Background - University of Texas at Austin
Focus Area 4: Simulation of Multi-scale, Multi-physics, Heterogeneous Systems
Research Background and Significance
CFSES scientists and engineers pool their knowledge and resources by forming collaborative research partnerships. Supporting these partnerships, and in many ways making them possible, are mathematical modeling and computer simulation. In fact, predictive computational simulation may be the only practical means available for the human mind to grasp the complexities of the subsurface environment.
Focus area Area 4 concerns the developspment of cutting-edge techniques for simulating complex processes in underground reservoirs and aquifers that govern the disposal of CO2, radionuclide wastes, and other byproducts of energy production. An overarching principle in the research is to bridge the scales in both space and time from fine molecular scales to the basin scale for the multitude of physical processes that affect CO2 and nuclear waste movement, reaction, and storage.
The basin scale is the coarsest, and it is the scale on which the simulation must be made. A typical reservoir or aquifer is extremely large in space. It may cover an area the size of several counties or even an entire state. In time, migration of CO2 may last hundreds of years, and nuclear contaminants millions of years.
The physical, chemical, and even biological processes within the subsurface operate over sub-millimeter space and sub-second time scales. These fine-scale processes have profound effects on the large and long-time basin-scale behavior, so they must be modeled so that they can be accurately simulated on the coarse basin-scale. The world’s fastest supercomputers, even well into the foreseeable future, are inadequate to resolve directly these fine scales over the basin-scale. Thus Focus Area 4 uses insight gained from the other Focus Areas operating on finer scales and develops upscaled computational technologies suitable for coarse, basin-scale simulation that bridge the multiple scales inherent in the problem.
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Illustration 1: Fig. XX – Water phase saturations (top figures) and speciesCO2 concentration (bottom figures) profiles at 50 and 100 days using enhanced velocity multiblock. The CO2 moves in complicated and unexpected ways.
IPARS has the capability of applying the enhanced velocity Mixed Finite Element method where non-matching multiblock grids are used to take advantage of different discretizations in different parts of the simulation domain and maintaining the flux continuity along the interfaces. Figure XX gives an example of such simulation that resulted in a significant computational time saving of about 81%. We are in the process of testing this methodology for CO2 injection where we can use very fine grids near injection wells and larger cell sizes further away from the wells to benefit from more accuracy of the fine grid near wells
Research Questions
1. How can we conduct simulations that capture the essential features of basin-scale behavior that emerge from fine-scale phenomena, such as flow, transport, reaction, and deformation in the Earth’s subsurface, without resolving all fine-scale features? In many ways, this is our first fundamental research question.
2. Can we devise accurate models and robust numerical algorithms for simulating events that are driven by far-from-equilibrium conditions? This is our second fundamental research question.
3. What are appropriate basin-scale reaction dynamic parameters, and how do they change in time as reservoir conditions change? Reactions take place on the molecular scale, but we cannot resolve individual molecules in a basin-scale simulation. In fact, computational limitations dictate that we must average all the molecules within perhaps a 10-meter3 volume, and yet still account for the fact that most of them are not in contact with each other to react. A particular challenge is to understand reactive mineral surface-area factors and how they change as diagenesis proceeds along a reaction front.
4. How do coarser-scale interactions such as changing temperature, or nutrient introduction via fracture flow, influence the growth of microbes? How does microbial growth feed back to change basin-scale properties, such as permeability, surface chemistry, reactive surface area, and microbe-mineral interactions? Coarse scale phenomena affect fine-scale processes, which in turn feed back to modify coarse-scale phenomena. These must be modeled carefully, or erroneous conclusions may be drawn.
5. How can the averaging that underlies a continuum basin-scale model be improved to account for underlying spatially correlated properties? The subsurface environment is highly heterogeneous, meaning that the physical properties of rocks change greatly from one point to another over fine-scales. Again, these fine scales cannot be resolved by the fastest supercomputers, and some upscaling is required.
6. How do we reliably model changes in mechanical properties of rocks due to chemical interactions between the injected fluid and rock mass, and how do we incorporate such transient rock property models in accurate geo-mechanical models for fracture initiation and propagation? Compaction of the reservoir rock and the overlying cap rock seal integrity are key issues in the sequestration of CO2.
