Application of the Multiscale Finite Element Method to the numerical modeling of regional land subsidence

Transcription

1 2th International Congress on Modelling and Siulation, Adelaide, Australia, 1 6 Deceber Application of the Multiscale Finite Eleent Method to the nuerical odeling of regional land subsidence Shuun Yea, Yuqun Xuea, Jichun Wua, Hanei Wangb, Zixin Weib and Xuexin Yanb a Key Laboratory of Surficial Geocheistr Ministry of Education; School of Earth Sciences and Engineering, Naning Universit Naning, P.R. China Eail: sye@nu.edu.cn b Shanghai Institute of Geology Surve Shanghai, China Abstract: For a regional land subsidence proble, there are difficulties for the traditional finite eleent ethod to solve the regional groundwater flow equation involved. A new finite eleent ethod - ultiscale finite eleent ethod (MsFEM) applied to resolve three diensional groundwater flow proble- is introduced. Three advantages ake the MsFEM very useful for groundwater flow siulation in a large region with highly heterogeneous porous edia. One advantage is that this ethod can significantly save coputing efforts by coarse grid esh. The variation of paraeters in an eleent is brought to the ultiscale base functions. Second one is that this ethod can to a large extent avoid large aspect ratios of eleents by cobining aquifers and aquitards in one eleent. The third advantage is that the hydraulic heads in different depths of aquitards or aquifers could be rather accurately interpolated using the ultiscale base functions and the nodal heads of the eleent by MsFEM. The third advantage is useful for the subsidence odeling. A nuerical exaple is done to verify the advantages of MsFEM. Finall the MsFEM is applied to solve the regional land subsidence odel of Shanghai, which includes a three-diensional groundwater flow atheatical odel and a one-diensional subsidence odel. The traditional two-step approach is followed in this stud with the hydrodynaics of the puped ultiaquifer syste first siulated by a 3-D groundwater flow odel and the subsidence then coputed with a 1-D subsidence odel with the pore pressure field specified as an external distributed source of strength within the porous ediu. The siulation period is fro March, 1986 to Deceber, Using MsFEM, the aquifer syste in Shanghai is divided into six layers, which eans one aquifer and one aquitard are in one layer with an average thickness of about 5 eters. Every eleent in MsFEM is subdivided into 7 sub-layers in the vertical direction and every horizontal layer in an eleent is divided into 4 hexahedral eleents. So each eleent is subdivided into 28 hexahedral sub-cells, which allows the heterogeneity of aquifers and aquitards to be considered by ultiscale base functions. The aquifers syste in total is discretized into 19,996 eleents with a total of 25, nodes. The hydraulic heads in any depth of aquitards or aquifers could be interpolated using the ultiscale base functions and the nodal heads of the eleent. It is very useful for the settleent calculation for the aquitards or aquifers after hydraulic heads at nodes of eleents are obtained by MsFEM. The application of MsFEM significantly saves coputing efforts, effectively decreases the aspect ratios of eleents and obtains good results, which proves the MsFEM can be applied to solve the real regional 3D groundwater flow probles, and it is an effective nuerical ethod for regional land subsidence odeling. Keywords: Multiscale finite eleent ethod, Nuerical odeling, Regional Land subsidence 2744

2 1. INTRODUCTION Land subsidence caused by excessive groundwater withdrawal has occurred in any regions in the world. In China, land subsidence is becoing a serious environental proble, occurring ainly in 17 provinces (cities) located in the eastern and central regions, with a total subsidence area of ore than k 2 (Xue et al. 25). The Yangtse Delta and the Huang-huai-hai Plain are the two biggest and ost serious land subsidence areas in China. By the end of the last centur the area where the cuulative subsidence exceeded 2 in the Yangtse Delta was approxiately 1, k 2. Land subsidence has occurred over alost all of 5, k 2 of the Huang-huai-hai Plain. Such large land subsidence regions have never been reported out of China. Land subsidence odel includes ground water flow odel and subsidence odel. In this paper the traditional two-step approach is followed, with the hydrodynaics of the puped ultiaquifer syste first siulated by a 3-D groundwater flow odel and the subsidence then coputed with the aid of a 1-D subsidence odel with the pore pressure field specified as an external distributed source of strength within the porous ediu. Regional groundwater flow odeling is the basis for siulating the regional land subsidence, because the pore pressure change fro the groundwater flow odel is used as input