Biaxial Anisotropy: Its Occurrence and Measurement With Multicomponent Induction Tools

Abstract

For years petrophysicists interpreted logs as if all formations were locally homogenous and isotropic. They did this even though they well understood that sedimentary deposition occurs layer by layer, making transversely isotropic, TI, media the rule rather than the exception in sedimentary geology. With the widespread availability of multi-component induction logging technology petrophysicists are more willing to use TI resistivity models (i.e., R h and R v ). Generally, sediments buried a few hundred meters or more are subject to a principal stress aligned with the overburden. In many environments, depending on mechanical properties, this leads to bedding perpendicular fractures, which can give rise to biaxial anisotropy in the macroscopic petrophysical properties.We have developed resistivity models for laminated sand-sale formations in the presence of drilling-induced and natural fractures. The principal resistivities R x , R y , R z are expressed as a function of horizontal and vertical resistivities of unfractured formation, fracture porosity and resistivity of fracture which is the resistivity of mud filtrate in the case of drilling induced fractures or the connate water resistivity and saturation in the case of natural fractures. These models allowed us to study the effect of fractures on the three principal resistivities as a function of laminated shale content and as a function of fracture porosity for both drilling induced and natural fractures. The combined three-step analysis of measured principal resistivities based on the modular concept allows a separation and quantification of the two anisotropy sources: fracturing and lamination. The analysis based on the three principal resistivities provides an estimate of fracture porosity, resistivity of the porous sand fraction (without fractures) and water saturations. We have also developed an inversion algorithm for the biaxially anisotropic formations that reliably determines three principal resistivities from multi-component induction measurements. First, in the least-square inversion the multi-frequency focused magnetic field tensor is rotated to the principal coordinate system. This rotation is characterized by three angles: ϕ, θ, and ψ; and three principal, diagonal magnetic field components H xx , H yy , and H zz . After the three angles and three principal components are determined, we recover principal formation resistivities using a fast inversion algorithm based on a look-up table approach. To validate the theory and algorithms described above, we have simulated the multi-component induction measurements in thick biaxially anisotropic layers for various sets of parameters and used the software to process the data. The new approach was also applied to the real measurements acquired in the well with drilling induced fractures and as a result principal formation resistivities, fracture orientation, and fracture porosity were reliably determined.

Tensor Green's function evaluation in arbitrarily anisotropic, layered media using complex-plane Gauss-Laguerre quadrature

“…Next, by making the substitution r ± x = r x τ ± , subsequently dropping the "±" superscripts in r ± x , and defining l ± = t ± o /τ ± , one obtains the following pair of integrals suitable for 12 We use the convention τ ± = ∆x sin γ ± + ∆z cos γ ± to compactly denote the relations τ + = ∆x sin γ + + ∆z cos γ + and τ − = ∆x sin γ − + ∆z cos γ − simultaneously. An analogous comment applies for other expressions bearing this plus-minus superscript type of convention.…”

“…First, we show a data set related to the use of induction-regime electromagnetic sondes for resistivity well-logging (hydrocarbon prospection) in layered geologic formations exhibiting uniaxial or biaxial resistivity in their effective [4,5,12] resistivity tensors. To facilitate comparison of accuracy of the computed field solution between the CGLQ and CPMWA methods, we choose the same set of results used for the CPMWA algorithm in [23].…”

“…Applications regularly encountering problem scenarios approximated by planar-layered, anisotropic media include hydrocarbon well-logging using radar and induction instruments [2][3][4][5][6][7][8][9][10][11][12][13], analysis and design of both microwave circuits and antennas [14][15][16], plasma physics [17,18], atmospheric studies [19], ground penetrating radar (GPR) [20,21], and optical field manipulation [22].…”

Full-wave algorithm to model effects of bedding slopes on the response of subsurface electromagnetic geophysical sensors near unconformities

“…In the interest of obtaining a good trade-off between the forward modeler's solution speed while still satisfactorily modeling the EM behavior of the environment's dominant geophysical features, a layered-medium approximation of the geophysical formation often proves very useful. Indeed cylindrical layering, planar layering, and a combination of the two (for example, to model the cylindrical exploratory borehole and invasion zone embedded within a stack of planar formation beds) are arguably three of the most widely used layering approximations in subsurface geophysics [2,4,5,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], for both onshore and offshore geophysical exploration modeling [23][24][25][26][27][28][29][30][31][32][33]. The prevalence of layered-medium approximations stems in large part, at least from a computational modeling standpoint, due to the typical availability of closed-form eigenfunction expansions to compute the EM field [34][Ch.…”

“…For example, with respect to formation conductivity properties, diverse geological structures can embody macro-scale conductive anisotropy in the induction frequency regime, such as (possibly deviated) sand-shale micro-laminate deposits, clean-sand micro-laminate deposits, and either natural or drilling-induced fractures. In the sub-2MHz frequency regime, the electrical conduction current transport characteristics of such structures indeed are often mathematically described by a uniaxial or biaxial conductivity tensor exhibiting directional electrical conductivities whose value range can span in excess of four orders of magnitude [2,7,12].…”