7. Can we design more efficient computational algorithms for solving the model equations, so as to take full advantage of modern supercomputer technology? Modern computers achieve high rates of computational power by working in parallel. New algorithms are needed to coordinate the computations of thousands of small processor nodes to keep each productively employed at all times during the simulation.
8. How do we assimilate seismic and other measured data into high-resolution models for simulating CO2 storage? During a CO2 injection process, data will be continually collected. It must be compared to simulation predictions to refine and improve the underlying model so it is both consistent with measured data and a more reliable predictive tool.
9. How do we represent and quantify uncertainty in as to the fate of injected fluids in the subsurface due to approximate modeling, data measurement errors, and incomplete geologic uncertaintycharacterization? The answer to this question is needed to assess risks to human health and the environment associated with human interaction with geo-systems.
Research Innovations and Goals
The main goal of research Focus Area 4 is to develop cutting-edge techniques for accurate and reliable simulation of CO2 sequestration and other energy waste storage phenomena techniques in the subsurface at reservoir and basin scales. The work involves:
1. Developing new and improved models of tightly interconnected physical processes, including multi-phase compositional flow, reactive transport, the behavior of multiple phases, hysteretic relative permeability and capillary pressure relations, and geo-mechanics;
2. Coupling multi-physics processes across spatial and temporal scales, including innovative multi-scale techniques, multi-physics couplings of physical processes, and multi-scale time-stepping and linear solver techniques;
3. Uncertainty quantification and validation, including more efficient data assimilation and uncertainty assessment tools, and characterization and validation against data from actual field sites.
Expected Outcome
CFSES will develop an underlying scientific knowledge base and prototypical simulator codes. These will facilitate the science and engineering community in developing the technology needed to reliably predict, control, and manage human interaction with geo-systems. Through computational simulation, it is possible to assess the efficacy of storage designs and operation protocols in a virtual world. Failure in this context causes no damage to the environment. By simulating various scenarios, uncertainty and risk to human health can also be quantified and minimized. Once proper designs and protocols have been developed, they can be implemented to achieve the societal objective: the safe and long-term storage of CO2 or nuclear waste. Spin-off of the technology might be used to protect underground water resources from compaction and contamination, to enhance fossil fuel production, and to foster groundwater sustainability.
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Highlights: Recent Results
GriddingMeshing and Discrete Approximations
At the University of Texas at Austin, research groups in the Bureau of Economic Geology (BEG) conduct laboratory and field experiments at CO2 injection demonstration sites. Collaboration with these groups provides data for calibration, verification, and validation of models. Figure 1 shows a hexahedral mesh from the BEG Frio Pilot Site. This mesh has elements with non-planar faces, some of which are highly distorted, that represent geological structures such as layers, faults, and pinch outs.
The geological data
Geological data of thedescribing reservoirs, and subsurface aquifers formations in general, is often poorly suited for numerical simulations. The goal of the gridding efforts is to transform the data to improve the quality of the numerical simulations. The focus has been in describing the given geometry by a grid or mesh of points in a more accurately way and with fewer and reducing the number of degrees of freedommesh points. At the same time, the quality of each individual mesh cell or element is kept highas nearly as possible to a cube, as illustrated in Figure 2. All this translates into shorter simulation times and, the most important, more accurate results.
Discrete Approximations
At UT-Austin, there are research groups in the Bureau of Economic Geology (BEG) working on CO2 laboratory and field demonstration sites. Collaboration with these groups provides data for calibration, verification, and validation of models. For example, the hexahedral grids being tested came from the BEG Frio Pilot Site. This grid has elements with non-planar faces, some of which are highly distorted, that represent geological structures such as layers, faults, and pinch outs.
SIt is well known that standard methods, such as mixed finite element methods, allow us to approximate the pressure field within the reservoir. The gradient of the pressure field is proportional to the fluid velocity, which determines where CO2 migrates. On general hexahedral on general hexahedrmeshes,ons standard methods can may not converge to the true pressure distribution as the mesh spacing is refined. be non-convergent unless the hexahedral meshes asymptotically close to parallelepipeds. This mesh restriction is no longer needed Iin a recently developed cell-centered numerical method, CFSES researchers have no convergence issues for general meshes.. This new computer model gives accurate prediction of CO2 flow in deep saline aquifers, which is crucial in preventing the leakage of CO2.
