Tara C. LaForce

Bayesian Reservoir History Matching Considering Model and Parameter Uncertainties

This paper presents a consistent Bayesian solution for data integration and history matching for oil reservoirs while accounting for both model and parameter uncertainties. The developed method uses Gaussian Process Regression to build a permeability map conforming to collected data at well bores. Following that, an augmented Markov Chain Monte Carlo sampler is used to condition the permeability map to dynamic production data. The selected proposal distribution for the Markov Chain Monte Carlo conforms to the Gaussian process regression output. The augmented Markov Chain Monte Carlo sampler allows transition steps between different models of the covariance function, and hence both the parameter and model space are effectively explored. In contrast to single model Markov Chain Monte Carlo samplers, the proposed augmented Markov Chain Monte Carlo sampler eliminates the selection bias of certain covariance structures of the inferred permeability field. The proposed algorithm can be used to account for general model and parameter uncertainties.

Nonmonotonic traveling wave solutions of infiltration into porous media

[1]xa0In uniform soils that are susceptible to unstable preferential flow, the water saturation may exhibit a nonmonotonic profile upon continuous infiltration. As this nonmonotonicity (also known as saturation overshoot) cannot be described by the conventional Richards equation, there have been proposed possible extensions to the unsaturated flow equations, including a nonmonotonic capillary pressure–saturation curve and a second-order hyperbolic term. Here, we present analytic traveling wave solutions to the extended Richards equation. These new solutions indeed display a nonmonotonic saturation profile, similar to previous simulation results. We show that these extensions need a regularization term to produce a unique solution. We develop complete analytic solutions using a relaxation regularization term, and we discuss the results in terms of recent measurements of saturation overshoot.

Robust optimisation of CO2 sequestration strategies under geological uncertainty using adaptive sparse grid surrogates

Geologic CO2 sequestration in deep saline aquifers is a promising technique to mitigate the effect of greenhouse gas emissions. Designing optimal CO2 injection strategy becomes a challenging problem in the presence of geological uncertainty. We propose a surrogate assisted optimisation technique for robust optimisation of CO2 injection strategies. The surrogate is built using Adaptive Sparse Grid Interpolation (ASGI) to accelerate the optimisation of CO2 injection rates. The surrogate model is adaptively built with different numbers of evaluation points (simulation runs) in different dimensions to allow automatic refinement in the dimension where added resolution is needed. This technique is referred to as dimensional adaptivity and provides a good balance between the accuracy of the surrogate model and the number of simulation runs to save computational costs. For a robust design, we propose a utility function which comprises the statistical moment of the objective function. Numerical testing of the proposed approach applied to benchmark functions and reservoir models shows the efficiency of the method for the robust optimisation of CO2 injection strategies under geological uncertainty.

CO2 injectivity in saline aquifers: The impact of non‐Darcy flow, phase miscibility, and gas compressibility

A key aspect of CO2 storage is the injection rate into the subsurface, which is limited by the pressure at which formation starts to fracture. Hence, it is vital to assess all of the relevant processes that may contribute to the pressure increase in the aquifer during CO2 injection. Building on an existing analytical solution for immiscible and spatially varying non-Darcy flow, this paper presents a mathematical model that accounts for combined effects of non-Darcy flow, phase miscibility, and gas compressibility in radial two-phase displacements. Results show that in low-permeability formations when CO2 is injected at high rates, non-Darcy simulations forecast better displacement efficiency compared to flow under Darcy conditions. This will have a positive effect on the formation CO2 storage capacity. This, however, comes at the cost of increased well pressures. More favorable estimations of the pressure buildup are obtained when CO2 compressibility is taken into account because reservoir pressures are reduced due to the change in the gas phase properties. Also, non-Darcy flow results in a significant reduction in halite precipitation in the near-well region, with a positive effect on CO2 injectivity. In the examples shown, non-Darcy flow conditions may lead to significantly different pressure and saturation distributions in the near-well region, with potentially important implications for CO2 injectivity.

