Compositional Flow Simulation Research Articles

A parallel framework for a multipoint flux mixed finite element equation of state compositional flow simulator

Mathematical models of physical problems are becoming increasingly complex and computationally intensive. At the same time, computing hardware is becoming more parallelized with an increasing number of cores promoting simultaneous tasks. In this work, we present a parallel, equation of state (EOS), compositional flow simulator for evaluating CO2 sequestration, enhanced oil recovery techniques such as gas flooding, and other subsurface porous media applications. Using the multipoint flux mixed finite element (MFMFE) method for spatial discretization, it can handle complex reservoir geometries using general distorted hexahedral grid elements, as well as satisfy local mass conservation and compute accurate phase fluxes. A parallel framework for the MFMFE is presented that has been extended to the highly non-linear, EOS, compositional flow model. Much of the non-linearity is due to the local flash and stability calculations associated with interphase mass transfer and phase behavior. Parallel multigrid linear solver libraries such as HYPRE are utilized to solve the algebraic problems on each Newton step. We perform a variety of strong and weak parallel scaling studies up to 10 million elements and 1024 processors, and discuss possible load balancing issues.

Non-equilibrium phase behavior of hydrocarbons in compositional simulations and upscaling

Numerical models widely used for hydrocarbon phase behavior and compositional flow simulations are based on assumption of thermodynamic equilibrium. However, it is not uncommon for oil and gas-condensate reservoirs to exhibit essentially non-equilibrium phase behavior, e.g., in the processes of secondary recovery after pressure depletion below saturation pressure, or during gas injection, or for condensate evaporation at low pressures. In many cases, the ability to match field data with equilibrium model depends on simulation scale. The only method to account for non-equilibrium phase behavior adopted by the majority of flow simulators is the option of limited rate of gas dissolution (condensate evaporation) in black oil models. For compositional simulations, no practical yet thermodynamically consistent method has been presented so far except for some upscaling techniques in gas injection problems. Previously reported academic non-equilibrium formulations have a common drawback of doubling the number of flow equations and unknowns compared with the equilibrium formulation. In the paper, a unified thermodynamically consistent formulation for compositional flow simulations with non-equilibrium phase behavior model is presented. Same formulation and a special scale-up technique can be used for upscaling of an equilibrium or non-equilibrium model to a coarse-scale non-equilibrium model. A number of test cases for real oil and gas-condensate mixtures are given. Model implementation specifics in a flow simulator are discussed and illustrated with test simulations. A non-equilibrium constant volume depletion algorithm is presented to simulate condensate recovery at low pressures in gas-condensate reservoirs. Results of satisfactory model matching to field data are reported and discussed.

Impact of eliminating fracture intersection nodes in multiphase compositional flow simulation

AbstractAlgebraic elimination of nodes at discrete fracture intersections via the star‐delta technique has proven to be a valuable tool for making multiphase numerical simulations more tractable and efficient. This study examines the assumptions of the star‐delta technique and exposes its effects in a 3‐D, multiphase context for advective and dispersive/diffusive fluxes. Key issues of relative permeability‐saturation‐capillary pressure (kr‐S‐Pc) and capillary barriers at fracture‐fracture intersections are discussed. This study uses a multiphase compositional, finite difference numerical model in discrete fracture network (DFN) and discrete fracture‐matrix (DFM) modes. It verifies that the numerical model replicates analytical solutions and performs adequately in convergence exercises (conservative and decaying tracer, one and two‐phase flow, DFM and DFN domains). The study culminates in simulations of a two‐phase laboratory experiment in which a fluid invades a simple fracture intersection. The experiment and simulations evoke different invading fluid flow paths by varying fracture apertures as oil invades water‐filled fractures and as water invades air‐filled fractures. Results indicate that the node elimination technique as implemented in numerical model correctly reproduces the long‐term flow path of the invading fluid, but that short‐term temporal effects of the capillary traps and barriers arising from the intersection node are lost.