data to the subsidence odel. Several issues have ade the large regional groundwater odeling quite a difficult task. One issue is the study areas are very large, any aquifers and aquitards are to be odeled, and the aquifers and aquitards are highly heterogeneous. Three diensional (3D) groundwater flow atheatical odel is usually developed for the siulation of regional land subsidence because of the anisotropic and highly heterogeneous porous edia, different copressibility of aquifers and aquitards and also the 3D groundwater flow. It is difficult to apply the FDM or traditional FEM to solve the 3D groundwater flow atheatical odel for land subsidence regions like the Yangtse Delta and the Huang-Huai-Hai Plain. Because very any coputer resources and long coputational tie are required. Second issue is the uch greater horizontal extent than the vertical extent of the large regional ground water flow odeling. Another iportant proble of the large regional land subsidence odeling is the suitable way to accurate calculate the hydraulic heads in aquitards. Mostly the land subsidence is caused by the settleent of aquitards which are usually coposed by clay edia. The aquitards can be very heterogeneous along the vertical direction and have to be divided into several soil layers according to its lithology. For exaple, the first aquitard in Yangtse Delta area was the ain copressible layer there in the history. It is 6-15 thick, coposed of two different soft soil layers, and a hard soil layer. The aquitard have to be divided into several different layers because of the heterogeneity when the traditional FEM is applied, which will further worsen the two probles of a lot of coputation efforts and eleents with large aspect ratios. As entioned above, it is difficult to apply the traditional finite eleent ethod to solve the regional three diensional groundwater flow odel when doing the regional land subsidence siulation. Therefore it s necessary to find a new nuerical ethod that can decrease the nuber of eleents, overcoe the proble of large aspect ratios of eleents, and give accurate results. The ultiscale finite eleent ethod (MsFEM) (Ye et al. 24) applied in this paper can serve these purposes. 2. ADVANTAGES OF THE MSFEM IN SOLVING GROUNDWATER FLOW MODELS The MsFEM has the advantages of decreasing the nuber of eleents by including heterogeneous edia in one eleent to save the coputing efforts and overcoing high aspect ratios of eleents to gain accurate results. Another advantage of MsFEM is that the hydraulic heads in different depths of aquitards and aquifers could be rather accurately interpolated using ultiscale base functions and nodal heads of the eleent. Details of MsFEM can refer to the paper by Ye et al. (24). One nuerical exaple is done below to verify these advantages. The study area is a rectangular doain covering 1k 1k 28 (Figure 1a). The siulated aquifer syste is coposed of seven confined aquifers and seven aquitards (Figure 1b). The aquifers and aquitards are distributed in turns in the vertical direction. All the aquifers and aquitards are 2 eters thick and are all heterogeneous. Hydraulic conductivities fro left to right in all aquifers vary gradually fro 1/d to 196/d, which eans they increase 5/d every 25 eters fro left to right. Hydraulic conductivities fro left to right in all aquitards vary gradually fro.5/d to.98/d, which eans they increase.25/d every 25 eters fro left to right. The left and right sides of the area are odeled as first type boundaries with the head on the left side is 1, and head on the right side is 1. Other sides are odeled as ipereable boundaries. MsFEM and traditional finite eleent ethod (FEM) are used to solve this 2745

3 Ye et al., Application of the Multiscale Finite Eleent Method to odeling of regional land subsidence exaple proble. The atheatical odel has no analytical solution, so the fine scale results obtained by the traditional finite eleent ethod with fine eleents (FEM-F) are regarded as the exact solutions. Using FEM-F, the study area is divided into 14 layers in the vertical direction, which eans the aquifers are separated fro the aquitards. Every horizontal layer is divided into 1,6 hexahedral eleents. Therefore the study area is divided into 22,4 hexahedral eleents and 25,215 nodes. Using MsFEM, the study area is divided into 2 layers in the vertical direction, which eans 4 aquifers and 3 aquitards in the first layer and 4 aquitards and 3 aquifers in the second layer. Every horizontal layer is divided into 4 hexahedral eleents. Therefore the study area is divided into 8 hexahedral eleents and 1,323 nodes. In order to copute the base function nuericall every eleent in MsFEM is subdivided into 7 layers in the vertical direction, which eans the aquifers are separated fro the aquitards, divided into 4 hexahedral eleents for each