Multiple Well Systems with Non-Darcy Flow

Optimization of groundwater and other subsurface resources requires analysis of multiple-well systems. The usual modeling approach is to apply a linear flow equation (e.g., Darcys law in confined aquifers). In such conditions, the composite response of a system of wells can be determined by summating responses of the individual wells (the principle of superposition). However, if the flow velocity increases, the nonlinear losses become important in the near-well region and the principle of superposition is no longer valid. This article presents an alternative method for applying analytical solutions of non-Darcy flow for a single- to multiple-well systems. The method focuses on the response of the central injection well located in an array of equally spaced wells, as it is the well that exhibits the highest pressure change within the system. This critical well can be represented as a single well situated in the center of a closed square domain, the width of which is equal to the well spacing. It is hypothesized that a single well situated in a circular region of the equivalent plan area adequately represents such a system. A test case is presented and compared with a finite-difference solution for the original problem, assuming that the flow is governed by the nonlinear Forchheimer equation.

Design of Carbon Dioxide Storage in Oil Fields

We extend our study of the design of carbon dioxide, CO2, storage in aquifers (Qi et al., 2007) to oilfields. We demonstrate that pore-scale capillary trapping is an effective and rapid mechanism to render the CO2 immobile in oil reservoirs. We construct analytical solutions to the transport equations, accounting for relative permeability hysteresis. We use this to design an injection strategy where CO2 and brine are injected simultaneously followed by chase brine injection. We study field-scale oil production and CO2 storage using a streamline-based simulator that captures dissolution, dispersion, gravity and ratelimited reactions in three dimensions. While injecting at the optimum WAG ratio gives the fastest oil recovery, this allows CO2 to channel through the reservoir leading to rapid CO2 breakthrough and extensive recycling of the gas. We propose to inject more water than optimum. This allows to CO2 to remain in the reservoir, increases the field life and leads to improved storage of CO2 as a trapped phase. A short period of chase brine injection at the end of the process traps most of the remaining CO2. Introduction Carbon Capture and Storage (CCS), the collection of CO2 from industrial sources and its injection underground, could contribute significantly to reductions in atmospheric emissions of greenhouse gases (IPCC, 2005). Possible sites for injection include coalbeds, deep saline aquifers, and depleted oil and gas reservoirs. Although aquifers have the greatest storage potential, injecting CO2 into depleted oil and gas reservoirs can provide additional hydrocarbon production and improved storage security. The principal concern with CO2 storage is its long-term fate: can it be guaranteed that the CO2 will remain underground for hundreds to thousands of years? Since oil and gas has been trapped for geological time in hydrocarbon reservoirs, they should all contain impermeable seals preventing escape. However, the top seal could leak, have gaps or be penetrated by wells through which CO2 could migrate to the surface (Bruant et al., 2002). Dissolution in water and reaction with rock could also contribute to safe CO2 storage, but both are slow processes taking thousands to billions of years (Ennis-King and Paterson, 2005; Hesse et al., 2006; Xu et al., 2003). Capillary trapping (residual non-wetting phase trapping) has been recognized as the most rapid method to immobilize CO2 with time scales in the order of years to decades (Ennis-King and Paterson, 2002; Kumar et al., 2005; Obi and Blunt, 2006; Juanes et al., 2006; Taku Ide et al., 2007; Ghomian et al., 2008). Our previous study on the design of CO2 storage in aquifers (Qi et al., 2007) has recommended an injection strategy in which CO2 and brine are injected simultaneously followed by chase brine injection to render more than 85% of injected CO2 immobile in a very short period. In the oil industry, CO2 flooding has been used worldwide as a tertiary Enhanced Oil Recovery (EOR) mechanism for more than 30 years, particularly for reservoirs with pressures above the minimum miscibility pressure (MMP) where miscible displacement would occur. The ideal reservoirs for miscible CO2 flooding usually have oil densities ranging from 29o to 48o API (882-788 kg/m 3 ) and reservoir depths from 760m to 3700m below the surface (Taber et al., 1997). CO2 flooding has the disadvantage that the unfavorable mobility ratio between the oil and CO2 can result in early CO2 breakthrough because of channeling of CO2 through the reservoir fluids. Water alternate gas (WAG) injection can be successfully applied to improve the sweep efficiency and delay early CO2 breakthrough (Lake, 1989). However, CO2 flooding EOR projects have been designed to minimize the amount of CO2 injected to recover the oil, since the CO2 costs money to transport and inject, while for CCS, injection needs to maximize both CO2 storage and oil recovery. It has been recommended by other researchers that the fluids be injected at the optimal WAG ratio, the injection gas composition be adjusted to reach the MMP, while the well type and completions are designed to maximize both oil recovery and CO2 storage (Jessen, et al., 2004; Kovscek and Cakici, 2004; Malik and Islam, 2000).

