Permeability tensor

Fluid extraction from the reservoir occurs simultaneously with re-injection of cold liquid and air. Atmospheric nitrogen and Ar propagate very fast through the fracture network in the reservoir rocks and arrive at producing regions with negligible thermal interference. This may be used as a simple and cheap method to estimate the three-dimensional permeability tensor. 

A bed of aligned fibers is approximately transversely isotropic and flow parallel to the fibers is greatly favored over perpendicular flow. One could thus argue that Equation 11.7 should be generalized to incorporate a permeability tensor. Due to its lack of scientific... 

Ky // -component of the permeability tensor [m2] n, normal vector to the limits of the mold cavity (seals) [m]... 

This defines the magnetic field intensity H /x is the 3x3 linear magnetic permeability tensor. Equation (2.7.13), linking B to H, is the "second constitutive equation." The magnetic field H is expressed explicitly by... 

Formally, if one has the experimental values of the dielectric tensor e, the magnetic permeability tensor /jl, and the optical rotation tensors p and p for the substrate, one can construct first the optical matrix M, then the differential propagation matrix A, and C, which, to repeat, is the x component of the wavevector of the incident wave. Once A is known, the law of propagation (wave equation) for the generalized field vector ift (the components of E and H parallel to the x and y axes) is specified by Eq. (2.15.18). Experimentally, one travels this path backwards. 

From equations (1) and (15), the change in magnetic permeability tensor due to a strong magnetic field is ... 

In contrast to the SHG coefficient, the electro-optic coefficient r,y/t is symmetric in its first two indices for a lossless, not optically active, material. This follows from Eq. (11), which relates the electro-optical coefficient to the impermeability tensor Tiy (the matrix inverse of the dielectric permeability tensor y), a real and symmetric tensor under the stated conditions [3],... 

We can introduce now the transformed conductivity and magnetic permeability tensors as... 

In the last formulae we assume that (Txx,( yy,crzz and Myyi Mzz are the principal values of the conductivity and permeability tensors, and the axes of the grid are oriented along the principal axes of these tensors. 

For anisotropic media, in general, these two vectors are not parallel and a linear transformation can be made using the permeability tensor K. 

Zhou C and Xiong W. 1996c. Permeability tensor for jointed rock. Chinese J. of Rock Mech. Engng., Vol. 15, pp. 338-344. 

The permeability tensor, being dependent upon the stress stale, changes with evolution of rock damage. The expressions for permeability tensor can be obtained for fractures subjected to different stress states and failure modes. 

Zhou Chuangbing and Xiong Wenlin,1996, Permeability tensor for Jointed rock masses in coupled seepage and stress field, Chinese J. Rock Mech. Eng., Vol.l5, No.4, pp.345-352... 

The first term of the equation results from a generalization of Darcy s law of water flow in variably saturated porous media. In this term, k/j is the saturated permeability tensor (m ) K, ( non-dimensional) is the relative permeability of unsaturated media and is a function of the degree of saturation S (For S=l, K,= l) p (kg/m/s) is the viscosity of water p is the density of water, which are both functions of temperature. 

Permeability Tensor A study by Long et al. (1982) suggested that rock mass behaves more like a porous medium if many fractures make a sufficient number of flow paths. Using a modified form of Oda s crack tensor theory (Oda 1986), the permeability of the homogenized material is given by ... 

Figure 2. Boundary conditions used to determine the permeability tensor with 3DEC.

After each computation, the equivalent vectorial flowrate is averaged from the flowrate value at different locations of the fracture network. The different terms of the equivalent permeability tensor are then computed from equation (1). 

The objective of this section is to compute, with the numerical approach already presented and validated, the equivalent permeability tensor of the fracture network of the WP3 case. The set of computation presented here concerned the fracture network of formation 1 (computation has also been done for formation 2 and the fault zone). 

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