Modified sequential fully implicit scheme for compositional flow simulation

The fully implicit (FI) method is widely used for numerical modeling of multiphase flow and transport in porous media. The FI method is unconditionally stable, but that comes at the cost of a low-order approximation and high computational cost. The FI method entails iterative linearization and solution of fully-coupled linear systems with mixed elliptic/hyperbolic character. However, in methods that treat the near-elliptic (flow) and hyperbolic (transport) separately, such as multiscale formulations, sequential solution strategies are used to couple the flow (pressures and velocities) and the transport (saturations/compositions). The most common sequential schemes are: the implicit pressure explicit saturation (IMPES), and the sequential fully implicit (SFI) schemes. Problems of practical interest often involve tightly coupled nonlinear interactions between the multiphase flow and the multi-component transport. For such problems, the IMPES approach usually suffers from prohibitively small timesteps in order to obtain stable numerical solutions. The SFI method, on the other hand, does not suffer from a temporal stability limit, but the convergence rate can be extremely slow. This slow convergence rate of SFI can offset the gains obtained from separate and specialized treatments of the flow and transport problems. In this paper, we analyze the nonlinear coupling between flow and transport for compressible, compositional systems with complex interphase mass transfer. We isolate the nonlinear effects related to transmissibility and compressibility from those due to interphase mass transfer, and we propose a modified SFI (m-SFI) method. The new scheme involves enriching the ‘standard’ pressure equation with coupling between the pressure and the saturations/compositions. The modification resolves the convergence problems associated with SFI and provides a strong basis for using sequential formulations for general-purpose simulation. For a wide parameter range, we show that the proposed m-SFI algorithm is as robust as the FI method with a comparable convergence rate.

Upscaling for Compositional Reservoir Simulation

Summary Compositional flow simulation, which is required for modeling enhanced-oil-recovery (EOR) operations, can be very expensive computationally, particularly when the geological model is highly resolved. It is therefore difficult to apply computational procedures that require large numbers of flow simulations, such as optimization, for EOR processes. In this paper, we develop an accurate and robust upscaling procedure for oil/gas compositional flow simulation. The method requires a global fine-scale compositional simulation, from which we compute the required upscaled parameters and functions associated with each coarse-scale interface or wellblock. These include coarse-scale transmissibilities, upscaled relative permeability functions, and so-called α-factors, which act to capture component flow rates in the oil and gas phases. Specialized near-well treatments for both injection and production wells are introduced. An iterative procedure for optimizing the α-factors is incorporated to further improve coarse-model accuracy. The upscaling methodology is applied to two example cases, a 2D model with eight components and a 3D model with four components, with flow in both cases driven by wells arranged in a five-spot pattern. Numerical results demonstrate that the global compositional upscaling procedure consistently provides very accurate coarse results for both phase and component production rates, at both the field and well level. The robustness of the compositionally upscaled models is assessed by simulating cases with time-varying well bottomhole pressures that are significantly different from those used when the coarse model was constructed. The coarse models are shown to provide accurate predictions in these tests, indicating that the upscaled model is robust with respect to well settings. This suggests that one can use upscaled models generated from our procedure to mitigate computational demands in important applications such as well-control optimization.

High-Performance Modeling of Carbon Dioxide Sequestration by Coupling Reservoir Simulation and Molecular Dynamics

Summary The present work describes a parallel computational framework for carbon dioxide (CO2) sequestration simulation by coupling reservoir simulation and molecular dynamics (MD) on massively parallel high-performance-computing (HPC) systems. In this framework, a parallel reservoir simulator, reservoir-simulation toolbox (RST), solves the flow and transport equations that describe the subsurface flow behavior, whereas the MD simulations are performed to provide the required physical parameters. Technologies from several different fields are used to make this novel coupled system work efficiently. One of the major applications of the framework is the modeling of large-scale CO2 sequestration for long-term storage in subsurface geological formations, such as depleted oil and gas reservoirs and deep saline aquifers, which has been proposed as one of the few attractive and practical solutions to reduce CO2 emissions and address the global-warming threat. Fine grids and accurate prediction of the properties of fluid mixtures under geological conditions are essential for accurate simulations. In this work, CO2 sequestration is presented as a first example for coupling reservoir simulation and MD, although the framework can be extended naturally to the full multiphase multicomponent compositional flow simulation to handle more complicated physical processes in the future. Accuracy and scalability analysis are performed on an IBM BlueGene/P and on an IBM BlueGene/Q, the latest IBM supercomputer. Results show good accuracy of our MD simulations compared with published data, and good scalability is observed with the massively parallel HPC systems. The performance and capacity of the proposed framework are well-demonstrated with several experiments with hundreds of millions to one billion cells. To the best of our knowledge, the present work represents the first attempt to couple reservoir simulation and molecular simulation for large-scale modeling. Because of the complexity of subsurface systems, fluid thermodynamic properties over a broad range of temperature, pressure, and composition under different geological conditions are required, although the experimental results are limited. Although equations of state can reproduce the existing experimental data within certain ranges of conditions, their extrapolation out of the experimental data range is still limited. The present framework will definitely provide better flexibility and predictability compared with conventional methods.

