Summerschool Modelling of Mass and Energy Transport

Black Box Analogy i: current, e(t): input voltage.

Characteristic Volume

Material Mix

Storage Capacity - Porosity

Grainsize, Ordering, Secundary Processes

Internal Structure

Anisotropy + Secondary Pathways

Mass and Energy Transport Exercise Given a flux vector approaching an oblique oriented surface element (line) of your choice. Separate it graphically into the flux components parallel and oblique to the surface

Normal Vector and Fluxes The normal is perpendicular to the surface. The normal vector points to the outside of the closed are. The closed area is always surrounded anticlockwise. The fluxes for the left case are or

Fluxes Derived from a Potential change = = =.

Gauss Theorem

Green‘s Formulae are functions of (x,y,z), In case the vector v(F,G,H) is the gradient of a scalar field.

Mass and Energy Balance Continued In this case there is no gain or loss within the control volume

Gain or loss within the control volume

Key Question Under which conditions will the equality in the equations and hold, independent of the size of the control volume?

Remark The equation where Q may be zero, provides a weak formulation of the problem to solve transport processes. For that purpose subdivide the area under consideration into aerial subsets, by example triangles as illustrated in the sketch Then we can evaluate the volume integral by using the Gauss theorem and we can formally evaluate the fluxes through the bounding lines or surfaces: FV-Method

The Differential Equation If the flux is derived from a potential or Without temporal change within the black box

Boundary and Initial Conditions 1 st kind or DIRICHLET’s condition: Find a harmonic solution for the interior of the considered area so that the solution 2 nd kind or NEUMANN’s condition: Find a harmonic solution for the interior of the considered area so that the solution satisfies given values of the normal derivative 3 rd kind or mixed condition: Find a solution for the interior of the considered area so that the solution satisfies the equation and satisfies given values of

Exercises 1)Find (qualitatively) the solution for the 1-D problem with u=a at x=0 and u=b at x=1. 2) What would be the solution over a square with values of u given at the four corner points? 3) How can the boundary condition be interpreted (2 possibilities). 4) Consider a well located within a homogeneous aquifer with free surface. The well has diameter r and you are pumping continuously Q [m3 ] water. The free water surface within the well than will have dropped to height H. Through the surface area of the well water will flow v [m/s]which is proportional to the slope of the free water surface or with y the height of the free water surface. At the boundary of the well the water level in the aquifer will be equal to the free water level within the well. Derive the differential equation, the required boundary conditions and the solution. Sketch the solution.

Solution Exercise 4 Through any cylindrical surface area surrounding the well we have the flux and the total flux becomes. By experience, the flux is We find the differential equation With the boundary conditions y=H for x=r Solution: Differential equation: Forandfor

Fick‘s 1st law D is mostly given as the diffusion coefficent in pure water. Provide an estimate for porous media. The flux is related to the temporal change by Or if chemical reactions are involved within the porous medium

Heat conduction The change of energy within the control volume becomes Or if sources or sinks are vailable (chemical reactions, radioactive heat production) Observe that and, and that the specific volume will also depend on the temperature. In detail the size of the control volume changes [J/s]

Exercises 1)Linearize the equation by partially differentiating the time derivative with respect to the parameters, observing that c=c(T) and. 2) Consider the heat flow from the interior of the earth which mostly is considered stationary, i.e. it is governed by the equation At the surface an average thermal gradient of 30°C is observed. At about 100 km depth the boundary between the lithosphere and the astenosphere is reached, the latter behaves like a fluid due to seismological data, the temperature at this boundary is approximately 1300°C, due to experiments. a)Compute the expected temperature at the base of the lithosphere based on the average surface gradient. b)Compute the expected temperature gradient throughout the lithosphere, based on a surface temperature of 0°C. c)Discuss for a layered crust qualitatively how the thermal conductivity should vary with depth. d)Determine for a homogenous lithosphere the heat production required to fit the boundary conditions or

Darcy‘s Law [kg m/s] [m/s] or or´ [m/s]

Coupled equations D=D(T)

Mathematical formulation of the thermohaline flow problem in FEFLOW

Stability criteria Solutal Rayleigh number Thermal Rayleigh number The solutal and thermal Rayleigh numbers are related by Buoyancy ratio (Turner) Lewis number

The monotonic instability (or stationary convection) boundary is a straight line defined by is the critical Rayleigh number.. The region delimited by is a stable regime characterized by pure conduction and no convection. In a range between steady state convective cells develop For the convection regime is unstable

Cross section through the Büsum diapir and adjacent rim synclines with temperature isolines (left) and vitrinite reflectance isolines (right)

Type of large-scale gw flow

Type of thermohaline flow Thermally induced brine plumes developping on a deep salt sheet Brine lenses, gravitational convection from a shallow salt sheet Dasehd lines: isotherms (°C)

Thermohaline flow in the NE German Basin Stratigraphic units

Thermohaline flow: Zoom salt dome

Mixed convection in the NE German Basin Bold vectors represent the topography-driven flow (i.e. regional flow)

Thermal convection in a shallow salt dome

Thermohaline flow in a shallow salt dome Dashed lines: isotherms (°C)