A computer model to simulate the tree-foundation system Vicente Navarro*, Miguel Mena, Miguel Candel, Ángel Yustres and Beatriz García Departamento de Ingeniería Civil y de la Edificación Universidad de Castilla-La Mancha

Abstract A new tool for characterizing the soil-moisture changes caused by trees growing close to foundations is presented. The movements induced in foundations in the proximity of roots, related with shrinkage/swelling caused by those changes in soil moisture, can be evaluated this new tool.

Key Words: Unsaturated flow, trees, foundations, shrinkage.

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Idealizated problem: the numerical solver 2NSAT

In 2NSAT, the new tool proposed, the water flow is described by a generalized Darcy’s equation: K ⋅κ (∇P + γ ) q=− (1.1)

µ

where q is the seepage vector, P is the water pressure, K, is the intrinsic permeability [L2], κ is the relative permeability, and µ and γ are, respectively, dynamic viscosity and specific weight of water. The relative permeability is computed combining the Brooks and Corey (1964) and the Burdine (1953) models:

κ =Se3+2/λ

(1.2)

where the effective saturation Se is defined as: Se =

(θ − θ R ) (θ S − θ R )

(1.3)

θ being the volumetric water content, and θR and θS the residual and the saturated water content, respectively. The effective saturation is calculated through van Genuchten’s (1980) water retention relationship:

(

Se = 1 + (α ⋅ s ) *

n

)

−m

Correpondence to: Vicente Navarro, Escuela de Caminos (Edificio Politécnico), Universidad de Castilla-La Mancha, Avda. Camilo José Cela s/n, 13071, Ciudad Real, Spain.

(1.4)

Vicente Navarro Miguel Mena, Miguel Candel, Ángel Yustres and Beatriz García

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where s is the matric suction (s=PG-P, being PG the gas pressure ), α is a fitting parameter related to air entry pressure, and both n and m are fitting parameters related to pore size distribution. The relationship m=1-1/n is assumed. Also is assumed the Carsel and Parrish (1988) equivalence between the fitting parameter λ of equation 3 and the van Genuchten parameter n, i.e. λ = n-1. Parameters K, θS , θR , α and m may be estimated using the mean values proposed by Meyer et al. (1999). Isothermal flow is considered. Moreover, we assume that air voids are interconnected in field. Therefore, each build-up in gas pressure rapidly vanishes, so that gas pressure PG remains at a constant atmospheric pressure, PATM. Consequently, only the water flow must be solved, being P the state variable of the problem. In order to calculate P, the mass-balance in 2NSAT is formulated as follow ∂ ∂t

((1 + e) θ ) = − (1 + e) (∇ ⋅ q + r )

(1.5)

where e is the void ratio (volume of voids/volume of solid), “∇·” defines the divergence operator, and r is a sink term related with the water uptake Q. During the calculation, e is not considered as a constant. A logarithmic state surface model is taken into account (see, for example, Lloret and Alonso, 1985) in order to define the volumetric strain, εV, by means of the matric suction:

εV=Ch·log(s+PATM)

(1.6)

where Ch is the suction compression index In order to obtain the strains εx and εz (horizontal and vertical), it has been assumed that the variations of the suction only produce volumetric strain, and not deviatoric strain. Consequently, and further assuming plane strain in the flow plane, taking the strain principal directions as the axis x and z, the relation εx= εz= εV/2 is obtained. Integrating the strain field the movements are calculated. A FTCS (Forward Time Centered Space) finite-difference approximation was implemented in 2NSAT to discretize and solve the boundary value problem. The solver was programmed using the language for macros “visual Basic” of the well-known computer code Microsoft©Excel, and can be used like a simple spreadsheet. The computation flow follows the hierarchy assumed for the water flow. Hence, the calculation is done by columns, starting by the column under the tree trunk and ending by the one located under the building façade. As the calculation proceeds, the updated values of P in the points located in the upper left with respect to the point where its water pressure is being calculated at that time are used for such computation. For the analyzed cases, limiting the maximum timestep to 10000 seconds, the code has probed to be stable, and 10-year simulations were performed in 5 minutes using a conventional laptop. Therefore the FTCS scheme was selected as the core of our numerical solver. The simplicity of this scheme, which will be provided upon request to the first author, will ease future modifications by new users, fulfilling their simulation requirements.

A computer model to simulate the tree-foundation system

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3

An application case study

The stabilization of 25 buildings in La Rondilla Street of Alcazar de San Juan (center of Spain) was analyzed with our code. During the summer of 1997 several improvements of urban services were carried out, including the excavation of a wide trench (1.3 m width× 2.5 m depth) in the center of the street intended to restore a sewage pipeline. The excavation of two small service trenches (0.6 m width× 0.7 m depth) in which gas and water supply pipelines were installed was also carried out. Furthermore, 67 chinaberry trees (Melia Azedarach) were planted. Since the summer of 2003, which was particularly dry and hot, the neighbourhood residents started to notice the formation of cracks in their walls. According to their experience, the cracks accelerated during summer. This fact made unlikely, if not impossible, that the damage was associated to shear mechanisms that usually are activated during wintertime, when the soil moisture is at its top. Only the shrinkage induced by the tree roots coincides with this seasonal character. Intact samples were obtained from the borehole and three swelling tests were carried out. On the basis of these test results, by means of a “grid search” (Neumaier 2004) we were able to identify the material parameters used in the simulation performed and from which we obtained figures 1 y 3. As it can be seen in figure 2 there is an acceptable correlation between model results and field values The model clearly shows that the trees did not change in a noticeable way the soil water content in the proximity of the building façades. Consequently, the settlements (vertical movements) of the footings were small in magnitude. On the contrary, the shrinkage due to tree transpiration during the summertime produced lateral movements which, in turn, were the ultimate cause of the structural damage.

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Conclussion

A new tool for characterizing the soil-moisture changes caused by trees growing close to foundations has been presented. Although an idealization of the problem has been adopted, the model is able to predict the building-tree system behaviour in a significant amount of cases.

Acknowledgements The authors would like to thank the Town Countil of Alcázar for providing the means and the financial support to carry out this study. This research was also financed in part by a Research Grant awarded authors numbers 2 and 4 by the Education and Research Department of the Castilla-La Mancha Regional Government and the European Social Fund within the framework of the Integrated Operative Programme for Castilla-La Mancha 2000-2006, approved by Commission Decision C(2001) 525/1. The support provided by the staff of Intedhor is also appreciated.

Vicente Navarro Miguel Mena, Miguel Candel, Ángel Yustres and Beatriz García

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Figure 1: Estimation of the water content distribution around a chinaberry tree placed close to the building #136 in La Rondilla Street. The results correspond to the summer of 2004.

Figure 2: Correlation between the water content obtained from the simulation detailed in the text and the field measured values. The water content values at 0, 1.5 and 3.0 m from the trunk and at 10, 20, 30, 40 y 50 cm depth are shown.

A computer model to simulate the tree-foundation system

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Figure 3: a) Horizontal deflections u obtained from the simulation corresponding to figure 1. b) Vertical movements v obtained in the same simulation.

References BROOKS, R.H. and COREY A.T. (1964). Hydraulic Properties of Porous Media. Hydrologic Paper nº 3. Colorado State Univ., Fort Collins, CO. BURDINE, N.T. (1953). Relative permeability calculations from pore-size distribution data. Petr. Trans. Am. Inst. Mining Metall. Eng. 198, 71-77. CARSEL, R.F. and PARRISH R.S. (1988). Developing joint probability distributions of soil water retention characteristics. Water Resour. Res. 24, 755-769. GEO-SLOPE. (2002). Seep/W for finite element seepage analysis. User's guide (Version 5). Canada: Geo-Slope International, 549 pp. LLORET, A. and ALONSO, E.E. (1985). State surfaces for partially saturated soils. Proc. 11th. Int. Conf. Soil Mech. and Fnd. Engng. San Fracisco, 2: pp. 557-562. MEYER, P.D., GEE, G.W. and NICHOLSON. (1999). Information on hydrological conceptual models, parameters, uncertainty analysis, and data source for dose assesment and decommissioning sites. NUREG/CR-6656. U.S. Nuclear Regulatory Comission. Washington, DC, USA. NEUMAIER A. (2004). Complete search in continuous global optimization and constraint satisfaction. In Acta Numerica 2004. Iserles A, editor. Cambridge University Press. pp. 271-369. VAN GENUCHTEN, M.T. (1980). A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. American Journal. 44, 892-898.

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