Effects of hetergeneities on Fluid Flow and Borehole Permeability Measurements

by
Xiaomin Zhao

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on May 1, 1994 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

ABSTRACT

One of the prominent features of a porous formation is the presence of heterogenities of various scales. The effects of the heterogenities on the fluid transport properties are of primary concern for basic scientific studies, petroleum production, ground water hydrology, and environmental characterization. This thesis is concerned with the effects of porous formation heterogenities on the steady, transient, and dynamic fluid flow behavior, and their influence on borehole permeability measurements.

From fluid flow point of view, heterogeneity and anisotropy are two closely related properties. We model steady fluid flow in media with different heterogenities, especially the lineated continuous and discontinuous permeability heterogenities. It is found that the lineation of heterogeneities results in macroscopic permeability anisotropy. The permeability contrast between the low- and high-permeability regions is the key factor controlling the degree of permeability anisotropy. Strong anisotropy exists only when the contrast is large. The effects of heterogeneity scale on permeability anisotropy are also studied. Numerical simulation indicates that large size heterogeneities along the lineation direction produce strong anisotropy. The anisotropy decreases with decreasing heterogeneity scale sizes.

In the laboratory transient tests, both the early time and the late time portions of the pressure transient are used to obtain rock permeability. Our numerical simulation results show that the early time behavior of the transient pressure pulse is controlled by the rock heterogeneity and can be used to characterize the permeability heterogeneity of the rock, while late time behavior is mainly controlled by the effective permeability of the sample. As in the steady fluid flow case, lineation of permeability heterogeneity results in anisotropy. The degree of anisotropy is controlled by the contrast between low- and high-permeability regions in the porous media.

This thesis is primarily concerned with dynamic (or frequency-dependent) fluid transport properties in heterogeneous porous media and its application to acoustic logging in heterogeneous porous formations. The theory of dynamic permeability is modified by introducing spatially varying permeability into the theory. As a result, a complex permeability as a function of spatial coordinates and frequency is used to describe the dynamic fluid transport properties in heterogeneous porous media. An iterative finite difference technique is developed to compute the fluid motion in the frequency domain.

The important application of the dynamic fluid flow modeling is in the study of borehole Stoneley wave propagation in heterogeneous permeable porous formations. By formulating the finite difference scheme for the cylindrical coordinates, we have modeled the effects of radial and azimuthal permeability heterogeneity variation on borehole Stoneley wave propagation. Our models show that random permeability variation has only minimal effects on the Stoneley wave propagation. However, in the case of a damaged borehole wall, where the wall has a much higher permeability than the surrounding formation, the effects of the heterogeneity can be detected by the significant delay in Stoneley wave arrival and the attenuation peak in the frequency range of common Stoneley wave measurements. The modeling results provide a theoretical basis for determing the borehole wall damage from measuring the Stoneley wave propagation characteristics.

To further study the problem of acoustic logging in heterogeneous porous formations, we look at the case where the formation permeability varies in the borehole axial and radial directions. This is a very important problem because vertical heterogeneity variations are commonly encountered in acoustic logging applications. Our numerical simulation results show that continuous permeability variations in the formation have only minimal effects on the Stoneley wave propagation, whereas the discontinuous variation can have significant effects on the Stoneley wave propagation. However, when the Stoneley wavelength is considerably larger than the scale of heterogeneity variations, the Stoneley wave is sensitive only to the overall fluid transmissity of the formation. To demonstrate the effects of heterogeneity on the Stoneley propagation, an experimental data set (Winkler et al., 1989) has been modeled using randomly layered permeability models. The heterogeneous permeability model results agree with the data very well.

The numerical technique for calculating Stoneley wave propagation across permeability heterogeneities has been applied to interpret the acoustic logging data across a heterogeneous fracture zone (Paillet, 1984). The modeling technique, in conjunction with variable permeability models, successfully explains the non-symmetric patterns of the Stoneley wave attenuation and reflection at the top and bottom of the fracture zone, whereas it is difficult to explain these patterns using a homogenous permeable zone model. The technique developed in this study can be used as an effective means for characterizing permeability heterogeneities using borehole Stoneley waves.

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