Local–Global Upscaling for Compositional Subsurface Flow Simulation

Compositional flow simulation is required for the modeling of many important subsurface processes including enhanced oil recovery and geological carbon storage. These simulations can be very demanding computationally, which motivates the need for efficient alternatives to detailed fine-scale flow modeling. In this work, a local–global upscaling procedure for oil–gas compositional problems is developed. With this approach, the key upscaling computations are performed over local (rather than global) fine-scale regions. The pressure boundary conditions required for these computations are determined through an approximate global coarse-scale simulation. Gas saturation and component molar fraction boundary conditions are prescribed based on the injected fluid conditions. An ‘attribute-based’ approach, which enables the efficient assignment of coarse-scale relative permeabilities and compositional functions based on quickly computed (static and single-phase flow) features, is introduced. Numerical results are presented for two- and three-dimensional models (containing eight and four components, respectively) with flow driven by wells. These results demonstrate that the local–global upscaling method, with or without the attribute-based property assignment, provides significantly more accurate predictions than standard (single-phase-parameter) upscaling methods, though accuracy is not as high as that achieved using more expensive global compositional upscaling. Speedups of about an order of magnitude, relative to fine-scale simulations, are observed using the attribute-based local–global procedure. The robustness of the upscaled models to variation in well bottom-hole pressures is also assessed. These computations demonstrate that the upscaled models retain accuracy over a range of well settings, which suggests that they will be useful in computationally demanding applications such as well control optimization.

Ensemble level upscaling for compositional flow simulation

Uncertainty quantification is typically accomplished by simulating multiple geological realizations, which can be very expensive computationally if the flow process is complicated and the models are highly resolved. Upscaling procedures can be applied to reduce computational demands, though it is essential that the resulting coarse-model predictions correspond to reference fine-scale solutions. In this work, we develop an ensemble level upscaling (EnLU) procedure for compositional systems, which enables the efficient generation of multiple coarse models for use in uncertainty quantification. We apply a newly developed global compositional upscaling method to provide coarse-scale parameters and functions for selected realizations. This global upscaling entails transmissibility and relative permeability upscaling, along with the computation of a-factors to capture component fluxes. Additional features include near-well upscaling for all coarse parameters and functions, and iteration on the a-factors, which is shown to improve accuracy. In the EnLU framework, this global upscaling is applied for only a few selected realizations. For 90 % or more of the realizations, upscaled functions are assigned statistically based on quickly computed flow and permeability attributes. A sequential Gaussian co-simulation procedure is incorporated to provide coarse models that honor the spatial correlation structure of the upscaled properties. The resulting EnLU procedure is applied for multiple realizations of two-dimensional models, for both Gaussian and channelized permeability fields. Results demonstrate that EnLU provides P10, P50, and P90 results for phase and component production rates that are in close agreement with reference fine-scale results. Less accuracy is observed in realization-by-realization comparisons, though the models are still much more accurate than those generated using standard coarsening procedures.

Modeling of capacitance flow behavior in EOS compositional simulation

Abstract Gas injection is a widely used method for enhanced oil recovery. Oil bypassing by gas occurs at different scales because of micro and macroscopic heterogeneities, gravity segregation, and front instability. Part of the bypassed oil can be recovered by crossflow between the bypassed and flowing regions. This characteristic of reservoir flow is referred to as capacitance flow behavior in the literature. Modeling of such flow behavior at the sub-grid scale is challenging in the conventional flow simulation since fluids are perfectly mixed and in equilibrium within individual grid blocks under the local equilibrium assumption. This research investigates capacitance flow behavior in compositional reservoir simulation. An efficient two-step method is presented to model bypassed oil recovery in multiphase compositional flow simulation of gas floods. The oil bypassing is first quantified by use of the dual-porosity flow with two dimensionless groups; bypassed fraction and throughput ratio. To represent bypassed oil recovery in single-porosity flow, a new flow-based fluid characterization is applied to part of the heavy fractions of the fluid model used. Properties for pseudo components can be determined based on the throughput ratio estimated in the dual-porosity flow. Case studies for various reservoir/fluid properties show that single-porosity flow with the new method reasonably represents bypassed oil recovery observed in core floods and fine-scale heterogeneous simulations.

