0]. For diffusion-dominated problems, there are several algorithms that yield accurate results. An inversion algorithm developed by de Hoog et al. [1982] was used in this study. This algorithm has been used for both numerical models and analytical solutions and has been applied to a broad spectrum of conditions, ranging from pure diffusion to almost pure advec-

4 ._ 1522 IBARAKI: TECHNICAL NOTE tion [e.g., Sudicky and McLaren, 1992]. The details of de Hoog et al. [1982] algorithm and implementation are given by Neville [1992]. The final solution (16) is implemented in a FORTRAN program, and a copy of the code is available from the author, free upon request Laboratory Experiment The experiments involved three sets of measurements, two with similar blocks of sandstone and one one with a kiln-fired clay block. The rock blocks were cut into slabs, 2 cm thick, 10 cm wide, and 10 cm high. A slab was suspended in a plastic reservoir with a volume of 190 cm 3 in each measurement. For the experiments to work in a reasonable time the slab size and reservoir volume needed to be matched. The size and the volume were determined by conducting preliminary numerical simulation trials based on the final solution (16). The experiments require the slabs to be saturated with deionized water. This procedure involved replacing the air in the pores of the slabs with nitrous oxide (NO2) gas that has high solubility in water. The porous slabs were placed in a chamber and maintained in a vacuum for 24 hours. After vacuum operation was completed, the chamber was flooded with NO2, and the slabs were kept in the chamber for 24 hours. Because the diffusion coefficient for the gas is >2 orders of magnitude greater than that for the solute, preliminary numerical simulation results showed that 24 hours were sufficient to 1.0 ( 0.9 o.--q 0.8 a) 1.0 ' 0.9 Measurem ent... ' Mathematical Simulation 0.70 ' ' (a) Time (sec) Figure 2. Measured and simulated relative specific conductance versus time for (a) sandstone and (b) kiln-fired clay. saturate the slabs completely with nitrous oxide. The porous slabs were then submerged so that saturation with deionized water would occur. The water-saturated porous slab was placed in the cell with a specified volume of water. Potassium chlorine solution was mixed instantaneously into the cell. The resulting dilution had be measured by applying this experimental method. These numerical simulations involve two hypothetical sets of rock samples with porosity values equal to 5 or 20%. In these trials, values of the diffusion coefficient are varied from D = 10-6 a specificonductivity of 1490/aS/cm. The specificonductance cm2/s to D = 10- o cm2/s. The relative concentration versus of the solution in the cell was measured continuously with data time calculation for various values of the diffusion coefficient recorded using a laptop PC. The solution was stirred in the reservoir using magnetic stirrers placed at top and bottom of the porous slab. This mixing maintained a uniform concentration distribution. The temperature was kept constant at 25øC. The experiments were conducted until the system reached steady state. With a slab of this size, it took days. The reader should note that there is no effect of solute convection into the slab because hydraulic head within the reservoir is constant and the permeability of the slab is low. D with porosities of 20 and 5 % are shown in Figures 3a and 3b, respectively. There are obvious differences in time response depending on the diffusion coefficient. There is much less differentiation for the calculations with 5% porosity (Figures 3b). However, the differences are appropriately large at 106 S (11.6 days) to make possible diffusion experiments feasible for blocks having this range in diffusion coefficient. The reader should remember that changes in relative concentration in the reservoir over Figures 2a and 2b illustrate measured specific conductance time solely depends on the magnitude of the diffusion coeffiwith time, plotted as relative specific conductance for sandstone and kiln-fired clay blocks, respectively. Using a trial and error process, it is possible to fit the experimental data using the solution obtained by numerical inversion of (16). The fitting process assumes that all parameters except the diffusion coefficient D are known. As is evident in Figures 2a and 2b, the analytical solutions fit the data well. The calculated diffusion coefficient for sandstone sample is 2.7 x 10-6 cm2/s. The results from the second experiment for sandstone are not cient if all other parameters, such as effective porosity, are specified in advance. For example, in a separate experiment, effective porosity can be calculated by using (18) and measuring the final relative concentration of solute in a reservoir that contains small pieces of a rock sample. Hence I can estimate the magnitude of a diffusion coefficient as low as 10-9 cm2/s within a measuring period of 12 days with an accuracy that depends on concentration monitoring technique employed and the magnitude of porosity. shown here; however, the second trial yields a diffusion coefficient of 3.1 x 10-6 cm2/s. The calculated diffusion coefficient for kiln-fired clay sample is 4.2 x 10-6 cm2/s. 4. Conclusion This paper describes a simplified technique for measuring 3. Limitation of Approach diffusion coefficients in porous block and an illustrative application that involves measuring the diffusion coefficient for Clearly, this experimental approach may not be appropriate sandstone and kiln-fired clay samples. This technique has three for all types of rocks, especially those with a small diffusion coefficient. A series of numerical simulations are conducted in order to investigate the range of diffusion coefficient which can advantages compared to existing approaches, such as sample sectioning or wafer diffusion experiments. The experiment setup is much simpler, and it is easier to prepare samples.

