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forces have the effect of deforming a volume of fluid. A measure of this deformation is given by the deformation tensor D. It is defined by flow are assumed, corresponding to the POISEUILLE fluid model (Wollrath, 1990). Between the parallel plates the velocity profile has a parabolic shape,indicating laminar flow. The upper and lower plates have values of , respectively, whilst the vertical distance between, the plates, b, can be considered to have a value of The NA VIER-STOKES equation(NSE) for the laminar single-phase flow of an incompressible Newtonian fluid yields the following equation for the velocity profile between two parallel plates (Snow 1969; White 1999). where is the acceleration due to gravity. The mean three-dimensional velocity can be determined by stating that the maximum velocity , is reached at , and for parabolic shaped profiles, the mean velocity is given as . (Dietrich P et al., 2005) therefore, It can then be seen that the hydraulic conductivity , and the permeability , have the following relationship: and Now the flow field can be determined by assuming there is steady flow between the parallel plates and no external forces act on the plates. We can now introduce Cartesian co-ordinate system, where the plates can be taken to be and . Allow the lower plate to be fixed while the upper plate moves with a velocity . Hence we need to solve the NSE and the Continuity equation, respectively. These are given by: Here it is observed that symmetric tensor which is dependent on the fluid velocity gradients of the fluid. Let The trace of elements of matrix containing all the elements of . A particular medium is characterized by the constitutive equation, which is the relationship between stress and strain . This stress-strain relationship is given by: When this relationship is linear, the fluid is said to be viscous, classical and Newtonian, and is shown by where A and B are constants (tensors), and is the identity matrix. By further analysis of the equation and using the facts that both are symmetric, and that the frame- indifference principle, (the constitutive equation does not depend on the orientation of the frame of reference), the eventual field equation, known as Navier-Stokes equation, is obtained as stated below. where the viscous coefficient is represented by .This equation is very useful when investigating fluid flows, since it is solved together with the continuity equation for incompressible flow, i.e. 1.3 Special case: Parallel Plate Concept The rock matrix bounds naturally occurring fractures on all sides. An identical profile does not exist for the fracture walls, hence the normal tension is carried by contact zones between the walls. A frequently used model for frature representation is the parallel plate concept. Tsang and Tsang 1987 showed that preferential flow paths exist, hence channeling effects may have significant influence on the flow and therefore also on the transport processes. For the application of the parallel plate concept it is assumed that the length scale, l, of the plates is much larger than the distance between them b (l>>b). Furthermore, hydraulically smooth walls and laminar then is a second order , is the summation of all the along the leading diagonal of a where Equations & simplify to The above equation are then subjected to the following boundary conditions where fluid at any point It is then assumed that is the velocity field of the in the flow field.PDF Image | CONCEPTUAL MODEL FOR GEOTHERMAL ENERGY CARIBBEAN
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