Moraes, R. (Department of Geoscience and Engineering, TU Delft) | Rodrigues, J. R. P. (Petrobras Research and Development Center) | Hajibeygi, H. (Department of Geoscience and Engineering, TU Delft) | Jansen, J. D. (Department of Geoscience and Engineering, TU Delft)
A multiscale gradient computation method for multiphase flow in heterogeneous porous media is developed. The method constructs multiscale primal and dual coarse grids, imposed on the given fine-scale computational grid. Local multiscale basis functions are computed on (dual-) coarse blocks, constructing an accurate map (prolongation operator) between coarse- and fine-scale systems. While the expensive operations involved in computing the gradients are performed at the coarse scale, sensitivities with respect to uncertain parameters (e.g., grid block permeabilities) are expressed in the fine scale via the partial derivatives of the prolongation operator. Hence, the method allows for updating of the geological model, rather than the dynamic model only, avoiding upscaling and the inevitable loss of information. The formulation and implementation are based on automatic differentiation (AD), allowing for convenient extensions to complex physics. An IMPES coupling strategy for flow and transport is followed, in the forward simulation. The flow equation is computed using a multiscale finite volume (MSFV) formulation and the transport equation is computed at the fine scale, after reconstruction of mass conservative velocity field. To assess the performance of the method, a synthetic multiphase flow test case is considered. The multiscale gradients are compared against those obtained from a fine-scale reference strategy. Apart from its computational efficiency, the benefits of the method include flexibility to accommodate variables expressed at different scales, specially in multiscale data assimilation and reservoir management studies.
When simulating foam floods, uncertainties exist in both the foam and reservoir parameters however the combination of these uncertainties are rarely incorporated in forecasting. Foam flooding is an effective enhanced oil recovery method that controls mobility, reduces gas relative permeability, delays gas breakthrough and helps improve sweep efficiency. Thus it is often used in highly heterogeneous reservoirs where significant subsurface uncertainties exist. Foam uncertainties exist as (a) foam stability is controlled by a number of factors such as critical water and surfactant concentrations, brine salinity, and oil saturation which are unknown in the subsurface spatially and (b) foam flood simulation requires the accurate description of multiple parameters used in the foam flood models which are unknown.
This study quantifies and compares the impact of uncertainties associated with foam model parameters with the heterogeneity of a fractured carbonate reservoir, an analogue to the highly prolific Arab D formation. Foam model parameters are not known a-priori but can be tuned to experimental data, which ideally represent a range of foam regimes and reservoir conditions. Geological heterogeneities in fractured carbonate reservoirs are complex and include, matrix wettability, fracture density/orientation and initial saturation distribution.
To quantify uncertainties geological uncertainties in fractured carbonate reservoirs, an automated framework was used to history match the production response of a fractured carbonate field by varying geological parameters. This accounts for the geological uncertainties during a waterflood, which are then combined with foam uncertainties from experimental analysis in the optimisation step, by optimising the mean response of the model to foam flooding across a range of geological and foam scenarios. Our workflow used a combination of Particle Swarm Optimisation for history matching and manual optimisation, the final results of which show a wide range of possible impacts of foam flooding from different but equally well matched reservoirs.
The novelty of our work is that it demonstrates how parameters that control foam stability and hence effectiveness in mobility control are related to both foam properties and geological uncertainty. Carrying these uncertainties into foam model properties from core to field scale will translate into considerably more robust estimates of uncertainty when predicting field-scale recovery compared to simulations that only consider uncertainty in the reservoir model.
A comprehensive methodology for gridding, discretizing, coarsening, and simulating discrete-fracture-matrix models of naturally fractured reservoirs is described and applied. The model representation considered here can be used to define the grid and transmissibilities, at either the original fine scale or at coarser scales, for any connectivity-list-based finite-volume flow simulator. For our fine-scale mesh, we use a polyhedral gridding technique to construct a conforming matrix grid with adaptive refinement near fractures, which are represented as faces of grid cells. The algorithm uses a single input parameter to obtain a suitable compromise between fine-grid cell quality and the fidelity of the fracture representation. Discretization using a two-point flux approximation is accomplished with an existing procedure that treats fractures as lower-dimensional entities (i.e., resolution in the transverse direction is not required). The upscaling method is an aggregation-based technique in which coarse control volumes are aggregates of fine-scale cells, and coarse transmissibilities are computed using a general flow-based procedure. Numerical results are presented for waterflood, sour gas injection, and gas condensate primary production. Coarse-model accuracy is shown to generally decrease with increasing levels of coarsening, as would be expected. We demonstrate, however, that by using our methodology, two orders of magnitude of speedup can be achieved with models that introduce less than about 10% error (with error appropriately defined). This suggests that the overall framework may be very useful for the simulation of realistic discrete-fracture-matrix models.