A Multi-scale Preconditioner
The simulation of complex subsurface phenomena such as CO2 sequestration requires highly specialized solution techniques, leveraging high performance computation on massively parallel supercomputers. In order to minimize computation time and maximize solution accuracy, it is advantageous to allow portions of the domain to contain different types of physics, scales, and numerical methods. For example, a compositional flow model in a salt dome could be coupled with a basin scale geo-mechanical model to assess the feasibility of long termlong-term carbon storage. To meet these goals, a computational framework has been developed that , which uses non-overlapping domain decomposition to allow multi-scale and multi-physics coupling using mortar finite elements. Our approach is referred to as the multi-scale mortar mixed finite element method.
This approach is both efficient and flexible: A reduced number of unknowns are consolidated on the interfaces and the resulting algebraic system is solved with an iterative method. In this way, the interface operator need not be formed since only its action is needed; each interface iteration requires the solution of sub-domain problems that can be performed in parallel. The solution can be resolved to a very fine level of detail around critical areas of interest, as well as adaptively resolved using mesh refinement approaches. Moreover, the physical models can also be placed in different areas throughout the domain where they are deemed necessary.
A new type of preconditioner has recently been developed to improve the efficiency of the multi-scale mortar method, known as the frozen Jacobian multi-scale preconditioner. Multiphase flow simulations are highly nonlinear, transient systems, which are typically solved using a Newton-Krylov algorithm. The basic idea of the preconditioner is to pre-compute the solutions to several sub-domain problems with various boundary conditions in parallel for a fixed state of the system, meaning for a fixed Jacobian and time step. These form what is known as a multi-scale basis, which can be used to approximate the action of the interface operator for several time steps. As the dynamics of the simulation change, the effectiveness of the preconditioner degrades in time. Our results demonstrate that the multi-scale preconditioner can be recomputed sparingly throughout the simulation to balance the overhead of its construction with its potential reduction in interface iterations.
Figure 1: Oil concentration in a two-phase multi-scale water flood simulation, using two sub-domains connected with high order mortar elements at their interface. It is a 400 day simulation where a heterogeneous reservoir is initially saturated with an oil phase, water is injected in the upper left corner, and oil is produced in the lower right corner from an aerial perspective.
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Figure 2: The effectiveness of the multi-scale preconditioner at reducing the amount of work necessary to solve this problem. The graph represents the number of interface iterations per time step, where the red curve is unpreconditioned, the blue curve is with a preconditioner computed at the initial time, and the black curve is where the preconditioner is recomputed periodically throughout the simulation. In this example, the use of the multi-scale preconditioner directly translated to a reduction in runtime by over 40% over the original approach.
Dynamically Coupled Flow and Mechanics for Geo-systems
Flow through porous materials is coupled to solid mechanics by virtue of Terzaghi’s principle of effective stress, wherein stresses in porous materials are borne by both pore pressure and the stress field in the solid skeleton. Coupled flow and mechanics in heterogeneous media is of particular importance to simulation of subsurface activities related to US energy security, including, for example, studies of geologic sequestration of anthropogenic CO2 and geologic disposal of heat-generating high-level nuclear waste. We are developing simulation tools to enable modeling of multiphase, multi-component flow in heterogeneous media (see Figure 1) and of nonlinear structural response of geo-materials. We have recently developed a new computational capability for coupling of multiphase, multi-component porous flow with nonlinear geo-mechanics. The geo-mechanics and porous flow models exist as separate, stand-alone codes, and are coupled via a solution control approach. Geo-mechanics is coupled to flow via the variation in the fluid pore pressures, whereas the flow problem is coupled to mechanics by the concomitant material strains which alter the pore volume (porosity field) and hence the permeability field. To facilitate coupling with disparate flow and mechanics time scales, the coupling strategy allows for different time steps in the flow solve compared to the mechanics solve. If time steps are synchronized, the controller allows intra-time-step iterations. The coupling is dynamically controlled by monitoring a norm measuring the degree of variation in the deformed porosity, thereby controlling the frequency of mechanics solves compared to flow solves. The implementation was verified by solving a subsidence problem discussed by Dean et al., (SPE-79709, 2003), as depicted in Figure 2 below.
