Mahyar Madadi

Modelling elastic anisotropy of dry rocks as a function of applied stress

We propose an analytical model for seismic anisotropy caused by the application of an anisotropic stress to an isotropic dry rock. We first consider an isotropic, linearly elastic medium (porous or non-porous) permeated by a distribution of discontinuities with random (isotropic) orientation (such as randomly oriented compliant grain contacts or cracks). The geometry of individual discontinuities is not specified. Instead, their behaviour is defined by a ratio B of the normal to tangential excess compliances. When this isotropic rock is subjected to a compressive stress (isotropic or anisotropic), the specific surface area of cracks aligned parallel to a particular plane is reduced in proportion to the normal stress traction acting on that plane. This effect is modelled using the Sayers-Kachanov non-interactive approximation, which expresses the effect of cracks on the elastic compliance tensor as an integral over crack orientations. This integral is evaluated using the Taylor expansion of the stress dependency of the specific surface area of the cracks. This allows the analytical solution previously derived for small anisotropic stresses to be extended to large stresses. Comparison of the model predictions with the results of laboratory measurements shows a reasonable agreement for moderate magnitudes of uniaxial stress (up to 50 MPa). While the model contains five independent parameters, the variation of the anisotropy pattern (which can be expressed by the ratios of Thomsen’s anisotropy parameters e, δ and γ) with normalized stress is controlled by only two parameters: Poisson’s ratio ν of the unstressed rock and the compliance ratio B. The model predicts that the e/γ ratio depends on both ν and B but varies only mildly with stress, while the e/δ ratio varies between 0.8–1.1 in a wide range of values of ν and B. The latter observation implies that the anisotropy remains close to elliptical even for larger stresses (within the assumptions of the model). The proposed model of stress-induced anisotropy may be useful for differentiating stress-induced anisotropy from that caused by aligned fractures. Conversely, if the cause of seismic anisotropy is known, then the anisotropy pattern allows one to estimate P-wave anisotropy from S-wave anisotropy.

A reproducible framework for 3D acoustic forward modelling of hard rock geological models with Madagascar

A special challenge of hard rock exploration is to identify targets of interest within complex geological settings. Interpretation of the geology can be made from direct geological observations and knowledge of the area, and from 2D or 3D seismic surveys. These interpretations can be developed into 3D geological models that provide the basis for predictions as to likely targets for drilling and/or mining. To verify these predictions we need to simulate 3D seismic wave propagation in the proposed geological models and compare the simulation results to seismic survey data. To achieve this we convert geological surfaces created in an interpretation software package into discretised block models representing the different lithostratigraphic units, and segment these into discrete volumes to which appropriate density and seismic velocity values are assigned. This approach allows us to scale models appropriately for desired wave propagation parameters and to go from local to global geological models and vice versa. Then we use these digital models with forward modelling codes to undertake numerous 3D acoustic wave simulations. Simulations are performed with single shot and with exploding reflector (located on extracted geological surface) configurations.

Estimation of elastic anisotropy from three-component ultrasonic measurements using laser Doppler interferometry

Ultrasonic measurements using laser Doppler interferometry (LDI) have been reported to provide robust estimates of elastic anisotropy of rock samples. In this approach, an ultrasonic wave is emitted by a piezo-electric source and detected by the LDI, which can be configured to measure three components of the particle velocity in a very small area (~1 mm2) of the sample. Repeating these measurements for a dense array of points on the sample’s surface gives a distribution of traveltimes and polarisation fields on the surface. Anisotropy is then obtained by inverting these fields using analytical expressions or numerical algorithms for computing phase and group velocities. The existing implementation of this approach involves the inversion of direct compressional (P) and shear (S) wave arrivals only. A previous study showed that this approach produces stable results if only a small range of source–receiver offsets is included in the inversion. This limitation resulted in a relatively large uncertainty of the result. This uncertainty can be reduced by inverting the entire traveltime field. To this end, we numerically simulate the wavefield in the sample. Analysis of the computed wavefield reveals the presence of P- and S-waves as well as a critically refracted converted PS-wave. Hence, the inversion of the entire traveltime field must include these three waves. We implement this inversion using global minimisation of the traveltime misfit function, coupled with numerical computation of ray velocities. Application of this algorithm to laboratory LDI measurements on a transversely isotropic phenolic sample provides stable anisotropy estimates consistent with previous studies. Laser Doppler interferometers were previously employed to detect ultrasonic waves propagating in different directions and to estimate elastic anisotropy from these measurements. Our numerical simulations and laboratory measurements show that the recorded wavefield contains converted PS-waves, which need to be taken into consideration to obtain robust estimates of anisotropy.