White, Deandra (The University of Texas at Austin) | Ganis, Benjamin (The University of Texas at Austin) | Liu, Ruijie (The University of Texas at San Antonio) | Wheeler, Mary F. (The University of Texas at Austin)
Permanent deformations in the solid matrix can be caused by many field scenarios, such as high injection rates. A pressure differential in the field can create geomechanical loading of large magnitude that may increase stress from an elastic regime to a plastic regime. Simple geomechanical models based on linear elasticity are insufficient in predicting these types of effects. To accurately predict rock formation damage and failure responses, nonlinear analyses based on geomaterial plasticity models should be included in modeling frameworks through rigorous coupling with reservoir flow simulators.
In this work we integrate an implementation of the Drucker-Prager plasticity model into the parallel compositional reservoir simulator, IPARS (Integrated Parallel Accurate Reservoir Simulator). Fluid flow is formulated on general distorted hexahedral grids using the multipoint flux mixed finite element method. The mechanics and flow systems are solved separately and coupled using a fixed-stress iterative coupling algorithm. This allows multiple flow models to be used with nonlinear mechanics without modification, and allows each type of physics to employ the best preconditioner for its linear systems. The fixed-stress iteration converges to the fully coupled solution on each time step.
With these components in place, we conduct a study on wellbore stability using different flow and geomaterial models. We demonstrate the capabilities of our integrated simulator in predicting near-wellbore plastic strain development and its effect on multiphase component concentrations. Our simulations run efficiently in parallel using MPI on high performance computing platforms up to hundreds or thousands of processors. The results of the simulations are useful in predicting wellbore failure.
Our integrated simulator has several distinctive features. The use of general hexahedral finite element grids is particularly well-suited to handle domain specific applications such as near-wellbore studies. The multipoint flux scheme is an accurate and convergent method, it is locally conservative, and its linear systems are efficiently solved with multigrid methods. The use of a fixed-stress iterative coupling scheme is novel for coupling nonlinear mechanics with compositional fluid flow. Finally, to achieve fast convergence rates for solving nonlinear solid mechanics problems, a material integrator has been consistently formulated to give quadratic convergence rates.
Luo, Haishan (The University of Texas at Austin) | Delshad, Mojdeh (The University of Texas at Austin) | Pope, Gary A. (The University of Texas at Austin) | Mohanty, Kishore K. (The University of Texas at Austin)
Unstable floods and resulting viscous fingers remain a big challenge for reservoir simulation as the gridblock size is usually many orders larger than the viscous finger wavelength. This problem becomes especially pronounced with increasing applications of polymer and other chemical floods in the development of heavy oil reservoirs. Traditional reservoir simulators do not consider sub-grid viscous fingering effects and tend to overestimate the waterflood oil recovery. Using extremely fine grid models with centimeters size is unrealistic for field-scale simulations.
While some researchers disregard viscous fingering by claiming that channeling dominates at the large scale for heterogeneous reservoirs, they miss the existence of viscous fingering at the small scale, which affects the displacement efficiency. To overcome this limitation, an effective-fingering model was developed to upscale fingering effects. The model divides each gridblock into three dynamic regions: two-phase flow, single phase oil flow, and bypassed-oil regions. Model parameters represent the maximum fraction of viscous fingering and the growth rates of different regions, which are used to modify flow functions. Model parameters from history match of a set of laboratory experiments show clear power-law correlations with a dimensionless viscous finger number, a function of viscosity ratio, velocity, permeability, interfacial tension, and core cross-sectional area.
The correlation was achieved in the lab scale by considering homogeneous cores, and we extended it further to the field scale by performing high-order spatial accuracy numerical simulations at the intermediate scale using fine gridblock sizes roughly the same as that of the core. Geostatistical realizations of the permeability field were generated with various variances and correlation lengths. In a statistical way, we were able to quantify the viscous finger number valid for a gridblock at the field scale affected by various heterogeneities using the effective-fingering model. We also observed that channelized permeability distributions increase the viscous finger number drastically, showing the important role of channeling in such cases. This new model was applied to a field case with high heterogeneity undergoing water/polymer floods. We observed that the oil recovery was improved by the polymer slug because of the enhancement in both local displacement efficiency and sweep efficiency.
In summary, we developed an upscaling model that provides a fresh-new insight on how to simulate unstable water/polymer floods at the field scale, which effectively accounts for the interplay of viscous fingering and channeling.