Speeding up the flash calculations in two-phase compositional flow simulations – The application of sparse grids

Flash calculations have become a performance bottleneck in the simulation of compositional flow in subsurface reservoirs. We apply a sparse grid surrogate model to substitute the flash calculation and thus try to remove the bottleneck from the reservoir simulation. So instead of doing a flash calculation in each time step of the simulation, we just generate a sparse grid approximation of all possible results of the flash calculation before the reservoir simulation. Then we evaluate the constructed surrogate model to approximate the values of the flash calculation results from this surrogate during the simulations. The execution of the true flash calculation has been shifted from the online phase during the simulation to the offline phase before the simulation. Sparse grids are known to require only few unknowns in order to obtain good approximation qualities. In conjunction with local adaptivity, sparse grids ensure that the accuracy of the surrogate is acceptable while keeping the memory usage small by only storing a minimal amount of values for the surrogate. The accuracy of the sparse grid surrogate during the reservoir simulation is compared to the accuracy of using a surrogate based on regular Cartesian grids and the original flash calculation. The surrogate model improves the speed of the flash calculations and the simulation of the whole reservoir. In an experiment, it is shown that the speed of the online flash calculations is increased by about 2000 times and as a result the speed of the reservoir simulations has been enhanced by 21 times in the best conditions.

Negative-saturation approach for (non)-isothermal compositional three-phase flow simulations in porous media

This article describes the extension of the two-phase (non)-isothermal negative-saturation solution approach to the three-phase (non)-isothermal negative-saturation (NegSat3) solution approach. The NegSat3 solution approach solves efficiently any (non)-isothermal compositional flow problem in porous media that involves phase transitions between different phase states when the maximum number of phases is less than four. The solution approach circumvents using different equations and primary variables for single-phase, two-phase, and three-phase regions in porous media. Consequently, the NegSat3 solution approach avoids switches or variable substitutions at phase transitions where the convergence of the Newton-Raphson procedure is hampered by oscillations between m-phase and n-phase states. The NegSat3 solution approach can be implemented efficiently in numerical simulators to deal with modeling issues for thermal recovery processes, CO2 sequestration, and for multicontact miscible gas injection in oil reservoirs if the number of phases is less than four. We illustrate the NegSat3 solution approach by way of example to steam injection in a 1D heavy-oil reservoir. However, the NegSat3 solution approach can be used for any n ≤ 3-dimensional, multicomponent three-phase problems. The example solution is compared with a standard numerical solution that is analytically verified by the method of characteristics and shows excellent agreement. The results show that the oil recovery depends critically on whether the boiling temperature of the volatile oil is around the water boiling temperature, or much below or above it. These boiling-temperature ranges give rise to three different types of wave structures. When the boiling temperature of the volatile oil is near the boiling temperature of water, the striking result is that the speed of the evaporation front is equal or somewhat larger than the speed of the steam-condensation front. Thus, the volatile oil condenses at the location where the steam condenses too, yielding virtually complete oil recovery. Conversely, if the boiling temperature is too high or too low, there is incomplete recovery.

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

An adjoint formulation for the gradient-based optimization of oil–gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford’s Automatic Differentiation-based General Purpose Research Simulator, AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil–water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented—a formal treatment within the optimizer and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. Results for four example cases of increasing complexity are presented. Improvements in the objective function (cumulative oil produced) relative to reference solutions range from 4.2 to 11.6 %. The heuristic treatment of nonlinear constraints is shown to offer a cost-effective means for obtaining feasible solutions, which are, in some cases, better than those obtained using the formal constraint handling procedure.

