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

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

The quantification of pressure and saturation changes in clastic reservoirs using 4D seismic data

The problem of quantifying pressure and saturation changes from 4D seismic data is an area of active research faced with many challenges concerning the non-uniqueness of seismic data inversion, non-repeatability noise in the data, the formulation of the inverse problem, and the use of appropriate constraints. The majority of the inversion methods rely on empirical rock-physics model calibrations linking elastic properties to expected pressure and saturation changes. Model-driven techniques indeed provide a theoretical framework for the practical interpretation of the 4D seismic response but pressure and saturation separation based on this approach are inconsistent with the observed 4D seismic response and insights from reservoir engineering. The outcome is a bias in estimated pressure and saturation changes and for some a leakage between the two. Others have addressed some of this bias using the causality between the induced-production and the observed 4D seismic response to formulate a direct, quick and less compute-intensive inversion - characterised by data-driven techniques. But challenges still remain as to the accuracy of the causality link- as defined by the reservoir’s sensitivity to production effects, and in defining appropriate constraints to tackle non-uniqueness of the seismic inversion and uncertainties in the 4D seismic data. The main contributions of this thesis are the enhancement of data-driven inversion approach by using multiple monitor 4D seismic data to quantify the reservoir’s sensitivity to pressure and saturation changes, together with the introduction of engineering-consistent constraints provided by multiple history-matched fluid-flow simulation models. A study using observed 4D seismic data (amplitudes and times shifts) acquired at different monitor times on four producing North Sea clastic fields demonstrates the reliability of the seismic-based method to decouple the reservoir’s sensitivity specific to each field’s geological characteristics. A natural extension is to combine multiple monitor 4D seismic data in an inversion scheme that solves for the reservoir sensitivity to pressure and saturation changes, the pressure and saturation changes themselves and the uncertainties in the inversion solution. At least two monitor 4D seismic datasets are required to solve for the reservoir’s sensitivity, and offset stacks (near, mid, and far) are required to decouple pressure, water and gas saturation changes. The generation and use of geologically-constrained and production-constrained multiple simulation models provided spatial constraints to the solution space, making the inversion scheme robust. Within the inversion, the fitness to spatial historical data, i.e. 4D seismic data acquired at different monitor times is analysed. The added benefit of using multiple monitor data is that it allows for a soft “close-the-loop” between the engineering and the 4D seismic domain. One step in the inversion scheme is repeated for as many history-matched simulation models as generated. Each model provides pressure and saturation input to the inversion to obtain maps of the reservoir’s sensitivity. By computing the norm of residuals for each inversion based on each model input, the best model (having the lowest norm of residuals) can be identified, besides the use of a history-matching objective. The inversion scheme thus marks the first step for a seismic-assisted history matching procedure, suggesting that pressure and saturation inversion is best done within the history-matching process. In addition, analysis of uncertainties in quantitative 4D seismic data interpretation is performed by developing a seismic modelling method that links the shot timings of a real field towed streamer and a permanent reservoir monitoring (PRM) acquisition to the reservoir under production. It is found that pressure and saturation fluctuations that occur during the shooting of monitor acquisitions creates a complicated spatio-temporal imprint on the pre-stack data, and errors if 4D seismic data is analysed in the post-stack domain. Pressure and saturation changes as imaged across the offset stacks (near, mid and far offset) are not the same, adding to the problems in separating pressure and saturation changes using offset stacks of 4D seismic data. The approximate modelling relay that the NRMS errors between offset stacks (up to 7.5%) caused by the intra-survey effects are likely at the limit of 4D seismic measurements using towed streamer technology, but are potentially observable, particularly for PRM technology. Intra-survey effects should thus be considered during 4D survey planning as well as during data processing and analysis. It is recommended that the shot timestamps of the acquisition is used to sort the seismic data immediately after pre-stack migration and before any stacking. The seismic data should also be shot quickly in a consistent pattern to optimise time and fold coverage. It is common to relate the simulation model output to a specific time within the acquisition (start, middle or end of survey), but this study reveals that it is best to take an average of simulation model predictions output at fine time intervals over the entire length of the acquisition, as this is a better temporal comparison to the acquired post-stack 4D seismic data

