A Study of the Fracture Impedance Method

Abstract The hydraulic impedance method is a method for estimating the dimension of a downhole fracture by using the reflected waves measured at the wellhead. The method is studied in this paper. By considering the wave motions in Q fractured borehole, the fracture impedance and the wave reflection coefficient are derived. It has been demonstrated that the shape of the reflected wave at wellhead can be calculated numerically by carrying out the inverse Fourier transform on the product of the input pressure pulse and the wave reflection coefficient. The calculated waveforms are in good agreement with the laboratory measurements. The results have also demonstrated that the hydraulic impedance method could provide an accurate estimation on the length and other characteristics of a downhole vertical hydraulic fracture if the input pressure pulse could produce a resonant motion of the fluid in the fracture. Introduction An interesting method for assessing the dimension of a downhole hydraulic fracture was introduced by Holzhausen and his co-authors1,2. The method was based on the phenomenon that the existence of a hydraulic fracture along the surface of a wellbore would alter the acoustic impedance of the well bore and change the pressure oscillation characteristics at the wellhead. Thus, the dimension and other characteristics of the downhole hydraulic fracture can be assessed by analyzing the pressure oscillations measured at the wellhead. The method requires one to generate a wellhead pressure pulse propagating downward along the wellbore and to measure the reflected pressure pulse at the wellhead. Since both the pressure pulse generation and measurement are made at the wellhead, it is a convenient and economical method for assessing the characteristics of downhole hydraulic fractures. By solving the one-dimensional fluid wave equation in the wellbore, Holzhausen has shown that the hydraulic impedance, which is defined as a ratio of the pressure and the volumetric flow rate, can be written as Equation (1) (Available in full paper) where, i = ν-1, is a complex number, Zch= vf/gAw is the characteristic hydraulic wave impedance in the borehole (vf is the wave speed in the fluid, Aw is the area of borehole section, and g is the gravitational acceleration), k is the wave number, and r is the wave reflection coefficient. By placing a hydraulic fracture at the bottom (x =L) of wellbore and letting the fracture has an impedance Zf, the reflection coefficient,τs, at the wellhead (x = 0) can be derived from the above equation to give Equation (2) (Available in full paper) where, the coefficient τfw is the reflection coefficient at the junction of fracture and wellbore. In Holzhausen's model, the characteristics of hydraulic fracture are represented by a fracture capacitance Cf and a fluid flow resistance Rf in series, i.e., Equation (3) (Available in full paper) where ω is the angular frequency of the wave. By defining the fracture capacitance as a ratio of the fracture volumetric change ΔV and the fluid pressure head change ΔH and using formulas from a static fracture analysis, the fracture capacitance Cf and fracture resistance Rf are derived by Holzhausen as follows: