Earth Resources Laboratory :: Abstracts

Acoustic Logging in Fractured and Porous Formations

Xiaoming Tang

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences in June 1990 in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Abstract

This thesis is concerned with the dynamic fluid transport properties of fractures and porous media and their application to the estimation of formation hydraulic properties using borehole acoustic logging techniques. In the first part of the thesis, the dynamic response of a viscous fluid in a borehole fracture to the oscillatory pressure excitation of borehole acoustic waves is investigated, which leads to the theory of fracture dynamic conductivity. The distinction between this dynamic conductivity and the conventional cubic law conductivity is whether the viscous skin depth, d= (2n/w)1/2, is large or small compared to the thickness of the fracture. Although this characteristics of dynamic fluid flow is obtained using the simple plane parallel fracture model, the physics involved is universally true for dynamic fluid flow in hydraulic conduits of rocks. The theory of fracture dynamic conductivity is compared with the theory of dynamic permeability of a general porous medium. It is found that the latter theory, when applied to the fracture case, is in excellent agreement with the theory of conductivity. This points to the general behavior of frequency-dependent fluid motion through conduits in rocks, regardless whether they are fractures or pores. Consequently, in acoustic logging measurements performed in a typical range of [2-20] kHz, the dynamic fluid flow theory, instead of the conventional Darcy’s law, is the appropriate theory for the fluid flow in the formation induced by logging acoustic waves.

In the second part of the thesis, the concept of dynamic permeability is applied to the important problem of acoustic logging in a permeable porous formation using borehole Stoneley waves. The interaction of the Stoneley wave with the porous formation is decomposed into two parts. The first is the interaction of the Stoneley with an equivalent elastic formation composed of the saturated porous matrix. The second is the interaction with pore fluid flow governed by the dynamic permeability. In this manner, a simple dynamic model is obtained for the Stoneley propagation in permeable boreholes. This simple model is compared with the complete model of the Biot-Rosenbaum theory for the effects of a porous formation on the Stoneley propagation characteristics. It is found that the results from the two models agree very well for a hard formation, although they differ at higher frequencies for a soft formation because of the increased formation compressibility. The simple model is also tested with recently published laboratory experimental data of Stoneley wave measurements. The theory and experiment are in excellent agreement. As a result, the application of the dynamic fluid flow theory not only clearly points to the physical process involved in wave propagation in permeable boreholes, but also yields a much simplified Biot-Rosenbaum model that can be applied to the problem of acoustic logging in porous formations, especially to an inverse problem to extract formation permeability from the Stoneley wave measurements.

In the third part, the problem of acoustic logging in a fluid-filled borehole with a vertical fracture is investigated both theoretically and experimentally. The Stoneley wave is used to probe the borehole. The propagation of this wave excites fluid motion in the fracture and the resulting fluid flow at the fracture opening perturbs the fluid-solid boundary condition at the borehole wall. The dynamic conductivity is applied to measure the fluid flow into the fracture and a boundary condition perturbation technique is developed to study the effects of the change in the boundary condition on the Stoneley propagation. The results indicate that the fracture has significant effects on the Stoneley waves, especially in the low frequency range. Significant Stoneley wave attenuation is produced and the Stoneley phase velocity is drastically decreased with decreasing frequency. Ultrasonic experiments are performed to measure Stoneley propagation in laboratory fracture borehole models. Cases of both hard and soft formations are studied. For both formations, the experimental results are found to agree well with the theoretical predictions. The important result of this study is that, a quantitative relationship between the Stoneley propagation and the fracture character is found. This relationship can be used to provide a method for characterizing a vertical borehole fracture by means of Stoneley wave measurements.

In the last part, the guided wave propagation in a fluid-filled borehole with a horizontal fracture is investigated. For the solution of the problem, a hybrid method is used to generate wave modes for the two regions separated by the fracture. The modes are then summed to match the boundary conditions at the fracture surfaces. A singularity problem arises in matching the surface conditions and is regularized by balancing borehole fluid flow across and into the fracture. The latter flow is characterized using the fracture dynamic conductivity. The results show that a low frequencies, the Stoneley wave attenuation across a fracture is controlled by the fluid flow into the fracture. As the frequency increases, mode conversion at the fracture becomes important. Above the cut-off frequency of the first pseudo-Rayleigh mode, the Stoneley wave is strongly coupled with pseudo-Rayleigh waves, which is demonstrated by synthetic microseimograms. The pseudo-Rayleigh wave is strongly attenuated and reflected by thin as well as thick fractures. These effects are more pronounced toward the cut-off frequencies than away from the frequencies. Consequently, in acoustic logging measurements, the lack of wave energy across a borehole fracture may be very good indication of the existing fracture. The substantial effects of a fracture on a pseudo-Rayleigh waves has been verified in the laboratory by experimenting with thin and thick fracture models. The experimental results demonstrate the guided wave characteristics across a fracture and confirm the theoretical analysis on these effects. The wave characteristics in the vicinity of a fracture, as described in this study, can be used to provide useful information for the detection and characterization of borehole fractures using an acoustic logging techniques.