Fracture characterization from attenuation of Stoneley waves across a fracture Sudhish K. Bakku∗, Michael Fehler and Daniel R. Burns, Earth Resource Laboratory, MIT SUMMARY mated fracture spacing and orientation using a modified scat- tering index method. Numerical simulations (Grandi, 2008) show that intensity of scattering is proportional to the frac- Fractures contribute significantly to the permeability of a for- ture compliance. Knowing the average fracture compliance of mation. It is important to understand the fracture distribution a region from borehole measurements we may be able to as- and fluid transmissivity. Though traditional well logs can im- sign fracture compliance to regions away from borehole based age fractures intersecting the borehole, they provide little in- on relative scattered energy. Moreover, previous lab studies formation on the lateral extent of the fractures, away from the (Pyrak-Nolte and Morris, 2000) suggest that fracture compli- borehole, or the fluid transmissivity. Experiments in the past ance and fluid conductivity are influenced by the same micro- demonstrated that fracture compliance can be a good proxy scopic features and are related. In the future, fracture compli- to fracture fluid conductivity. We describe a method to es- ance can be a key link to estimate fracture conductivity and timate fracture compliance from the attenuation of Stoneley help us predict the permeability of the formation. In addition, waves across a fracture. Solving the dispersion relation in the fracture compliance can be a good measure to test the effec- fracture, transmission coefficient of Stoneley waves across a tiveness of hydro-fracing. We can use compliance values to fracture is studied over all frequency ranges. Based on the estimate the relative fluid transmissivity of different fractured observations from the model, we propose that measuring the zones. Thus, it is important to be able to estimate in-situ frac- transmission coefficient near a transition frequency can help ture compliance. constrain fracture compliance and aperture. Comparing atten- uation across a finite fracture to that of an infinitely long frac- In this paper, we develop a model to study the attenuation of a ture, we show that a bound on the lateral extent of the frac- Stoneley wave as it passes a fracture intersecting the borehole, ture can be obtained. Given the limitation on the bandwidth to estimate fracture compliance and aperture, and constrain the of acoustic logging data, we propose using the Stoneley waves lateral extent of fractures. Attenuation of Stoneley waves was generated during micro-seismic events for fracture characteri- studied earlier by Mathieu (1984), Hornby et al. (1989), Tang zation. and Cheng (1993) and Kostek et al. (1998a,b). Mathieu (1984) assumed Darcy flow in the fracture, a low frequency approxi- mation, and studied attenuation of Stoneley waves across the fracture. However, the assumption of Darcy flow is not valid INTRODUCTION for typical logging frequencies. Hornby et al. (1989) and Tang and Cheng (1993) solved the problem under a high-frequency Fractures are one of the main conduits for fluid flow in the sub- approximation, which is a valid assumption for kHz range of surface and characterizing them is important for economic pro- frequencies. Later, Kostek et al. (1998b) extended the theory duction of hydrocarbons or geothermal energy. Borehole tele- to include the elasticity of the formation. None of the studies viewer (BHTV) and formation micro imager (FMI) logs are above accounted for the fracture compliance that plays an im- the most popular tools for characterizing fractures intersecting portant role in the Stoneley wave attenuation. We study Stone- boreholes, in-situ. These logs provide the location and orienta- ley wave attenuation over all frequency ranges, considering the tion of fractures intersecting the borehole. However, from this effects of fracture compliance. data it is hard to differentiate between fractures with high or low fluid conductivity, and it is not possible to estimate the lat- eral extent of the fractures. Some of the fracture like features THEORY seen in the logs could be drilling induced. On the other hand, pressure transient tests can give an estimate of fluid conduc- When Stoneley waves in a borehole cross a fracture intersect- tivity. But, it is a macroscopic measurement averaging over ing the borehole, part of the energy is spent in pushing the a conducting region. On a reservoir scale, fracture networks fluid into the fracture and part of the energy is reflected at the are characterized by applying methods like amplitude varia- interface. As a result, the transmitted wave is attenuated. The tion with offset and azimuth (AVOA) (Sayers and Kachanov, attenuation of the Stoneley wave depends