Modelling flow and solute transport in fractured porous media

Transcription

1 Modelling flow and solute transport in fractured porous media A. Peratta & V. Popov Wessex Institute of Technology, Souuthampton, UK Abstract A BEM numerical model is presented for the flow and solute transport in fractured porous media, which is based on the dual reciprocity method - multi domain scheme (DRM-MD). The model solves the 3D flow equation based on the Darcy law in the coupled system of porous media, fracture network and fracture intersections. Unlike the discrete fracture models, which for simplicity use Fick's law in the porous media, the present model solves the complete 3D advection-dispersion equation with reaction, which is then coupled with the other two models, the fracture network and the fracture intersections. In this work the developed model is used for long-term risk assessment of hazardous waste disposal in underground abandoned mines. The following sections describe three different models: the continuum model (3D), the discrete-fracture model (2D) and the fracture intersectionlpipe model (ID). The three models are integrated into a single modevsimulation tool, which accounst for flow and transport through a system consisting of porous matrix, fractures and fracture intersections. The continuum, the fracture and the pipe models are tested and their performance verified using analytical solutions. This is done separately for each model, before the integration, and for the complete model after the three models are integrated. The results show that the integrated simulation tool has sufficient capability for modelling geometries and processes of complexity that are encountered in a real-world mine repositories. 1 Introduction The importance of the flow and transport processes through fractured porous media has influenced development of different approaches to predict the

2 64 Boutrdary Elcmatr~~ XXV outcome of these phenomena. A review of some of the current modelling trends can be found in [l], [2]. The main objective is to determine the capacity of reference mine repositories to provide safe and permanent isolation of hazardous waste. Because of the possibility to havc large fracture zones intersecting the repository, the continuum approach is coupled with the discrete fracture approach [3]. Thc inclusion of fractures and fracture intersections in the model are important as they represent a fast way for transport of contaminants. The numerical technique that is used to solve the partial differential equations defined by the models is based on the dual reciprocity method - multi domain approach (DRM-MD) [4]. 2 Continuum (3D) model 2.1 Continuum flow model The flow in the porous matrix is dcscribed by where P = - i- z - hydraulic head [L] Pg a@ v.(k,vq>)+& =S,- at KR - hydraulic conductivity of porous matrix [Lit] S, =pg(a+nd) - specific storativity [IL] a=-- I an 2 - coefficient of rock matrix compressibility [L /h!] I-n3P n - porosity [%l P - prcssure [NL'] 1 JP 2 p = -- - coefficient of compressibility of the fluid [L m] P JP p - density [M&'].g = 9.81 [m/s2] z - elevation [L] Q,( - sources or sinks [Ilt] Equation (1) is valid under the following assumptions: - isothermal conditions (T=const.) - homogeneous fluid (c = const.) - Kn is practically independent of pressure changes

3 - SR and Kfi are unaffected by variations in n - spatial variations in p are much smaller than the local temporal ones The flow fi'ield is described by Darcy's law Between two sub-domains the lollowing matching conditions must be satisfied (see Figurc 1 ): For hydraulic hcad: 0, (3) Figure 1 : Interface between two kedia with different properties. In the case when: p,=pz=const; pi=p2=const (4) reduces to k, V@/,. ii, =-k,v@12. Z2 When c = const and KR = const inside each sub-domain, (1) becomes 2.2 Continuum transport model The following equation is often used when the solute transport obeys advectiondispersion equation: where c, is the concentration of chemical i, D is dispersion coefficient D = ljla + D~ (8) and a: is the dispersivity [L], D, is the molecular diffusion coefficient [~'/t], V is the average water velocity [L/t] in the voids of the rock, which we assume to be described by Darcy's law. The flow equation must be solved first to find V