horizontal layer, and then subdivided into 28 hexahedral eleents (Figure 1c). Using FEM, the study area is divided into 14 layers in the vertical direction. Every horizontal layer is divided into 4 hexahedral eleents. Therefore the study area is divided into 5,6 hexahedral eleents and 6,615 nodes. The eleents in FEM are hoogeneous by averaging the hydraulic conductivities in the eleents. The relative errors of the results of FEM and MsFEM copared to these of FEM-F in the interface y=5, and z=-14 are shown in Figure 2. All the results in Figure 2 are the calculated heads at the nodes of eleents. Table 1 shows the coparison of the nuber of nodes, CPU tie ( by the PC of Intel(R) Core 2 Quad CPU 2.39GHz, 3.23GB Meory), ratio of horizontal extent to vertical extent of eleents, and relative errors of results of MsFEM, FEM and FEM-F. Fro Figure 2 and table 1, one can see that accurate results are gained by MsFEM with uch less coputing eory -19 ties less nuber of nodes, uch less CPU tie - 64 ties faster, and 3.5 ties less aspect ratios of eleents for the proble. Although 4 aquifers and 3 aquitards are put in one eleent in the first layer and 4 aquitards and 3 aquifers are put in one eleent in the second layer by MsFEM, the hydraulic head in any point in the eleent can be rather accurate interpolated by the ultiscale base functions which can capture the sall scale heterogeneity (a) (b) (c) Figure 1. Illustration of the nuerical exaple (a) 2-D horizontal diagra (b) 3-D diagra (c) subgrid of an eleent inforation. Figures 3 shows the relative errors between the interpolated heads by MsFEM and the nodal heads by FEM-F on section y=2,5 and z=-6. Figures 3 also shows the relative errors between the nodal heads by FEM and FEM-F on section y=2,5 and z=-6. The relative errors of the results of FEM are large as shown in Figures 2. It is caused by the coarse grid which siplifies the heterogeneity inforation and has large ratio of horizontal extent to vertical extent. We can find that the interpolated heads by MsFEM are very close to the nodal heads by FEM-F, which eans the head at any point in the eleents can be rather accurately interpolated by the nodal heads and the ultiscale base functions. Table 1. Coparison of different nuerical ethods Method Nuber Nodes of CPU tie (second) Ratio of horizontal extent to vertical extent of eleents Maxiu relative error (%) FEM-F 25, FEM 6, MsFEM 1,

4 So the hydraulic heads in different depths of aquitards or aquifers could be rather accurately interpolated using the ultiscale base functions and the nodal heads of the eleent by MsFEM without dividing the heterogeneous aquitards or aquifers into several layers of eleents. The interpolated heads can be further used to effectively calculate the settleent of aquitards or aquifers by the subsidence odel. 3 3 Relative error(%) 2 1 FEM MsFEM Relative error(%) 2 1 FEM MsFEM x() Figure 2. Relative errors of the results of FEM and MsFEM on section y=5, and z= x() Figure 3. Relative errors of the results of FEM and MsFEM on section y=2,5 and z= APPLICATION OF MSFEM TO LAND SUBSIDENCE MODELING IN SHANGHAI The land subsidence in Shanghai induced by excessive groundwater withdrawal was detected in Shanghai cit lying in the Yangtse Delta, is one of the cities which experienced the ost severe Figure.4 The geographic location of stu land subsidence in China (Figure 4). To siulate the land subsidence in Shanghai, 3-D groundwater flow atheatical odel and 1-D subsidence odel are developed. The MsFEM is applied to solve the 3-D groundwater flow proble, and then the calculated hydraulic heads is used as input data to 1-D subsidence odel Sedientary layers in Shanghai The study area is Shanghai, with an area of about 5,3 k 2.The aquifers syste is coposed of Quaternary unconsolidated sedients. The sedients are about 2-35 thick ust lacking in Figure 4. The geographic location of study area. soe local places. The bedrock is located under the sedients. The unconsolidated sedients are coposed of cla sandy clay and sand, in layers, in vertical direction. They are divided into aquifers and aquitards according to lithology. The sedients are divided into the phreatic aquifer (A), first aquitard which can be divided into first soft layer, second soft layer, second hard layer and sei-confined aquifer in soe places (B1), first confined aquifer (A1), second aquitard (B2), second confined aquifer (A2), third aquitard (B3), third confined aquifer (A3), fourth aquitard (B4), fourth confined aquifer (A4), fifth aquitard (B5), fifth confined aquifer (A5), and sixth aquitard (B6) (Figure 5). The aquifers and aquitards are not continuous in the study area, and do not exist in soe places Matheatical odel of land subsidence The traditional two-step approach (Gabolati and Freeze, 1973; Teatini et al., 211) is followed in the present analysis, with the hydrodynaics of the puped ultiaquifer syste first siulated by a 3-D groundwater flow odel and the subsidence of Shanghai then coputed with a 1-D subsidence odel with the pore pressure field specified as an external distributed source of strength within the porous ediu. 2747