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Figure 1: On the right is an illustration of CO2 injection into one realization of a heterogeneous aquifer, which contains an abandoned well. On the left is the distribution of leakage as a percentage of the injection rate for several realizations.
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Figure 2. Comparison of the time history of subsidence due to production in a porous layer, under two levels of geo-mechanics/flow coupling, with the solution of Dean et al. (SPE-79709, 2003). Also shown is the final porosity distribution and (exaggerated) subsidence on a cross section through the layer.
Discrete fracture generation
A new computational method for modeling fluid-induced discrete-fracture generation and growth has been developed. The method is based on modeling the solid continuum with polyhedral finite elements obtained from a randomly close-packed Voronoi tessellation (see Figure 1). Each cell of the Voronoi mesh is formulated as a conforming displacement-based finite element. New cohesive fracture surfaces are allowed to nucleate only at the inter-element faces of the Voronoi cells. New fracture surfaces are inserted by creating new nodes during the simulation. The RCP Voronoi mesh possesses a number of advantageous properties including large included angles within the cells (finite-element), and a space of statistically isotropic crack paths. The a priori crack paths of the RCP Voronoi mesh are viewed as instances of realizable random crack paths within a random field representation of the continuum material properties. Fluid flow within the fractures is modeled using a Reynold’s lubrication theory. An example simulation showing fluid-induced fracturing around a borehole is shown in Figure 2.
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Figure 1: Randomly close-packed Voronoi tessellation (finite element mesh).
Figure 2: A finite element simulation showing fluid-induced fracturing around a borehole. The maximum principal stress field is shown.
Parameter Estimation
The subsurface geological formations are subject to high uncertainty because of limited knowledge of rock properties. Therefore, consideration of the uncertainty calls for a stochastic description of the underground formations that necessitate using multiple realizations for uncertainty quantification. The prior uncertain geological models (model parameters) are obtained by integration of the static data from different sources such as well logs, core sample, and 3D seismic data. This prior uncertainty in model parameters is reduced by assimilation of dynamic (observed) data.
Our main focus has been on developing a general parallel framework for sequential data assimilation with implementation of ensemble Kalman filter (EnKF) algorithm. The effects of incorporation of dynamic data on calibration of model parameters (log-permeability) and saturation distribution for a synthetic case are studied (Figure). The assimilation of the observed data results in better model parameters and improves saturation distribution compared to the reference case. We will investigate these effects for more realistic cases such as Frio field.
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Figure: Horizontal log-permeability fields (left column) and CO2 saturation distribution after 1000 days (middle column) and after 2000 days (right column). The first row is the reference (true) case, the second row is from the prior model (our guess), and the third row shows the final results after assimilation of all dynamic data.
Convergence of Monte-Carlo Uncertainty Estimation
A statistical method has been developed for verifying the grid or mesh convergence of a sequence of statistical distributions generated by direct Monte Carlo sampling of stochastic partial differential equations. Example systems include those from fluid or solid mechanics, particularly those with instabilities and sensitive dependence on initial conditions or system parameters. The convergence assessment is based on demonstrating empirically that a sequence of cumulative distribution functions converges in the L(-norm. The effect of finite sample sizes is quantified using confidence levels from the Kolmogorov-Smirnov statistic. The statistical method is independent of the underlying distributions. The figure shows various statistical quantities used in the method: continuous cumulative distribution functions, their samples and resulting distances. This statistical method will help validate computational models of complex systems subjected to uncertain system parameters.
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Figure: Two cumulative distribution functions (black), the cumulative distribution functions of their samples (red), and their confidence bounds (blue) along with their L( distances (arrows).
Phase Behavior and Physical Properties
The Peng-Robinson cubic equation of state (EOS) is used for phase behavior of binary system of CO2 and water as a function of pressure and temperature followed by the flash algorithm to determine the mole fractions of CO2 and water in two equilibrium phases. The CO2 module of IPARSv-3 is non-isothermal compositional EOS and is coupled with biogeochemical reactions. Recent developments to improve the phase behavior and petrophysical properties are outlined as:
• An additional species in the aqueous phase is added to model the brine salinity expressed as total dissolved solids. The EOS variables of binary interaction coefficients and volume shift parameters are then modified according to the salt concentration and temperature using published correlations. These correlations proved to give more accurate CO2 solubility in brine and brine density.
