SOME NOTES ON MORPHODYNAMIC MODELING OF VENICE LAGOON Gary Parker, Spring, 2004 Venice Lagoon was formed by the action of deposition of mud and sand from rivers, sea level rise and compaction of fine-grained sediment under its own weight. Today the source of sediment has been cut off, but the deposit continues to compact, or consolidate under its own weight. Consolidation is made worse by groundwater withdrawal and possibly sea level rise caused by global warming.
EXNER EQUATION OF SEDIMENT CONSERVATION WITH COMPACTION Let p = porosity, and c = 1 - p fraction of bed volume that is solid (not pores). Exner from some level b below which only tectonic effects are felt to the water-sediment interface Apply Leibnitz where cD = solids fraction in freshly deposited material at z = and cb = solids fraction at interface below which only tectonics is felt.
EXNER EQUATION OF SEDIMENT CONSERVATION WITH COMPACTION contd. Define t = - b/t = tectonic subsidence rate. As a fine-grain layer compacts, c/t > 0, and so a subsidence rate due to compaction can be defined as Thus Exner becomes Now in general tcan be specified independently of the local process of deposition. In case of the deposition of fine-grained material, however, c is a function of the deposition itself: deposition induces compaction, which creates accomodation space for more deposition. Compaction progresses as water is slowly squeezed out of a mud layer by the process of consolidation.
QUICK REVIEW OF CONSOLIDATION Consider a layer of fine-grained material (mud) bounded by highly permeable sand below and above. The water table is located in the upper sand layer. The water supports the water above in hydrostatic balance, and the mud supports its weight (minus the buoyant weight) by means of the contacts between the mud grains.
QUICK REVIEW OF CONSOLIDATION contd. At some time t = 0 a load is placed on the surface (above the water table). The sand layers quickly respond to the load. Initially, however, the particles in the mud layer do not have enough contacts to support the added load, so an excess pore pressure above and beyond hydrostatic pressure is created.
QUICK REVIEW OF CONSOLIDATION contd. D’Arcy’s law assumes that groundwater flows from zones of high excess pore pressure to low excess pore pressure. Defining the excess piezometric head he as he = pe/(g), the relation takes the form where w is the groundwater flow velocity in the z direction and K is the hydraulic conductivity of the mud. Excess pore pressure in the mud layer is dissipated to the sand layers as illustrated below:
INTEGRATING CONSOLIDATION INTO A MORPHODYNAMIC MODEL Consider the subaqueous deposition of mud with consolidation
INTEGRATING CONSOLIDATION INTO A MORPHODYNAMIC MODEL contd. Relation between equilibrium solids concentration and depth in mud: Here cD denotes the concentration of solids in freshly-deposited surface mud. Note that in this linearized treatment cE increases linearly with vertical distance below the surface. Lc is a length scale associated with the linearization. Relation between excess piezometric head he and the difference between the actual solids concentration and the equilibrium value: where again Lh is a length scale associated with consolidation in a linearized treatment. Mass conservation of fluid phase:
REDUCTION Now then So or reducing, Thus using D’arcy’s law, Assuming water at hydrostatic pressure at z = and e.g. a porous sand layer at z = b , the boundary conditions become
REDUCTION contd. Reduce to
CONCLUSION The coupled groundwater-morphodynamic problem is: In order to finish the problem formulation, it is necessary to specify relations for D and E of mud as functions of flow conditions. For example, subsidence under compaction increases flow depth, which may in turn increase the overall rate of deposition of fine-grained material. The same model for hydrodynamics and erosion and deposition of mud should easily incorporate the effect of sea level rise, which will appear as a boundary condition on the morphodynamic model.