Thermal effects in a depleted gas field by cold CO2 injection in the presence of methane

Depleted gas fields are seen as promising options for geological storage of CO2. The advantage of hydrocarbon fields are that the characteristics, such as the storage capacity and the proven sealing capacity are known. This means that only limited uncertainty remains after a technical feasibility study. One important issue with depleted gas reservoirs is the low pressure, at the onset of injection. This may lead to adiabatic cooling (Joule-Thomson cooling or JTC). The temperature reduction associated with this JTC can lead to hydrate formation and freezing of the residual pore water, especially the injected fluid also has a low temperature. In a related paper, we report on the development of a simulator, which is capable of predicting the fate of transport of a CO2 in either the liquid gaseous and supercritical phase. The model only contains CO2 besides the aqueous phase. One of the remaining challenges is to predict the impact of the methane, which is still present in a depleted gas reservoir, on the phase behavior of CO2 and the thermal aspects. In this study simulations were performed on the P18-4 depleted gas field. The CO2 was injected at a minimum temperature of 12 °C into the reservoir formation with a temperature of 120 °C. The average injection rate during the demonstration phase was modelled with 1.1 Mton/yr over a period of 5 year. The injection of cold CO2 into P18-4 was modelled in TOUGH2 and a new module, which can handle not only the behaviour of CO2 but also of CH4. The original ECO2M module was initially designed for CO2 behaviour in gas reservoir and aquifers for brine-CO2 mixtures, including all possible phase transitions, in the absence of any other gas. The new TOUGH2 module, which we gave the name ECO2MG module, was extended in such a way that compositional flow simulations of two different gases can be modelled, namely CO2 and CH4. The ECOMG module can not only model the behaviour brine CO2 mixtures but also the behaviour of brine-CO2-CH4 mixtures including phase changes from gas to liquid and from super- And sub-critical conditions. Density, viscosity and enthalpy of the CO2-CH4 mixture is based on NIST data. The new TOUGH2 module was used to model the thermal aspects of CO2 injection including the presence of brine and CH4 initially in the reservoir. The reservoir model was initialized with a brine-methane mixture at 20 bar, which is the expected abandonment pressure of the reservoir. The prediction of thermal effects of this new tool were compared with those of the earlier version (CO2 only) for several times during the projected 5 year injection period. The thermal effects investigated in this study were the JTC and also the effect of evaporation of the aqueous phase. The heat of evaporation of the brine is very significant and cools down the near well area significantly depending on the initial water saturation and the injection temperature of the CO2 itself. In the new module we saw that after the CO2 injection was started at first instance an uniform front of CO2/CH4 was formed. After a few years gravity underride of the still gaseous CO2 becomes visible, which became stronger after the phase transition to liquid CO2. A free CH4 phase moved during the simulation in front of the CO2 plume.

Higher-order compositional modeling of three-phase flow in 3D fractured porous media based on cross-flow equilibrium

Numerical simulation of multiphase compositional flow in fractured porous media, when all the species can transfer between the phases, is a real challenge. Despite the broad applications in hydrocarbon reservoir engineering and hydrology, a compositional numerical simulator for three-phase flow in fractured media has not appeared in the literature, to the best of our knowledge. In this work, we present a three-phase fully compositional simulator for fractured media, based on higher-order finite element methods. To achieve computational efficiency, we invoke the cross-flow equilibrium (CFE) concept between discrete fractures and a small neighborhood in the matrix blocks. We adopt the mixed hybrid finite element (MHFE) method to approximate convective Darcy fluxes and the pressure equation. This approach is the most natural choice for flow in fractured media. The mass balance equations are discretized by the discontinuous Galerkin (DG) method, which is perhaps the most efficient approach to capture physical discontinuities in phase properties at the matrix-fracture interfaces and at phase boundaries. In this work, we account for gravity and Fickian diffusion. The modeling of capillary effects is discussed in a separate paper. We present the mathematical framework, using the implicit-pressure-explicit-composition (IMPEC) scheme, which facilitates rigorous thermodynamic stability analyses and the computation of phase behavior effects to account for transfer of species between the phases. A deceptively simple CFL condition is implemented to improve numerical stability and accuracy. We provide six numerical examples at both small and larger scales and in two and three dimensions, to demonstrate powerful features of the formulation.

