Laboratory flow-through experiments on artificially fractured phyllite (Kamali-Asl et al., 2021) showed that ultrasonic velocities, amplitudes, attenuation, and the time-frequency partitioning between body and interface waves change systematically with stress-induced evolution of fracture aperture and permeability, with the frequency partitioning tracking the hydraulic (flow-equivalent) aperture rather than the mechanical aperture. Turning these observations into a quantitative, mechanism-based relationship for wave-based monitoring of fractured reservoirs requires linked results that are still missing: (i) effective stress coefficients separating the roles of confining and pore pressure on the wave and hydraulic properties of fractured rocks; (ii) a physical mechanism explaining why interface-wave frequency partitioning encodes hydraulic aperture and is insensitive to purely mechanical aperture changes; together with an explanation of why the Power-Law creep model loses accuracy at elevated pore pressure.

Background

A fluid-bearing fracture in rock is governed by the geometry of the void between two rough wall surfaces. Two related but distinct measures of fracture openness are used. The mechanical aperture is the average geometric separation of the walls; the hydraulic aperture is the equivalent smooth parallel-plate opening that would pass the measured steady flow according to the cubic law of fracture flow. Because of wall roughness, tortuosity, and partial contact between the walls, the hydraulic aperture is generally smaller than the mechanical aperture and responds differently to stress. In subsurface reservoirs (geothermal, hydrocarbon), fracture aperture and permeability evolve during stimulation and production as the confining pressure PcP_c (total normal stress on the fracture) and the pore pressure PpP_p (fluid pressure in the connected void space) change, so remotely measurable proxies of this evolution are highly desirable.

Elastic waves are the leading candidate proxy. A fluid-infiltrated fracture can guide elastic interface waves: modes trapped at, and propagating along, the fracture, with phase velocities between the shear-wave and Rayleigh-wave velocities of the wall rock, existing in fast (roughly symmetric) and slow (roughly antisymmetric) variants; they were discovered and characterized by Pyrak-Nolte and co-workers (Pyrak-Nolte and Cook 1987; Pyrak-Nolte, Xu and Haley 1992). Their excitation and dispersion are governed by the fracture specific stiffness, the incremental normal stress required per unit closure of the fracture, which depends on stress state and contact topology and is itself frequency-dependent (Pyrak-Nolte and Nolte 1992). A parallel line of results connects mechanics to hydraulics: for fractures with spatially correlated aperture fields, fluid flow and fracture specific stiffness obey a near-universal scaling relationship (Petrovitch, Nolte and Pyrak-Nolte 2013; Pyrak-Nolte and Nolte 2016), which is the existing foundation for inferring hydraulic behavior from seismic measurables.

For intact porous rocks, the separate roles of PcP_c and PpP_p on an observable are conventionally expressed through an effective stress coefficient nn: the observable responds to the effective stress σe=Pc−nPp\sigma_e = P_c - n P_p rather than to a single differential stress. For compressional-wave velocities in low-porosity intact rocks such coefficients were measured long ago (Todd and Simmons 1972; Khaksar, Griffiths and McCann 1999) and deviate from unity in property- and pressure-dependent ways; for transport properties, Berryman (1992) showed theoretically that the coefficient can differ substantially from both unity and the Biot poroelastic coefficient (the coefficient relating pore-pressure change to bulk-volume change of a fluid-saturated porous solid in Biot's poroelasticity theory). Effective-stress laws for the permeability of fractures have recently been measured (Zhou et al. 2019 for a single shale fracture under high effective stress; Cheng et al. 2022 for a single fracture versus filling ratio and confining stress; Stanton-Yonge, Mitchell and Meredith 2023 for fracture intersections in basalt, who determined a permeability effective-stress coefficient). By contrast, coefficients for the elastic-wave properties of fractured rocks are largely unavailable.