4 66 Boutrda~y Elcmatr~~ XXV before the advection-dispersion equation can be solved. The last term on the right-hand side of (7), zc, represents the decay of the chemical and z= ln(2)/tlc? [llt], where Tin represents half-life of the chemical. During the decay other chemicals will be produced which may be necessary to be observed. In such case (7) is solved for each chemical of interest. The term that contains Sj [M/L~~] represents the sum of all the sources of the given chemical as a result of decay or chemical reactions of other chemicals. R represents retardation factor. 3 Fracture (2D) model The classical method in the previous section is not unreasonable if the material is with little anisotropy in the dispersion (uniformly distributed fractures of approximately same size). However, in practice, fractures will give rise to short cuts through the rock in which contaminant can rapidly spread (preferential paths inside large blocks of rock, perhaps further from the repositories). The continuum model describe above is not suitable for such problems. This problem requires that separate equations be used for the fractures and porous matrix, which are then integrated into a single system that describes the complete flow in the fractured porous media (see Figure 2) Cr; fracture - - Figure 2: Integrated model of porous matrix and fracture. 3.1 Fracture flow model The flow in the fractures is described by the following equation qn)p + qnll- f3aj V.&,V@)- + QF =SF (9) a where Q [L] is hydraulic head, SF =&l [1/L] is specific storativity of fracture, KF [Llt] is hydraulic conductivity of fracture, I-+ [L] is aperture of fracture, qn

5 [Llt] is fluid leakage flux across surfaces 1' or 1- (see Figure 3), and QF [l/t] represents sources or sinks. If the walls of the fracture are parallel and smooth, KF is given by in which case (9) becomes This problem is a 2D problem embedded in a 3D space. In other words, the Laplace operator depends on two co-ordinates only while the field variables are functions of three spatial co-ordinates (see Figure 3). It can be shown that equation (11) will preserve its form when transformed from 3D to 2D space providing that the 2D space is flat and the basis vectors for the 2D co-ordinate system are orthogonal. 3.2 Fracture transport model X 1 Figure 3: Simplified representation of a fracture. Thc transport equation for the fracture is given by where subscript "F' stands for "inside fracture" and thc same notation is used as in (7), RP includes the effect of sorption on the porous matrix wall, and q is the current of chemical crossing the fracturelporous matrix interface. 4 Fracture intersection 1D model Where two fractures intersect the flow would correspond more to flow through pipes, and this has to be accounted for as well. In Figure 4 an interscction of three fracturcs is shown that crcales three pipe-like structures.

6 68 Boutrdary Elcmatr~~ XXV 4.1 Flow in intersections of fractures Figure 4: Intersection of three fractures. The theory of flow and transport in "pipes" is similar to the one presented for fractures. The following equation describes the flow in apipc where: A,, [L'] is the area of the cross section of the pipe, Kp [Lltj is the hydraulic conductivity of the "pipe", H, [L] is the aperture of the J-th fracture that intersects in that "pipe", 9, [L/t] is fluid flux into thc pipc from the j-th fracture, and Sp [In] is the specific storativity of the "pipe". The relation between the diameter and thc hydraulic conductivity of the pipes can be obtained applying the solution of the Poiseuille equation. Adopting this approach yields the followmg expression for thc hydraulic conductivity: where u,, is the radius of the pipe and p is viscosity. Alternatively Kp can be given as input data. 4.2 Transport in intersections of fractures Thc transport equations is analogous to the one for hctures wherc 4 is the coordinate and V[ is the velocity along the pipe and q,. is chemical flux fracture/pipe. 5 Verification of the models The numerical technique that was used to solve the 3D and 2D equations is based on the DRM-MD [4]. The 1D problem was solved using domain