on the amount of 1995) and Scattering Index (Willis et al., 2006). AVOA esti- fluid squeezed into the fracture, which, in turn depends on the mates the anisotropy due to fracture sets, which is a function fracture transmissivity and compliance. We consider a circu- of fracture density, fracture compliance and orientation. This lar horizontal fracture, of radius ‘D’ and intersecting a vertical methodology is successful in determining the preferred frac- borehole of radius R (see Figure 1). Due to the axial symmetry ture orientation but falls short in estimating the fracture den- in the problem, we use cylindrical co-ordinates with ‘r’ de- sity and compliance. Knowing the in-situ fracture compliance noting the radial distance from the center of the borehole, and from borehole measurements, we may be able to estimate frac- z-axis along the borehole. The fracture top surface is located ture density. The Scattering index method is suitable for larger at z=0. discrete fracture networks and is based on analyzing the scat- tered coda from the fracture networks. Fang et al. (2012) esti- The pressure due to the incident Stoneley wave at the fracture Fracture characterization from Stoneley waves borehole, Z , and the impedance of the fracture, Z , as Axis of symmetry Borehole B F < PI > < PT > < PR > Incident Wave Reected wave ZB = = = (5) z = 0 < v > < v > < v > Flow into fracture I T R < PF (R) > < PT (R) > L0 ZF = = Fracture Transmitted Wave r dr < vF (R) > < vF (R) > D Solving equations 2 to 5, simultaneously, Mathieu (1984) ob- r r = R z tained the transmission coefficient as P 1 T = (6) PI 1 + X Figure 1: Schematic showing attenuation of Stoneley wave at with a fracture intersecting a borehole. f L I ( f R) Z X = 0 0 B (7) 2 I1( f R) ZF where, ZB = r f ct . He obtained ZF by estimating flow into the top, z=0, PI, can be written as (Cheng and Toksoz,¨ 1981) fracture, assuming Darcy flow in the fracture, a low frequency PI(r;w) = AI0( f r) (1) approximation, and did not consider the effect of the fracture v compliance. We estimate ZF for arbitrary frequency and ac- w u c2 f = u − t count for fracture complaince. t1 2 ct a f For simplicity, we assume the fracture to be a parallel plate L Z where, I0 is the modified bessel function of the first kind and with static aperture, 0, and normal compliance, . Here, order zero, ct is the phase velocity of the Stoneley wave, a f is we neglect the effect of roughness, tortuosity and actual con- the acoustic wave velocity in the fluid, w is the frequency of the tact area of fracture on the fluid motion in the fracture. Frac- incident Stoneley wave and A is a constant, respectively. The ture opening is proportional to the fracture compliance and the bar over the symbols denotes that the quantities are in the fre- fluid pressure in the fracture, above static equilibrium. Frac- quency domain. The variation of the pressure along the radial ture opening due to formation elasticity is negligible compared direction in the borehole is low for the range of frequencies of to that due to fracture compliance and is neglected. Thus, dy- interest and we use the pressure averaged over the borehole ra- namic fracture aperture, L(t), can be written as (Hardin et al., dius as the measure for the remainder of the paper and denote it 1987) as < PI >. Pressure for the reflected Stoneley wave and trans- L(t) = L0 + ZPF (t) (8) mitted Stoneley wave follow the same equation with different where PF (t) is the perturbation in the fracture fluid pressure constants. We denote the pressure averaged over borehole ra- due to fluid motion into the fracture. Fluid pressure and flow dius for reflected and transmitted waves as < PR > and < PT >, in the fracture are averaged over the aperture and only their respectively. For continuity of pressure at z=0, we require that radial variation is considered. The net flow out of a volume element, 2prL(t) dr, between r and r +dr from the axis of the < P >=< P > + < P > (2) T I R borehole, during a time increment dt, should equal the change Also, conservation of mass requires that the flow due to the in volume of the element, during the same time, due to pertur- incident wave should be equal to the sum of flow due to the bation in the aperture and the change in the fluid volume due reflected wave, transmitted wave and the flow into the frac- to compressibility of the fluid. Thus, we arrive at ture. Expressing average flow as the product of cross section area and the average particle velocity, the mass conservation ¶q q dL ¶P − + = + Lg F (9) equation is given by ¶r r dt ¶t pR2 < v > = pR2 < v > +pR2 < v > (3) I R T where g is the fluid compressibility and q is the radial flow +2pRL0 < vF (R) > per unit length. Flow in the above equation can be related to the pressure gradient through dynamic conductivity, C.