[Introduction]

Flow is considered over an obstacle placed on a coarse gravel layer that covers a sand layer. The pressure distribution as a function of position and time is obtained for partially saturated conditions in the subsoil. The flow in the open water and coarse layer is calculated using a lattice Boltzmann technique; the pressure in the subsoil is evaluated by means of an analytical solution using the lattice Boltzmann simulation as boundary condition. The results demonstrate where the greatest risk of damage to the subsoil may be located. Introduction A partially saturated subsoil, loaded by a falling external pressure, experiences a vertical pore pressure gradient. The magnitude of this gradient depends on the rate of external pressure variation, details of the soil permeability, as well as the degree of saturation. If the gradient is large enough, and the subsoil is non-cohesive, fluidization may take place, which obviously leads to erosion, if simultaneously a horizontal fluid flow is present in the system. Shallow, natural soils will always be slightly unsaturated: gas may develop as a result of biological activity, or air bubbles may adhere to the grains of the soil. Typically, erosion may be expected in cases where the external pressure fluctuates rapidly: in waterways due to propeller motion, in locks and sluices where water level changes rapidly and under coastal structures subjected to wave loading. A one dimensional analysis of this phenomenon has been presented by Köhler and Koenders [18]. In this paper it has been demonstrated that the theoretically predicted swelling of gas bubbles that is responsible for the creation of the pressure gradient is observed experimentally. A criterion has been devised that enables the evaluation of the various factors involved in the process for shallow conditions, see also Koenders [17]. The theory predicts that fluidization of an unprotected bed first takes place after a time 1 t reckoned from the moment when the external pressure had started to fall at a rate of . An estimate of 1 t is a key design parameter for unprotected beds. Basically it states: if one expects the external pressure fall at a rate to continue for less than a period of 1 t , then the bed will not fluidize. Corresponding author e-mail mast@kingston.ac. uk Davis, M. et al.: Unsaturated subsoil erosion protection in turbulent flow conditions page 2 of 20 Paper submitted (July 2003) to be printed with IAHR Journal of Hydraulic Research (JHR) If it goes on for a period of time that is greater than 1 t , then fluidization will occur. The estimate for the time 1 t is 2 0 1 a i y t w c , (1) where 0 y is the pressure-equivalent water depth, c i the critical gradient [26] (measured in engineering units as a fraction of the fluid specific weight w ) and a a system parameter with the dimension of the inverse of the square root of a velocity. The latter is the only variable that contains information about the particle/fluid/gas mixture. The presence of the gas bubbles is represented through the saturation measure, i s , stating the fraction of the fluid that is not in bubble form. Note that the bubbles tend to adhere to the grains; in fact it is very difficult to get rid of them and a typical figure for the saturation in natural deposits at water depths that are less than 10 m deep is some 95-99%. The saturation is embedded in the parameter a as follows k s n a i 1 4 2 , (2) where n is the porosity of the deposit and k its permeability. Protection of the subsoil can be introduced by means of a layer of a coarse, more permeable material. The analysis and design of the thickness of the layer has been discussed by Roussell et al [23]. This analysis is also one-dimensional. In this paper the analysis is extended to include both another important engineering geometry and external pressure fluctuations that arise from a turbulent flow field; the latter is associated with the flow over an obstacle. The subsoil is protected by a layer of coarse material. The obstacle is placed on the coarse layer. The geometry is sketched in Figure 1. The purpose of the analysis is to establish the likely damage areas in the subsoil. The analysis is twodimensional. The analysis is part of a wider research exercise. In this paper the analysis is demonstrated. In a forthcoming paper a series of experiments will be reported, as well as variations of the calculations. Davis, M. et al.: Unsaturated subsoil erosion protection in turbulent flow conditions page 3 of 20 Paper submitted (July 2003) to be printed with IAHR Journal of Hydraulic Research (JHR) Figure 1: Geometry of the problem. The height of the coarse layer is 0 y . In order to address the highly complex geometry presented by the combined open-water flow and flow in the coarse layer, a numerical method is used that is particularly well-suited to deal with complex geometries and a wide range of flow conditions: the lattice Boltzmann method. The length scale of the features of the turbulent flow are of the same order of magnitude as the grain size of the coarse layer. Therefore, no continuum approach (in which the permeability of the coarse layer is an average over many grains) is accurate for the combined open water/coarse layer domain. The grain size of the subsoil, on the other hand, is much smaller than the typical dimension of the turbulent flow features. So here the spatial and temporal development of the excess pore pressure in the subsoil may be determined by solving a half-space continuum problem, using the outcome of the pressure of the lattice Boltzmann calculation as a boundary condition at the top of the half-space. This approach is permissible as long as the flow in the subsoil does not significantly affect the flow in the coarse layer, which is the case while the permeability of the subsoil is negligible compared to the permeability of the protective material. Mixing analytical and numerical techniques in this manner is the most effective way to obtain results that give insight in the spatio-temporal properties of the system. The two-dimensionality of the problem implies that the particles in the coarse layer are modelled as discs. Estimate of the pressure gradient of a loaded half-space It was demonstrated by Köhler and Koenders [18] that for a partially saturated subsoil under shallow conditions, the excess pore pressure development t , y , x p as a function of position and time is approximately described by the following equation Davis, M. et al.: Unsaturated subsoil erosion protection in turbulent flow conditions page 4 of 20 Paper submitted (July 2003) to be printed with IAHR Journal of Hydraulic Research (JHR) t p p s n y p x p γ k i w 0 2 2 2 2 1 . (3) Here k is the soil permeability, n the porosity and w γ the specific weight of fluid (water); 0 p is the mean ambient pressure near the top of the subsoil. The position is identified by the horizontal co-ordinate x and the vertical co-ordinate y. The saturation is denoted by i s and is assumed to be a constant, independent of time and position. The equation is valid for finegrained soils in which the fluid flow is laminar. The further crucial approximation that has been made is that the stiffness of the soil is much greater than the stiffness of the fluid-gas mixture. This can only be true under unsaturated circumstances, but that is the limit considered here. As a rule of thumb the approximation is true for dense non-cohesive soils with gas content in the fluid > 1%. For convenience a parameter A is introduced as i w s n p γ k A 1 0 . (4) This parameter has the dimension of 1 2 s m and plays the role of consolidation coefficient. Equation (3) is solved in Appendix A; the solution is