5 The groundwater flow field presents obvious three-diensional flow in Shanghai. An anisotropic, heterogeneous and three-diensional groundwater flow atheatical odel is developed as follows: H H ( K i ) + ω = S i, = 1,2, 3 ( x, z) Ω (1) s x x t i H ( x, z,) = H ( x, z) Ω (2) H x, z, t) H = H ( x, z, t) K t,( x, z) Γ (3) ( = Γ H n i, ni = t Γ2 q( x, z, ) t,( x, z) Γ (4) where H is hydraulic head, K is hydraulic conductivity and S s is specific storage, ω is the source or sink ter, H is hydraulic head at initial tie, H 1 is hydraulic head on the first type boundaries, Γ 1 is the first type boundaries, q is the flow on the second type boundaries, Γ2 is the second type boundaries, n is the noral vector of the second type boundary.. There is no obvious horizontal displaceent of sedients in Shanghai, so one diensional vertical displaceent of sedients is assued. The traditional 1-D nonlinear subsidence odel is followed in the present analysis. The tie dependent deforation of any aquifer or aquitard is given by ΔL = =1 γ α ΔH ( t) ΔZ (5) where Δ L is deforation, is the total nuber of sedientary layers in an aquitard or aquifer, Δ Z is the thickness of the th sedientary layer, γ is specific weight of water, ΔH is the change of hydraulic head in a tie period, and α is volue copressibility of sedients which depends on the hydraulic head as follows α = α ke (H H p ) α = α kv (H < H p ) (6) Where α ke is elastic volue copressibilit α kv is plastic volue copressibilit and H p is the hydraulic head corresponding to the preconsolidation stress. The α is related to S s by S s = γ ( α + nβ ) (7) Where γ is specific weight of water, n is porosit β is volue copressibility of water. For this 1-D subsidence odel, it is very siple without using MsFEM. The deforation of any aquifer or aquitard is calculated according to equation (5) with the changes in hydraulic heads calculated by the 3-D groundwater flow atheatical odel Nuerical Modeling There are 6 aquifers and 6 aquitards which have different thicknesses, with the thin layers being less than 1 thick (Figure 5). The horizontal extent of the study area is uch larger than the vertical extent. Using MsFEM, the aquifer syste in Shanghai is divided into six layers, which eans one aquifer and one aquitard are in one layer with an average thickness of about 5 eters. The aquifer syste is discretized into hexahedral eleents. The hexahedral eleents are ostly square in the horizontal direction, but becoe irregular quadrangle in soe local places like at the observation wells and puping wells. The size of the eleent in the horizontal direction is about 1,2 1,2. The ratio of horizontal length to the vertical length of the hexahedral eleent is about 24. This ratio is a little large, however the aquifers syste is not divided into finer eleents due to the increase in calculation efforts. The ratio can reach several hundred for soe aquifers or aquitards with thicknesses of less than 1 if the traditional FEM is applied. Every eleent in MsFEM is subdivided into 7 sub-layers in the vertical direction and every horizontal layer in an eleent is

6 divided into 4 hexahedral eleents. So each eleent is subdivided into 28 hexahedral subcells, which allows the heterogeneity of aquifers and aquitards to be considered by ultiscale base functions. This is the pivotal difference between the MsFEM and the traditional FEM. Subcell grid of each eleent looks like the one shown in Figure 1c. The spatial distribution of aquifers and aquitards in each eleent could be different, which depends on the distribution of sedients. For exaple, the fifth layer of eleents include the fourth confined aquifer and the fifth aquitard, and the fourth confined aquifer accounts for 3 to 6 sub-layers and the fifth aquitard accounts for 1 to 4 sub-layers in an eleent. Each aquifer or aquitard has a different area, so the nuber of eleents in each layer is different as shown in table 2. The aquifers syste in total is discretized into 19,996 eleents with a total of Figure 5. Scheatic stratigraphic section in Shanghai. 25, nodes. The hydraulic heads in any depth of aquitards or aquifers could be interpolated using the ultiscale base functions and the nodal heads of the eleent. It is very useful for the aquitards or aquifers to calculate the settleent after hydraulic heads at nodes of eleents are obtained by MsFEM. The siulation period is fro March, 1986 to Deceber, The period is separated into 51 tie steps, with each tie step being 9 days. The odel calibration stage includes the first 35 tie steps while the odel verification stage includes tie step 37 to tie step 51. Tie unit is day. Table 2. 