Vertically averaged equations with variable density for CO2 flow in porous media

Carbon capture and storage has been proposed as a viable option to reduce CO 2 emissions. Geological storage of CO 2 where the gas is injected into geological formations for practically indefinite storage, is an integral part of this strategy. Mathematical models and numerical simulations are important tools to better understand the processes taking place underground during and after injection. Due to the very large spatial and temporal scales involved, commercial 3D-based simulators for the petroleum industry quickly become impractical for answering questions related to the long-term fate of injected CO 2 . There is an interest in developing simplified modeling tools that are effective for this type of problem. One approach investigated in recent years is the use of upscaled models based on the assumption of vertical equilibrium (VE). Under this assumption, the simulation problem is essentially reduced from 3D to 2D, allowing much larger models to be considered at the same computational cost. So far, most work on VE models for CO 2 storage has either assumed incompressible CO 2 or only permitted lateral variations in CO 2 density (semi-compressible). In the present work, we propose a way to fully include variable CO 2 density within the VE framework, making it possible to also model vertical density changes. We derive the fine-scale and upscaled equations involved and investigate the resulting effects. In addition, we compare incompressible, semi-compressible, and fully compressible CO 2 flow for some model scenarios, using an in-house, fully-implicit numerical code based on automatic differentiation, implemented using the MATLAB reservoir simulation toolkit

A Finite-Volume-Based Module for Unsaturated Poroelasticity

In this chapter, we present fv-unsat, a multipoint finite-volume–based solver for unsaturated flow in deformable and nondeformable porous media. The latter is described using the mixed form of Richards’ equation, whereas the former by the equations of unsaturated poroelasticity. The module aims at flexibility, relying heavily on discrete operators and equations, exploiting the automatic differentiation framework provided by the MATLAB Reservoir Simulation Toolbox (MRST). Our examples cover two numerical convergence tests and two three-dimensional practical applications, including the water infiltration process in a nondeformable soil column and a realistic desiccation process of a deformable clay sample using atmospheric boundary conditions. The resulting convergence rates are in agreement with previously reported rates for single-phase models, and the practical applications capture the physical processes accurately.publishedVersio

Upscaled modeling of CO2 injection and migration with coupled thermal processes

A practical modeling approach for CO2 storage over relatively large length and time scales is the vertical-equilibrium model, which solves partially integrated conservation equations for flow in two lateral dimensions. We couple heat transfer within the vertical equilibrium framework for fluid flow, focusing on thermal processes that most impact the CO2 plume. We investigate a simplified representation of heat exchange that also includes transport of heat within the plume. In addition, we explore available CO2 thermodynamic models for reliable prediction of density under different injection pressures and temperatures. The model concept is demonstrated on simplified systems.publishedVersio

Numerical studies of long-term wettability alteration effects in CO2 storage applications

The wettability of the rock surface in porous media has an effect on the constitutive saturation functions that govern capillary pressure and relative permeability. The term wettability alteration refers to the change of this property over time by processes such as CO2 interactions with the rock. In this work, we perform numerical simulations considering a two-phase two-component flow model including time-dependent wettability alteration in a two-dimensional aquifer-caprock system using the open porous media framework. Particularly, we study the spatial distribution over time of injected CO2 into the aquifer neglecting and including wettability alteration effects. The numerical simulations show that wettability alteration on the caprock results in a loss of containment; however, the CO2 front into the caprock advances very slow since the unexposed caprock along the vertical migration path also needs to be changed by the slow wettability alteration process. The simulations also show that wettability alteration on the aquifer results in an enhancement of storage efficiency; this since the CO2 front migrates more slowly and the capillary entry pressure decreases after wettability alteration.acceptedVersio

Modeling geomechanical impact of fluid storage in poroelastic media using precomputed response functions

When injecting CO2 or other fluids into a geological formation, pressure plays an important role both as a driver of flow and as a risk factor for mechanical integrity. The full effect of geomechanics on aquifer flow can only be captured using a coupled flow-geomechanics model. In order to solve this computationally expensive system, various strategies have been put forwards over the years, with some of the best current methods based on sequential splitting. In the present work, we seek to approximate the full geomechanical effect on flow without the need of coupling with a geomechanics solver during simulation, and at a computational cost comparable to that of an uncoupled model. We do this by means of precomputed pressure response functions. At grid model generation time, a geomechanics solver is used to compute the mechanical response of the aquifer for a set of pressure fields. The relevant information from these responses is then stored in a compact form and embedded with the grid model. We test the accuracy and computational performance of our approach on a simple 2D and a more complex 3D model, and compare the results with those produced by a fully coupled approach as well as from a simple decoupled method based on Geertsma’s uniaxial expansion coefficient.acceptedVersio