• Published correlation for interfacial tension (IFT) between water and supercritical CO2 is implemented that accounts for effects of pressure, temperature, and brine salinity on IFT.
• We have incorporated a scaling group (trapping number) that combines the forces of gravity, viscous, and capillarity that controls the flow or trapping of both water and CO2 phases.
• Relative permeabilities and capillary pressures are adjusted as a function of trapping number because of the shift in trapped phase saturations.
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Figure: Measured water/CO2 relative permeability curves at different pressures that cause the shift in interfacial tension (data from Bennion and Bachu, 2006)
Solvers
The modeling of CO2 sequestration requires the coupling of compositional flow, thermal effects, geo-mechanics, and transport of multiple reactive chemical species at multiple temporal and spatial scales. Moreover, unstructured gridding is generally needed for treating geological heterogeneities. Thus the development of accurate, efficient and robust solvers coupled with adaptive time stepping for solving the large nonlinear dynamical systems that arise from finite element discretizations represents a formidable challenge.
For many years, researchers from Focus Area 4 have been investigating algorithms for linear and nonlinear solvers to address this "bottleneck" of subsurface flow computations. Some of these efforts include fast banded-solvers, Krylov subspace methods with a recycling strategy, two-stage preconditioning, algebraic multigrid, and specialized domain decomposition methods.
Collaboration with the SciDAC Towards Optimal Petascale Simulations Center at Sandia National Laboratories has started recently. The goal is to implement an interface between the Trilinos library developed at Sandia and the CO2 simulator used in Focus Area 4 allowing to explore new software tools aimed at maximizing the solver performance and parallel scalability on emerging architectures.
High Performance Computing
Modeling of the long-term movement of CO2 will require assimilation of huge datasets into simulators that incorporate complex physical and chemical processes such as coupled multiphase flow with chemical transport and geo-mechanics. Therefore, performing a large-scale simulation run will take significant amount of time. Also, we require multiple simulations to quantify the uncertainty in the model parameters and prediction performances. Thus, use of high performance computing is advantageous in reduction of time cycle for reservoir characterization.
In our modeling experiments, the main simulation tool is the Integrated Parallel Accurate Reservoir Simulator (IPARS). IPARS provides multi-scale and multi-physics capabilities to model multiphase and multi-component (compositional) flow in porous media coupled with geo-mechanics and reactive transport. It is easy to implement IPARS in a high-performance, parallel computational framework. We have developed a two-level parallel EnKF framework in which multiple realizations are spawned off in parallel on several partitions of a parallel machine (cluster) each of which are further sub-divided among different nodes (processors) and communication performed at data assimilation time, between the partitions before proceeding again to next assimilation step. Our results, depicted in the Figure, (Figure 2) show that the high-performance computing process (parallel EnKF) significantly reduces the computation time.
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Figure: Parallel speedup as a function of number of processors with 150 realization
The Senior Research Team
The University of Texas at Austin
1. Todd Arbogast, PhD, Professor, Department of Mathematics (ProjectFocus Area 4 co- co-leader)
2. Mojdeh Delshad, PhD, Research Associate Professor, Department of Petroleum & Geosystems Engineering
3. Susan Hovorka, PhD, Senior Research Scientist, Bureau of Economic Geology
4. Gergina Pencheva, PhD, Research Associate, Institute for Computational Engineering and Sciences
5. Mary F. Wheeler, PhD, Virginia Cockrell Chair in Engineering Professor, Department of Aerospace Engineering & Engineering Mechanics and Department of Petroleum & Geosystems Engineering
Sandia National Laboratory
1. Joseph E. Bishop, PhD, Principal Member of the Technical Staff, Engineering Sciences Center (Project Focus Area 4 co-leader)
2.