Spatially varying fractional flow in radial CO2-brine displacement

[1]xa0In analytical modeling of two-phase flow problems in porous media, the saturation profile for a fixed time can be obtained by using the method of characteristics (MOC). One of the basic assumptions in the application of the MOC is that the fractional flow is a function of saturation only. However, when gas is injected, it is often flowing under nonlinear flow conditions and inertial losses are significant in the near-well region. Therefore, in a radial displacement non-Darcy flow applies at the injection well, but as the saturation front gets further away, its velocity will decrease and the fractional flow curve will vary with the distance along the streamline. This paper presents the extension of the Buckley-Leverett analytical solution when the injected gas phase flow is governed by the two-phase extension to the Forchheimer equation and the fractional flow function depends both on the saturation and radial distance from the well. The behavior of a gas-liquid system under non-Darcy flow conditions is shown for carbon dioxide injection into saline aquifers. Finally, this analytical solution is tested against the corresponding finite difference numerical model and the limitations of the approach are discussed.

A Robust Multi-criterion Optimization of CO2 Sequestration Under Model Uncertainty

Successful CO2 storage in deep saline aquifers relies on economic efficiency, sufficient capacity and long-term security of the storage formation. Unfortunately, these three criteria of CO2 storage are generally in conflict, and often difficult to guarantee when there is a lack of geological characteristics of the storage site. We overcome these challenges by developing: 1) multiwell CO2 injection strategies using a multi-criterion optimization to handle conflicting objectives; 2) CO2 injection management that is robust against model uncertainty. PUNQ-S3 model was modified as a leaky storage to study injection strategies associated with the risks of CO2 leakage under geological uncertainty. Based on our numerical results, the NSGA-II with the ASGI technique can effectively obtain a set of efficient-frontier injection strategies. For the uncertainty assessment, the impact of the model uncertainty to the outcomes is significant. Therefore, our findings suggest using the mixture distribution of the objective-function values, as opposed to the traditional Gaussian distribution to cover model uncertainty

Insight from analytical solutions for improved simulation of miscible WAG flooding in one dimension

In this work, the analytical and numerical solutions for modeling miscible gas and water injection into an oil reservoir are presented. Conservation laws with three levels of complexity are considered. Only the most complex model has the correct phase behavior for the example system, which is a multicontact miscible condensing gas drive with simultaneous water and gas injection. Example displacements in which one or both of the simpler models result in accurate simulations in a fraction of the computation time are presented, along with an example in which neither simplified thermodynamic model achieves a truly satisfactory result. A methodology is presented that can be used to establish the accuracy of simplified models in 1-D simulation based on convergence to analytical solutions for the full three-phase system.

6 – Carbon dioxide (CO2) injection design to maximise underground reservoir storage and enhanced oil recovery (EOR)

We propose designs for CO 2 injection to maximise storage in aquifers and to maximise both CO 2 storage and enhanced oil recovery (EOR) in oil reservoirs. A review of simulation and experimental studies suggests a carbon storage strategy where CO 2 and brine are injected into an aquifer together followed by brine injection alone. Based on simulation studies, this can render 80–95% of the CO 2 immobile in pore-scale droplets within the porous rock. The method does not rely on an impermeable cap rock to contain the CO 2 ; furthermore, the favourable mobility ratio between injected and displaced fluids leads to a more uniform sweep of the aquifer leading to higher storage efficiency than injecting CO 2 alone. We then consider CO 2 storage in oilfields. We propose to inject more water than the traditional optimum that maximizes only oil recovery. This causes the CO 2 to remain in the reservoir, increases the field life and leads to improved storage of CO 2 as a trapped phase. Again, a short period of chase brine injection at the end of the process traps most of the remaining CO 2 .