5 IBARAKI: TECHNICAL NOTE O cm2/s X Xx... 0-cm/s \ X x cm2/s ' cm2/s o-løcm2/s i i I IIIIII... I... I... I... I i i i i iii1... i... i... i... i Time ($ec) Figure 3. The relative specific conductance versus time for values of the diffusion coefficient for porosities of (a) 20% and (b) 5%. Solute concentration measurements are quite simple with no need to measure solute concentration within the porous block after a specified period. Finally, the magnitude of diffusion coefficient can be calculated as low as 10-9 cm2/s within a measuring period of 12 days if all other parameters are specified in advance. Acknowledgments. Funding for this research was provided by a grant award to M. Ibaraki and F. W. Schwartz from the National Science Foundation (EAR ). I appreciate the assistance of Alison Laughbaum in conducting the experiments. References Bradbury, M. H., and A. Green, Measurement of important parameters determining aqueous phase diffusion rates through crystalline rock matrices, J. Hydrol., 82(1-2), 39-55, de Hoog, F. R., J. H. Knight, and A. N. Stokes, An improved method for numerical inversion of Laplace transforms, SIAM J. Sci. Stat. Comput., 3(3), , Goltz, M. N., and M. E. Oxley, Analytical modeling of aquifer decontamination by pumping when transport is affected by rate-limited sorption. Water Resour. Res., 27(4), , Hubert, A., and R. Calvet, D6termination du coefficient de diffusion mo16culaire de solut6s dans des milieux poreux satur6s en eau, Ann. Inst. Natl. Rech. Agron., Ser. A, 30(3), , Ibaraki, M., and E. A. Sudicky, Colloid-facilitated contaminant transport in discretely fractured porous media, 1, Numerical formulation and sensitivity analysis, Water Resour. Res., 31(12), , Krom, M.D., and R. A. Berner, The diffusion coefficients of sulfate, ammonium, and phosphate ions in anoxic marine sediments, Limnol. Oceanogr., 25(2), , Moench, A. F., and A. Ogata, A numerical inversion of the Laplace transform solution to radial dispersion in a porous medium, Water Resour. Res., 17(1), , Neville, C. J., An analytical solution for multiprocess nonequilibrium sorption, MS thesis, Dep. of Earth Sci., Univ. of Waterloo, Waterloo, Ontario, Canada, Neville, C. J., M. Ibaraki, and E. A. Sudicky, Solute transport with multiprocess nonequilibrium: A semi-analytical solution approach, J. Contam. Hydrol., 44(2), , Novakowski, K. S., and G. van der Kamp, The radial diffusion method, 2, A semianalytical model for the determination of effective diffusion coefficients, porosity, arid adsorption, Water Resour. Res., 32(6), , Sudicky, E. A., and R. G. McLaren, The Laplace transform Galerkin technique for large-scale simulation of mass transport in discretely fractured porous formations, Water Resour. Res., 28(2), , van Rees, K. C. J., E. A. Sudicky, P.S. C. Rao, and K. R. Reddy, Evaluation of laboratory techniques for measuring diffusion- coefficients in sediments, Environ. Sci. Technol., 25(9), , M. Ibaraki, Department of Geological Sciences, The Ohio State University, 125 South Oval Mall, Columbus, OH (ibaraki. (Received June 26, 2000; revised December 12, 2000; accepted December 28, 2000.)