3. Mario J. Martinez, PhD, Principal Member of the Technical Staff, Engineering Sciences Center
4. C. Michael Stone, PhD (Retired)
Post-doctoral Associates
1. Ben Ganis, PhD.
2. Mika Juntunen, PhD.
3. Reza Tavakoli, PhD.
4. Guangri (Gary) Xue, PhD.
Graduate Students
1. Nick Alger
2. Mohammad Reza Beygi
3. Horacio Florez
4. Omar al Hinai
5. Zak Kassas
6. Xianhui Kong
7. Zhen (Jane) Tao
8. Bin Wang
9. Hailong Xiao
10. Changli Yuan
Events
• Drs. Daniil Svyatskiy
,
Mary F. Wheeler
, Guangri Xue
, and
Ivan Yotov organized the mini-symposium “Multiscale Methods for Porous Media Applications” for the 2011 SIAM Conference on Mathematical & Computational Issues in the Geosciences, March 20-24, 2011.
• Drs. Anozie Ebigbo
,
Rainer Helmig
, Mary F. Wheeler
, and
Guangri Xue organized the mini-symposium “Large Scale Simulations and Porous Media Applications” for the 2011 SIAM Conference on Mathematical & Computational Issues in the Geosciences, March 20-24, 2011.
• Drs. Mary F. Wheeler, Todd Arbogast, and Mojdeh Delshad hosted the Annual Center for Subsurface Modeling (CSM) Industrial Affiliates Meeting, on October 26-27 , at the University of Texas at Austin. It was attended by the Industrial Affiliates: Chevron, ConocoPhilips, Saudi Aramco, IBM, and British Petroleum.
• Drs. Mary F. Wheeler, Todd Arbogast, Mojdeh Delshad, and Ian Duncan hosted the workshop: “The Role of Computation in Protecting the Environment: A Workshop on Carbon Sequestration Simulation for High School Mathematics and Science Teachers,” June 15-16, 2010, University of Texas at Austin.
News
• Professor Mary F. Wheeler’s research was featured on the University of Texas at Austin Cockerel School of Engineering web site in the article “Engineer Reduces Big Problems To Manageable Numbers.”
• Professor Mary F. Wheeler was elected a fellow of the American Academy of Arts and Sciences in 2010.
-----------------------
From I. Duncan
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Figure 1: An illustration of the Frio Pilot CO2 injection test site. The grid on the right is of very poor quality for numerical simulation.
[pic]Illustration: On the left is the original Frio data from the Frio field test site. It has with 100,000 elements. In the middle is a straightforward but poor quality computational mesh with 26x26x21 elements. On the right is an improved mesh with 18x24x3 elements. The mesh on the right is was created with the GridPro. mesh generator.
Illustration 21: Movie simulation of a CO2 leak through an abandoned well. Deep saline aquifers are considered to be one of the most important formations because of their worldwide availability and generally low economic value. It is attractive to use the existing infrastructure of oil and gas wells to inject the CO2 into a formation. However, formations that have been explored for oil and gas production are typically perforated by a large number of wells, a fact that increases the risk of leakage through such wells. A simple leakage scenario involving one CO2 injection well, one leaky well, two aquifers and an aquitard are shown. The leaky well connects the two aquifers. As CO2 is injected underground in to the lower reservoir, it rises due to buoyancy. When it finds the old abandoned wellbore, it rises through it, contaminating the upper reservoir. Simulation helps scientists and engineers visualize and study such undesirable scenarios, so that they can devise prevention or response strategies.Simulation of a CO2 leak through an abandoned well. As CO2 is injected underground in to the lower reservoiraquifer, it rises due to buoyancy. When it finds the old abandoned leaky wellbore, it rises through it, contaminating the upper reservoiraquifer. Simulation helps scientists and engineers visualize and study such undesirable scenarios, so that they can devise prevention or response strategies.
Illustration 32: Movie Finite element simulation showing fluid-induced dynamic fracturing around a borehole. Such fracturing affects the flow of fluids and well-bore stability. Fracturing of the cap rock could create pathways for CO2 leakage.
[pic]IllustrationFigure 2: On the left is the original Frio data from the Frio Pilot Test field site. It has with 100,000 elements. In the middle is a straightforward but poor quality computational mesh with 26x26x21 elements. On the right is an improved mesh with 18x24x3 elements. The mesh on the right is created with the GridPro. mesh generator.
Leakage Curves
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