Geomechanical consequences of large-scale fluid storage in the Utsira Formation in the North Sea

-In this work we look at the geomechanical implications of injecting large volumes of fluid in the Utsira Formation. Our modelling is based on Biot's poroelasticity in combination with one-phase fluid flow. We study four different injection scenarios over 25 year: 1 Mt/year, 10 Mt/year, 100 Mt/year and 1000 Mt/year. We observe that the pore fluid pressure scales with the injection rate. The strain of the Utsira Formation and the related surface uplift can be estimated with simple a 1D model. A particular uncertainty with the modelling is the mechanical properties of the Utsira sand and the cap rock

Investigation of caprock integrity due to pressure build-up during high-volume injection into the Utsira formation

In this work, we investigate the pressure-limited storage capacity of the Utsira formation. We employ the use of a simple capacity estimate based on maximum sustainable pressure. Here, the pressure during injection or post-injection cannot exceed the least compressive stress at the base of the caprock at any location. Given the global capacity estimate, large-scale simulations are performed to determine if the global capacity can be reasonably attained given local injectivity constraints and long-term CO2 trapping. We find that the Utsira can withstand injection rates over 100 Mt/y for 50 years, which is equivalent to 8.3 Gt of CO2.publishedVersio

Dynamic estimates of extreme-case CO2 storage capacity for basin-scale heterogeneous systems under geological uncertainty

Geological CO2 storage is expected to grow dramatically in the coming decades to meet global climate targets. Assessment of worldwide storage resources using static methods indicates significant theoretical potential for large-scale deployment. Dynamic capacity estimates are needed at the basin-scale that fully capture the impact of geological uncertainty and account for regional limits on pressure buildup. Accurate quantification of the risk of low or critically low capacity under extreme occurrences of heterogeneity will be increasingly important. There are significant challenges associated with efficient computation of low probability capacity within Monte Carlo frameworks at these scales. In this paper, we propose a workflow for uncertainty quantification that is able to efficiently estimate increasingly outer percentiles of dynamic capacity such as P1, P0.1, or even lower probability events. Our approach is based on the rare-event methodology that uses a subset simulation approach to concentrate sampling of the parameter space in the tail regions of the capacity distributions. This approach greatly speeds up uncertainty quantification for very small probabilities compared to standard Monte Carlo. We demonstrate the method by introducing a correlated heterogeneity field to a highly prospective basin-scale system that can support regional injection rates of 100 million tons annually. We find that the outer quantiles are more sensitive to the underlying geostatistical model compared to the median P50 capacity. This implies that for large-scale systems, well characterized heterogeneity is essential to identify the likelihood of very rare yet still relevant dynamic estimates of storage capacity

An analytical plane-strain solution for surface uplift due to pressurized reservoirs

In this paper, we present an analytical plane strain solution for surface uplift above pressurized reservoirs. The solution is based on a Fourier representation of the reservoir pressure. The plane strain model is developed in two stages: First, an exact solution is derived for the displacement field for the reservoir alone subjected to a periodic overpressure distribution of one wavelength. This one-layer model forms the basis for the analytical plane strain solution for a two-layer model –a pressurized reservoir with an overburden. We give an example where numerically computed uplift is quite accurately estimated by a simple 1D estimate, except for in the near well area. The plane-strain solution is well suited to study conditions for when the simple 1D approximation of the uplift is accurate. A condition for the accuracy of the simple 1D approximation is first derived for just the reservoir expanded by a periodic overpressure distribution of one wavelength, which corresponds to one term in a Fourier series. The 1D estimate is accurate for wavelengths larger than 2π2π times the reservoir thickness. Then, a condition is derived for when the 1D estimate is accurate for the two-layer model. We show that the wavelength of the overpressure distribution must be larger than 2π2π times the maximum of the reservoir thickness and the overburden thickness for the 1D approximation to be accurate. We demonstrate how uplift is computed from a Fourier decomposition of the reservoir overpressure. The resulting uplift is analysed in terms of Fourier coefficients, using the knowledge of how a single wavelength behaves. The analytical results for the displacement field and the uplift are tested by comparison with finite element simulations, and the match is excellent.An analytical plane-strain solution for surface uplift due to pressurized reservoirssubmittedVersio