3D grid of the aquifers syste of Shanghai No. of layer Coposition Nuber of eleents 1 A and B1 3,476 2 A1 and B2 3,659 3 A2 and B3 3,651 4 A3 and B4 3,639 5 A4 and B5 3,59 6 A5 and B6 1,864 The initial tie of the flow atheatical odel is March 3, The groundwater levels at the nodes on the top of each eleent layer are obtained by Kriging interpolation of the groundwater level data of the corresponding aquifers at the initial tie, as those on the top of the fifth layer of eleents shown in Fig. 9. The initial groundwater levels in the fifth confined aquifer are also assigned to the nodes on the botto of the sixth layer of eleents. There is no observation groundwater level data in the phreatic aquifer prior to The groundwater levels in the phreatic aquifer on March 3, 1993 are approxiately considered as the initial water levels, because the groundwater level didn t change drastically over any years without withdrawal. The subsidence odel has the sae initial tie as the flow odel and the vertical displaceents of all the nodes are zero initially in the subsidence odel. The first type boundary conditions of each layer for the flow atheatical odel are obtained by Kriging interpolation of the observation well data close to the boundaries. The top boundary of the flow atheatical odel is the phreatic aquifer and the botto boundary is bedrock which is assued as a no flow boundary. The top of the vertical one diensional subsidence odel is land surface which is a free surface, and the botto of the subsidence odel is bedrock which is a confined surface without displaceent. The odel is calibrated and verified against historical records of hydraulic heads and land subsidence. This is done by trial and error with the hydroechanical paraeters of the groundwater syste accoodated as follows. First the groundwater flow odel is validated so as to reproduce the head behavior in tie. Then the head change fro the 3-D flow odel is used as the input data to the 1-D subsidence odel. The siulated land subsidence and sedient settleent are verified against bench ark and extensoeter records, with the procedure iteratively repeated until a satisfactory atch is achieved. The results fro the subsidence odel are copared with the available bench ark easureents in Figure 6 showing the coputed land subsidence and the easured settleent in Shanghai over the period

7 1998. The agreeent of land subsidence distribution is deeed to be satisfactory if considering soe land subsidence caused by high building construction or soe underground engineering activities land subsidence () 2 1 land subsidence () (a) Figure 6. Siulated land subsidence (a) and easure land subsidence (b) over the siulation tie. 4. CONCLUSIONS To siulate the regional land subsidence proble, the MsFEM is introduced to solve the regional threediensional groundwater flow atheatical odel of the regional land subsidence atheatical odel in this paper. The MsFEM has the advantages of decreasing the nuber of eleents by including heterogeneous edia in one eleent to save the coputing efforts and overcoing the high aspect ratios of eleents to gain accurate results. Another advantage of MsFEM is that the hydraulic heads in different depths of aquitards and aquifers could be rather accurately interpolated using ultiscale base functions and nodal heads of the eleent. It is iportant to fatherly obtain good results of the settleent of aquitards and aquifers with obvious heterogeneity along the vertical direction, and at the sae tie avoid the probles of significant coputational tie and eory and high aspect ratios of eleents of the traditional FEM. These advantages are verified by a nuerical exaple. The MsFEM is applied to solve the 3D groundwater flow atheatical odel of Shanghai which is the iportant part of land subsidence atheatical odel. It significantly saves coputing efforts, effectively decreases the aspect ratios of eleents and obtains good results, which proves the MsFEM can be applied to solve the real regional 3D groundwater flow probles, and it is an effective nuerical ethod for regional land subsidence odeling. (b) ACKNOWLEDGMENTS The paper is financially supported by NSFC No REFERENCES Gabolati, G. and Freeze, R.A. (1973). Matheatical siulation of the subsidence of Venice:1. Theory. Water Resources Research, 9(3): Teatini, P., Castelletto, N., Ferronato, M., Gabolati, G., and Tosi, L. (211), A new hydrogeologic odel to predict anthropogenic uplift of Venice. Water Resources Research, 47, W1257, doi:1.129/211wr19. Xue, Y.Q., Zhang, Y., Ye, S.J. et al (25). Land subsidence in China. Environental Geolog 48(6): Ye, S.J., Xue, Y. and Xie, C. (24). Application of the ultiscale finite eleent ethod to flow in heterogeneous porous Media. Water Resources Research,4, W922, doi:1.129/23wr