Nonlinear Formulation Based on an Equation-of-State Free Method for Compositional Flow Simulation

Summary Compositional simulation is necessary for modeling complex enhanced oil recovery (EOR) processes. For accurate simulation of compositional processes, we need to resolve the coupling of the nonlinear conservation laws, which describe multiphase flow and transport, with the equilibrium phase behavior constraints. The complexity of the problem requires extensive computations and consumes significant time. This paper presents a new framework for the general compositional problem associated with multicomponent multiphase flow in porous media. Here, adaptive construction and interpolation using the supporting tie lines are used to obtain the phase state and the phase compositions. For the parameterization of the full solution of a complex compositional problem, we need only a limited number of supporting tie lines in the compositional space. The parameterized tie lines are triangulated using Delaunay tessellation, and natural-neighbor interpolation is used inside the simplexes. Then, the computation of the phase behavior in the course of a simulation becomes an iteration-free, table look-up procedure. The treatment of nonlinearities associated with complex thermodynamic behavior of the fluid is based on the new set of unknowns—tie-line parameters that allow for efficient representation of the subcritical region. For the supercritical region, we use the standard compositional variable set based on the overall composition. The efficiency and accuracy of the method are demonstrated for several multidimensional compositional problems of practical interest. In terms of the computational cost of the thermodynamic calculations, the proposed method shows results comparable to those of state-of-the-art techniques. Moreover, the method shows better nonlinear convergence in the case of near-miscible gas-injection simulation.

Enhanced-Velocity-Multiblock Method for Coupled Flow and Reactive-Species Transport Through Porous Media: Applications to Bioremediation and Carbon Sequestration

Summary This paper presents a multiblock-discretization method—the enhanced-velocity mixed-finite-element method (EVMFEM) (Wheeler et al. 2002)—for coupled multiphase flow and reactive-species-transport modeling in porous-media applications. The method provides local mass balance and a continuous approximation of fluxes across interfaces of elements and subdomains. It can treat nonmatching grids, allowing for a flexible choice of grid refinements. Further, by distributing the blocks among processors such that each block has approximately the same number of elements, this method can be implemented efficiently in parallel, thereby offering further reductions in computational cost. The paper also presents recent application of EVMFEM to challenging problems such as compositional flow simulations of CO2 sequestration. Tests with EVMFEM suggest that it is advantageous to apply grid refinements around wells and to areas in which dynamics of chemical-species concentration is highest. Allowing for variable grid refinements greatly reduces the simulation cost, while preserving overall accuracy of the solution. For completeness, a few significant analytic results on convergence of the method are stated and referenced, omitting proof. This work is significant in advancing the discretization and application of EVMFEMs in reservoir-simulation development. Problems such as transport of chemical species in multiphase flow and CO2 sequestration have begun to assume significant importance in decisions regarding the preservation of our environment and in the safe and reliable means of delivering energy. This paper offers useful methods and some innovative future directions to address the huge computational costs involved in solving such complex problems.

Negative-Saturation Approach for Compositional Flow Simulations of Mixed CO2/Water Injection Into Geothermal Reservoirs Including Phase Appearance and Disappearance

Summary Cold mixed CO2/water injection into hot-water reservoirs can be used for simultaneous geothermal-energy (heat) production and subsurface CO2 storage. This paper studies this process in a 2D geothermal homogeneous reservoir, a layered reservoir, and a heterogeneous reservoir represented by a stochastic-random field. We give a set of simulations for a variety of CO2/water-injection ratios. In this process, often regions of two-phase flow are connected to regions of single-phase flow. Different systems of equations apply for single-phase and two-phase regions. We develop a solution approach, called the nonisothermal-negative-saturation (NegSat) solution approach, to solve efficiently nonisothermal compositional flow problems (e.g., CO2/water injection into geothermal reservoirs) that involve phase appearance, phase disappearance, and phase transitions. The advantage of this solution approach is that it circumvents using different equations for single-phase and two-phase regions and the ensuing unstable switching procedure. In the NegSat approach, a single-phase multicomponent fluid is replaced by an equivalent fictitious two-phase fluid with specific properties. The equivalent properties are such that the extended saturation of a fictitious gas is negative in the single-phase aqueous region. We discuss the salient features of the simulations in detail. When two phases are present at the injection side, heterogeneity and layering lead to more CO2 storage compared with the homogeneous case because of capillary trapping. In addition, layering avoids movement of the CO2 to the upper part of the reservoir and thus reduces the risk of leakage. Our results also show that heterogeneity and layering change the character of the solution in terms of useful-energy production and CO2 storage. The simulations can be used to construct a plot of the recovered useful energy vs. maximally stored CO2. Increasing the amount of CO2 in the injection mixture leads to bifurcation points at which the character of the solution in terms of energy production and CO2 storage changes. For overall injected-CO2 mole fractions less than 0.04, the result with gravity is the same as the result without gravity. For larger overall injected-CO2 mole fractions, however, the plot without gravity differs from the plot with gravity because of early breakthrough of a supercritical-CO2 tongue near the caprock. The plot of the useful energy (exergy) vs. the CO2-storage capacity in the presence of gravity shows a Z-shape. The top horizontal part represents a branch of high exergy recovery and a relatively lower storage capacity, whereas the bottom part represents a branch of lower exergy recovery and a higher storage capacity.

Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part II: numerical scheme and numerical results

We consider the modeling and simulation of compositional two-phase flow in a porous medium, where one phase is allowed to vanish or appear. The modeling of Marchand et al. (in review) leads to a nonlinear system of two conservation equations. Each conservation equation contains several nonlinear diffusion terms, which in general cannot be written as a function of the gradients of the two principal unknowns. Also the diffusion coefficients are not necessarily explicit local functions of them. For the generalised mixed finite elements approximation, Lagrange multipliers associated to each principal unknown are introduced, the sum of the diffusive fluxes of each component is explicitly eliminated and the static condensation leads to a “global” nonlinear system of equations only in the Lagrange multipliers also including complementarity conditions to cope with vanishing or appearing phases. After time discretisation, this system can be solved at each time step using a semi-smooth Newton method. The static condensation involves “local” nonlinear systems of equations associated to each element, solved also by a semismooth Newton method. The algorithm is successfully applied to 1D and 2D examples of water–hydrogen flow involving gas phase appearance and disappearance.

Open thermodynamic model for compressible multicomponent two-phase flow in porous media

The compressible compositional two-phase flow in porous media was considered. This system was characterized by two essential parameters: the perturbation parameter and the relative mobility parameter. The perturbation parameter represents the stabilization time of the pressure field after perturbation. The relative mobility parameter represents the interior characteristic of the system by the ratio between liquid mobility and gas mobility. For the contrast phase mobility and fast pressure relaxation process an open thermodynamic model was obtained by HT-splitting. This model includes several differential thermodynamic equations for characterization of the transport phenomena (Delta Law), i.e. describes the equilibrium in an open system. A new open thermodynamic simulator (OTS) was developed. The OTS simulations are validated by the full compositional flow simulations using Eclipse. However, compared to the full compositional simulator an evident advantage of the new OTS simulator is the fact that due to the independence from space and time, data construction demands an extremely short time of calculation. Moreover this new approach does not limit the number of chemical components in the system.The closed thermodynamic system was shown as behaving similarly to the open system for several test cases. In order to characterize the difference between the closed and the open thermodynamics we proposed to use Delta Law for one chemical component and a variation function for the combination of all components. This behavior depends on mass conservation in an individual volume. The closed description can be realistic if the incoming gas in an individual volume is not different from the leaving gas; otherwise the use of the open approach (new or classical) is suggested.

Negative Saturation Approach for Non-Isothermal Compositional Two-Phase Flow Simulations

This article deals with developing a solution approach, called the non-isothermal negative saturation (NegSat) solution approach. The NegSat solution approach solves efficiently any non-isothermal compositional flow problem that involves phase disappearance, phase appearance, and phase transition. The advantage of the solution approach is that it circumvents using different equations for single-phase and two-phase regions and the ensuing unstable procedure. This paper shows that the NegSat solution approach can also be used for non-isothermal systems. The NegSat solution approach can be implemented efficiently in numerical simulators to tackle modeling issues for mixed CO2–water injection in geothermal reservoirs, thermal recovery processes, and for multicontact miscible and immiscible gas injection in oil reservoirs. We illustrate the approach by way of example to cold mixed CO2–water injection in a 1D geothermal reservoir. The solution is compared with an analytical solution obtained with the wave-curve method (method of characteristics) and shows excellent agreement. A complete set of simulations is carried out, which identifies six bifurcations. The two main bifurcations are (1) when the most downstream compositional wave is replaced by a compositional shock and (2) when an extra Buckley–Leverett rarefaction appears. The plot of the useful energy (exergy) versus the CO2 storage capacity shows a Z-shape. The top horizontal part represents a branch of high exergy recovery/relatively lower storage capacity, whereas the bottom part represents a branch of lower exergy recovery/higher storage capacity.