Inversion for reservoir pressure change using overburden strain measurements determined from 4D seismic

When significant pore pressure changes occur because of production from a hydrocarbon reservoir the rocks both inside and outside of the reservoir deform. This deformation results in traveltime changes between reflection events on timelapse seismic data, because the distance between reflection events is altered and the seismic velocity changes with the strain. These traveltime differences are referred to as time-lapse time shifts. In this thesis, time-lapse time shifts observed in the overburden are used as an input to a linear inversion for reservoir pressure. Measurements from the overburden are used because, in general, time shift estimates are more stable, the strain deformations can be considered linear, and fluid effects are negligible, compared to the reservoirlevel signal. A critical examination of methods currently available to measure time-lapse time shifts is offered. It is found that available methods are most accurate when the time shifts are slowly varying with pressure and changes in the seismic reflectivity are negligible. While both of these conditions are generally met in the overburden they are rarely met at reservoir level. Next, a geomechanical model that linearly relates the overburden time-lapse time shifts to reservoir pressure is considered. This model takes a semi-analytical approach by numerical integration of a nucleus of strain in a homogeneous poroelastic halfspace. Although this model has the potentially limiting assumption of a homogenous medium, it allows for reservoirs of arbitrary geometries, and, in contrast to the complex numerical approaches, it is simple to parameterise and compututationally efficient. This model is used to create a linear inversion scheme which is first tested on synthetic data output from complex finite-element model. Despite the simplifications of the i inversion operator the pressure change is recovered to within ±10% normalised error of the true pressure distribution. Next, the inversion scheme is applied to two real data cases in different geological settings. First to a sector of the Valhall Field, a compacting chalk reservoir in the Norwegian Sea, and then the Genesis Field, a stacked turbidite in the Gulf of Mexico. In both cases the results give good qualitative matches to existing reservoir simulator estimates of compaction or pressure depletion. It is possible that updating of the simulation model may be assisted by these results. Further avenues of investigation are proposed to test the robustness of the simplified geomechanical approach in the presence of more complex geomechanical features such as faults and strong material contrasts

Experimental Rock Physics and Applied Geophysical Models for Long-Term Monitoring of Carbon Dioxide Injected Reservoirs

New methods of frequency and stress dependent petrophysical modeling are developed to link and predict laboratory, well log, and seismic scale pore fluid and pressure effects. These effects include pressure induced pore expansion, dissolution and material loss, and fluid effects on bulk properties. Petrophysical models that incorporate wave propagation at ultrasonic, well log, and seismic frequencies are produced with effective pressure and fluid dependent elements in reservoir limestone and sandstone for the purpose of reservoir monitoring. The petrophysical model introduces stress sensitivity elements into bulk and shear moduli to account for non-linear elastic behavior at the low effective pressure regimes. Stress effects are modeled by defining stiff and compliant pore classes with assigned stress sensitivities based on geometric properties. The c33 elastic constant is then modified to include frequency dependent attenuation in the P wave velocity model. The characteristic frequencies are defined by not only the passing wave frequency but also key properties including permeability, fluid viscosity, and bulk modulus. The model input parameters are derived from core measurements and multi-scale observations including core velocity measurements, scanning electron microscopy, and computed micro tomography. Limestone dissolution is observed in laboratory experiments performed with reactor vessels at in situ conditions using CO2-H2O mixes. The petrophysical models are updated to reflect the observed dissolution results. Further, the before and after µCT analysis of the samples reveal internal porosity gains, accompanied by decreases in pore surface area to volume ratios, which are seen to be limiters in chemical reaction rates. Finally, CO2 quantification techniques in reservoir pore space are explored. Modeled and observed properties are implemented to interpret repeat reflection seismic surveys in which changes in pore pressure and pore-filling fluid density occur. The Sandstone and limestone reservoir fluid substitution models are compared with seismic anomalies to delineate pressure effects from fluid property effects. Impedance models at the sandstone reservoir reveal a 25% maximum acoustic impedance decrease with a fluid substitution filling the reservoir with 75% CO2. This significant impedance difference leads to increased reflectivity, which is confirmed with actual 4-D reflection seismic surveying