7 integration BEM schcme. Comparisons between numerical and analytical results were made in a set of examples under different conditions such as domain size, mesh and boundary conditions. The 3D, 2D and ID models were first separately compared towards analytical solutions and then they were again compared to analytical solutions after the three models were coupled together. In all the cases the agreement with the analytical solutions was good. 5.1 Time step adaptation The calculation time step was continuously adapted in order to control the relative error of thc solution during the transient period. Figure 5 shows the time step adaptation for a transient case. To preserve the accuracy of the solution lower and upper limits have been imposed at 6t = 1 yr and 6t = 25 yrs respectively m b P E V).- E l Time [years] Figure 5: Timc stcp adaptation during the transient period. 5.2 Influence of the EDZ on the flow in the near-field The excavation-disturbed zone (EDZ) was included in the model by considering it as an extended layer in the vicinity of the room, which was modelled using the same 2D model used for fractures. The main assumption taken into account for modelling the following example is that the EDZ lhickncss is very small in comparison with its typical length and width. In this way, it can be seen as a two dimensional discontinuity embedded in a three dimensional space. In lhis example thc hydraulic conductivities used were: for the EDZ E-7 ds, for the clay E-l 0 m/s, for the rock E-9 mls, for the fraclurcs E-7 m/s and for the pipes E-6 m/s. Thc size of the domain is 300mx300mx450m and the size of the room is 100mx100mx150m. In Figure 6 the flow is shown around the room, which is filled with clay. In this case the EDZ is not includcd in the model. It can be seen that the velocity field around thc room has got a component that is directed away from

8 70 Boutrdary Elcmatr~~ XXV the room. The reason is the lower permeability inside the room, which diverts part of the flow around the room. The effect of the EDZ is obvious in Figure 7 where the flow velocity has got a component, which is directed towards the room. The reason for this effect is that the EDZ has got higher permeability than the surrounding rock so part of the fluid that would otherwise flow through the surrounding rock is diverted towards the EDZ. Figure 6: Flow around a room embedded in rock. The room is filled with clay and the EDZ is not included in the model. Figure 7: Flow around a room embedded in rock. The room is filled with clay and the EDZ is included in the model.

9 5.3 Transport from a room filled with clay and hazardous chemicals In the next example the solute transport is solved for the model from the previous section that includes the EDZ around the room. It is assumed that at t=o the clay is saturated with water. The initial concentration inside the room is c = 1. In the Figures S and 9 the transport of the hazardous chemicals is shown fort = 600 and 1200 years, respectively. Figure S: Dispersion of hazardous chemicals at t=600 years. Figure 9: Dispersion of hazardous chemicals at t=1200 years.

10 72 Boutrdary Elcmatr~~ XXV 6 Summary A BEM DRM-MD model has been developed for flow and solute transport in 3D fractured porous media. The model can be used for various environmental risk assessment purposes including underground repositories for hazardous waste. In the present case it is used to assess the long-term risk from disposal of hazardous waste in abandoned undergournd mines. The numerical scheme was tested for a case of a room in fractured rock filled with hazardous chemicals embeded in clay. The numerical scheme showed good perfomance for problems of flow and transport throught fractured rock. The approach has the possibility to include the excavation disturbed zone (EDZ) in the model. The analysis shows that the EDZ has significant influence on the.flow and transport in the near-field. Future improvements and tcsts of the model can be followed on the following www address: ht~p:llwww.wessex.ac.umresearch/divisions/lowriskdt.html. Acknowledgements This rcsearch was sponsored by the LowRiskDT projcct (Contract number EVGl-CT ) - part of the FP5, Energy, Environment and Sustainable Development European Commission Programme. References [l] Pruess, K., Faybishenko, B., Bodvarsson, G.S. (1999), "Alternative concepts and approaches for modelling flow and transport in thick unsaturated zones of li-actured rocks", Journal of Contaminant Hydrology 38, Bcar, J., Chin-Fu Tsang, Ghislain de Marsily (1993), Flow and contaminunt transport in fractured rock, Academic Press, Inc., San Diego. [3] Therrien, R., Sudicky, E.A. (1994), "Three-dimensional analysis of variablysaturated flow and solute transport in discretely-fractured porous media", Journal of Contaminant Hydrology 23, [4] Popov, V., Power, H. (1999), "The DRM-MD integral equation method: An ccficicnt approach for the numerical solution of domain dominant problems", International Journal for Numerical Methods in Engineering, 44,