Adjoint-state method for seismic AVO inversion and time-lapse monitoring

Abstract

This dissertation presents seismic amplitude versus offset (AVO) inversion methods to estimate water saturation and effective pressure quantitatively in elastic and viscoelastic media. Quantitative knowledge of the saturation and pore pressure properties from pre- or post-production seismic measurements for reservoir static or dynamic modeling has been an area of interest for the geophysical community for decades. However, the focus on the existing inversion methodologies and explicit expressions to estimate saturation-pressure variables or changes in these properties due to production or fluid injection has been based on elastic AVO models. These conventional methods do not consider the seismic wave attenuation effects on the reflection amplitudes and therefore can result in biased prediction. Numerous theoretical rock physics models and laboratory experiments have demonstrated the sensitivity of various petrophysical and seismic properties of partially fluid-filled porous media to seismic attenuation. This makes seismic wave attenuation a valuable time-lapse attribute to reliably measure the saturation (Sw) and effective pressure (Pe) properties. Therefore, in this work, I have developed two AVO inversion processes i.e., the conventional AVO inversion method for elastic media and the frequency-dependent amplitude versus offset (FAVO) inversion technique for the viscoelastic media. This dissertation first presents the inversion strategies to invert the pre-stack seismic data for the seismic velocities and density by using the conventional AVO equation and for the seismic velocities, density, and Q-factors by using the frequency-dependent AVO method. These inversion methods are then extended to estimate the dynamic reservoir changes e.g., saturation and pressure variables, and can be applied to predict the saturation and pressure variables at any stage e.g., before and during production, or fluid injection, or to estimate the changes in saturation (ΔSw) and pressure (ΔPe). The first part of the dissertation describes the theory and formulation of the elastic AVO inversion method while in the second half, I have described the viscoelastic inversion workflow. FAVO technique accounts for the dependence of reflection amplitudes on incident angles as well as seismic frequencies and P and S waves attenuation in addition to seismic velocities and density. The fluid saturation and pressure in the elastic and inelastic mediums are linked to the reflection amplitude through seismic velocities, density, and quality factors (Q). The inversion process is based on the gradient-descent method in which the least-square differentiable data misfit equation is minimized by using a non-linear limitedmemory BFGS method. The gradients of the misfit function with respect to unknown model variables are derived by using the adjoint-state method and the multivariable chain rule of derivative. The adjoint-state method provides an efficient and accurate way to calculate the misfit gradients. Numerous rock physics models e.g., the Gassmann substitution equation with uniform and patchy fluid distribution patterns, modified MacBeth’s relations of dry rock moduli with effective pressure, and constant Q models for the P and S wave attenuation are applied to relate the saturation and effective pressure variables with elastic and an-elastic properties and then forward reflectivity operator. These inversion methods have been defined as constrained problems wherein the constraints are applied e.g., bound constraints, constraints in the Lagrangian solution, and Tikhonov regularization. These inversion methods are quite general and can be extended for other rock physics models through parameterizations. The applications of the elastic AVO and the FAVO methods are tested on various 1D synthetic datasets simulated under different oil production (4D) scenarios. The inversion methods are further applied to a 2D realistic reservoir model extracted from the 3D Smeaheia Field, a potential storage site for the CO2 injection. The inversion schemes successfully estimate not only the static saturation and effective pressure variables or changes in these properties due to oil production or CO2 injection but also provide a very good prediction of seismic velocities, density, and seismic attenuation (quantified as the inverse quality factor). The partially CO2-saturated reservoir exhibits higher P wave attenuation, therefore, the addition of time-lapse P wave attenuation due to viscous friction between CO2-water patches helps to reduce the errors in the inverted CO2/water saturation variables as compared to the elastic 4D AVO inversion. This research work has a wide range of applications from the oil industry to carbon capture and storage (CCS) monitoring tools aiming to provide control and safety during the injection. The uncertainty in the inversion results is quantified as a function of the variability of the prior models obtained by using Monte Carlo simulation.