Joint inversion of P-, and S-wave travel times for characterisation of anisotropic materials using laser Doppler interferometry measurements

We used laser Doppler interferometer for measuring the displacement on the sample surface. These measurements allow us to clearly separate different wave types, whose picked travel times are used for estimation of VTI anisotropy parameters. One of the observations in this study is the very strong amplitude of critically refracted SP wave at the measurement surface. We confirmed the characteristics of this wave by numerical modelling. We used this wave to improve the estimates of the anisotropy. The observed strong amplitude of this wave can have strong implications for the interpretation of ultrasonic measurements.

Integration of Stratigraphic & Rock Physics Models to Generate Synthetic Seismic Data

Stratigraphic forward modelling (SFM) is an important subsurface modelling method. A numerical SFM program, such as the Sedsim software used in this study, is able to quantitatively model the sedimentation process with time in order to predict rock properties away from well data. Although numerical SFM is a powerful technique, it is important to quantify and minimise the uncertainty in the resultant stratigraphic model. This uncertainty can be reduced by producing synthetic seismic traces from the results of the stratigraphic model. This simulated seismic may then be compared to observed seismic over the same area and the parameters of the stratigraphic model modified based on the results of the comparison. In order to generate synthetic seismic from the results of a stratigraphic model, sediment properties from the stratigraphic model must be converted to acoustic properties. This becomes challenging at inter-well locations, or locations with little or no well control. Fortunately, such conversion can be achieved by the application of a suitable rock physics model even at those challenging locations. The integration of a Sedsim stratigraphic model and the Velocity-Porosity-Clay (VPC) rock physics model in the Cornea field, Browse Basin, Australia shows the importance of integrating geological and geophysical methods in order to reduce uncertainty when predicting subsurface properties.

Reciprocity principle in finite difference modelling of waves in elastic media

Reciprocity principle has been used in a number of seismic applications. This principle relates the two wave fields with interchanged source and receiver locations, where the radiation patterns of the source and receiver are interchanged as well. In extending this principle to be used in real-world scenarios where radiation patterns vary in different locations, a number of experiments to determine the validity of this principle were conducted. Given the proliferation of the numerical modelling in today’s geophysical data processing and imaging, the verification of validity of the reciprocity theorem for the modelling algorithms is important. We found that the reciprocity principle is not upheld for some instances of finite difference modelling due to the implementation of the free surface boundary condition. In the case of absorbing boundary conditions however, good reciprocity relation can be achieved.

Feasibility of Cross-well Seismic as CO2 Monitoring Tool

The next stage of CO2CRC Otway project involves exploration of the ability of various CO2 geosequestration techniques, including cross-hole seismic, to detect and monitor presence of CO2. Despite the limited spatial coverage of a cross-well survey, the acquired data could be used to improve reliability of the whole monitoring and verification program. Prior to any field experiment we evaluate the feasibility of cross-well seismic using computer modelling. We utilize finite-difference time-domain (FDTD) method for pre- and post- injection stages. Here we present the results of our study and validate the detectability of CO2/CH4 gas mixture on time-lapse cross-well seismic data on the direct as well as the reflected wave fields. We demonstrate that the presence of 15,000 t of a gas plume can lead to changes in transit times of up to 1.4 ms (in cross-well setting). The computed seismic tomography detects the difference in velocities up to 80 m/s. The difference caused by gas is also detectable in the migrated time section of reflected waves.

Modelling anisotropy pattern of dry rocks as a function of applied stress

We propose an analytical model for seismic anisotropy caused by application of an anisotropic stress to an isotropic dry rock. We first consider an isotropic linearly elastic medium (porous or non-porous) permeated by a distribution of discontinuities with random (isotropic) orientation (such as randomly oriented compliant grain contacts or cracks). Geometry of individual discontinuities is not specified. Instead, their behaviour is defined by a ratio of the normal to tangential excess compliances. When this isotropic rock is subjected to a small compressive stress (isotropic or anisotropic), the specific surface area of cracks aligned parallel to a particular plane is reduced in proportion to the normal stress traction acting on that plane. This effect is modelled using the Sayers-Kachanov non-interactive approximation. The integral over the orientation distribution is evaluated using Taylor expansion of the stress dependency of the specific surface area of cracks. Comparison of the model predictions with the results of laboratory measurements shows a reasonable agreement for moderate magnitudes of uniaxial stress (up to 30 MPa). The results suggest that the relations between anisotropy parameters do not change with increasing stress.