Frequency imposes a further constraint. Laboratory ultrasonic measurements (hundreds of kHz) sample a regime in which wave-induced pressure gradients between a fracture and the adjacent rock matrix relax only partially within a wave cycle (squirt flow, or more generally wave-induced fluid flow), whereas field seismic waves (tens of Hz) sample the fully pressure-equilibrated regime. Theory predicts characteristic transition frequencies set by fracture geometry and permeability (Brajanovski, Müller and Gurevich 2006; Alkhimenkov et al. 2020), but no validated empirical relationship carries laboratory ultrasonic signatures of fractured rock into the seismic band. This frequency gap is background context for the field applicability of any wave-based monitoring relationship; it is not posed as a required component of the problem below.

Kamali-Asl, KC, Ghazanfari and Falcon-Suarez (2021) performed flow-through experiments on artificially fractured phyllite from the Blue Mountain geothermal field (Nevada, USA) with concurrent ultrasonic P- and cross-polarized S-wave measurements, using two complementary stress paths. In the displacement-controlled test (DCT) the injected flow rate is held constant, so the hydraulic aperture stays fixed while the mechanical aperture changes as pore pressure varies from 1 to 40 MPa; in the pressure-controlled test (PCT) the differential pore pressure is fixed, so both mechanical and hydraulic apertures evolve with stress. They found that velocities, amplitudes, and fracture specific stiffness increase with confining pressure and decrease with pore pressure; that the wavelet-transform time-frequency maps split into a high-frequency body-wave band (about 650–800 kHz) and a low-frequency interface-wave band (about 200–250 kHz); and that the frequency partitioning between the bands changed in the PCT, where the hydraulic aperture varies, but not in the DCT, where it does not, even though the mechanical aperture changed substantially there. Time-dependent fracture closure (creep driven by chemo-mechanical processes at asperity contacts) was captured by Three-Element and Power-Law rheological models — the Three-Element model being the standard linear viscoelastic solid built from elastic and viscous elements, and the Power-Law form having creep compliance that grows as a power of elapsed time — with the Power-Law form losing accuracy at elevated pore pressure. The authors state that the effective stress coefficient for fractured rocks was not estimated and that the literature on it is lacking, and they attribute the frequency-partitioning behavior only qualitatively to flow characteristics, without a quantitative mechanism.

Problem Statement

The problem is to convert the empirical observations of Kamali-Asl et al. (2021) into a quantitative, mechanism-based relationship between the stress state of a fluid-bearing fracture and its elastic-wave signatures, sufficient to infer the evolution of hydraulic aperture from wave measurements. The common scientific objective is remote, wave-based monitoring of stress-induced changes in fracture permeability; the requested results are linked components of that single objective, not independent deliverables:

  1. Effective stress coefficients for fractured rock. Determine — experimentally, with independent control of PcP_c and PpP_p, and/or by validated theory — the effective stress coefficient(s) for the wave properties (velocities, amplitudes, attenuation, frequency content) of fractured rocks, and their relation to the coefficients for the hydraulic properties (aperture, permeability). This includes deciding whether a single coefficient per property suffices and characterizing how the coefficients depend on stress level, lithology, and temperature; that dependence may be established for one rock type over one stress and temperature range, with broader lithology, stress, and temperature coverage counting as progress toward the objective rather than a completion requirement.

  2. Mechanism of hydraulic-aperture encoding. Establish the physical mechanism by which the partitioning of transmitted-wave energy between the body-wave and interface-wave frequency bands encodes the hydraulic aperture and remains insensitive to changes in mechanical aperture at constant hydraulic aperture. An acceptable answer must quantitatively reproduce both contrasting observations — invariance under the displacement-controlled stress path (constant hydraulic aperture, varying mechanical aperture) and sensitivity under the pressure-controlled path (varying hydraulic aperture) — for example by coupling interface-wave dispersion to fracture specific stiffness, contact topology, and fluid effects such as wave-induced flow between fracture and matrix.

  3. Time-dependent component. Resolve why the Power-Law creep model for time-dependent fracture displacement loses accuracy at elevated pore pressure, since creep-driven aperture evolution is the process the wave signatures are meant to monitor.

Partial answers (for example, coefficients for one additional lithology, or a mechanism validated in the ultrasonic band only) constitute progress toward the objective; a complete answer resolves the components jointly in the sense that together they enable quantitative inference of hydraulic-aperture evolution from wave data. No restriction to a particular rock type, stress range, frequency band, or model class is intended.

References

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