In the presence of counter-current flow, nonlinear convergence problems may arise in implicit time-stepping when the popular phase-potential upwinding (PPU) scheme is used. The PPU numerical flux is non-differentiable across the co-current/counter-current flow regimes. This may lead to cycles or divergence in the Newton iteration. Recently proposed methods address improved smoothness of the numerical flux. The objective of this work is to devise and analyze an alternative numerical flux scheme called C1-PPU that, in addition to improving smoothness with respect to saturations and phase potentials, also improves the level of scalar nonlinearity and accuracy. C1-PPU involves a novel use of the flux limiter concept from the context of high-resolution methods, and allows a smooth variation between the co-current/counter-current flow regimes. The scheme is general and applies to fully coupled flow and transport formulations with an arbitrary number of phases.
Several complex heterogeneous multi-dimensional numerical examples under the three-phase black-oil formulation are presented. The examples include a miscible gas injection problem that involves significant variations in the total-velocity field over time. The proposed scheme is compared to the conventional PPU and the recently proposed Hybrid Upwinding schemes. We investigate three properties of these numerical fluxes: smoothness, nonlinearity, and accuracy. The results indicate that in addition to smoothness, nonlinearity may also be critical for convergence behavior and thus needs to be considered in the design of an efficient numerical flux scheme. Moreover, the numerical examples show that our C1-PPU scheme is both entropy-satisfying, and exhibits superior convergence properties for large time steps compared to the other alternatives.
A novel high-order, Double Control Volume Finite Element Method in which pressure is discretized using finite volumes is presented to model multiphase flow porous media simulations. This approach shows a better behavior to calculate the pressure field than the well-known Control Volume Finite Element Method. The latter, as many other formulations, have problems to calculate the pressure field when the elements conforming the mesh, triangles (2D) or tetrahedra (3D), have very obtuse angles. This limits the advantages that can be obtained by using dynamic adaptive meshing methods, as the meshes are constrained to have good quality elements rather than just optimizing the elements to adapt to the peculiarities of the flow. This paper presents a novel formulation that shows to be more resilient with regard to elements with very obtuse angles, enabling dynamic mesh adaptivity to be more effective.
A method is presented for the numerical simulation of wells with arbitrary trajectories using 3D unstructured grids, with a resolution that automatically adjusts to the simulation context: for pressure transient analysis (logarithmic time steps) or low permeability reservoirs a very fine grid is used. The grid is progressively coarsened when early-time transients can be ignored and/or mobility increases. Different discretization controls are applied for full late time consistency between the various resolutions.
A criterion is derived to determine the optimum grid resolution based on the mobility of the fluid in the vicinity of the wellbore and the smallest time step to simulate.
The finest grid uses a full 3D voronoi module around the wellbore, with cells of progressively increasing size away. This module is connected to a coarser background 2.5D voronoi grid for the bulk of the reservoir. Generalized transmissibility derivations accurately account for the well trajectory and estimate the fluxes across potentially non-orthogonal connections.
Progressively coarsened grids are then used to adapt to the problem. For relatively fine grids, generalized derivations of transmissibility and well index values capture the radial effects and avoid the numerical pseudo-skin typically observed with 2.5D grids.
As illustrated with various cases, such approach accurately and consistently simulates wells of complex geometries for all resolutions. The outcome is a new method for generating 3D unstructured grids that automatically adapt to the expected time resolution of the simulation, while ensuring consistency between transient and long-term simulations via original discretization controls.
History matching within the Bayesian framework in practice assumes perfect simulation models. However, for real field cases this assumption may lead to a spurious reduction in forecast uncertainty when a large number of data is used to constrain imperfect reservoir models. To mitigate this spurious uncertainty reduction, we propose a new approach to automatically and consistently inflate the standard deviation of measurement errors for the constraining field data. In previous work we applied the simple mitigation strategy of using a single inflation factor for all data. In this work we propose to use information from the Hessian matrix evaluated at the maximum a posteriori (MAP) points in parameter space: data are regrouped into different categories according to their sensitivities with respect to principal directions of the posterior Hessian matrix. For each group a suitable inflation factor can then be estimated from the number of data and observed mismatches in that group. The proposed procedure is applied to a synthetic as well as a field scale model. The truth of the synthetic model is selected from one unconditional realizations of a real field model with three facies. Synthetic measured production data are generated by adding Gaussian noise to those predicted from the true simulation model. During the process of history matching, a few uncertain model parameters are artificially fixed to values that are inconsistent with the truth to mimic the unknown real field case and make the model imperfect. Numerical results indicate that the proposed approach is able to give a balanced and reasonable range of forecast uncertainty for the cases considered.
Alkali-surfactant-polymer (ASP) floods can recover significant amounts of remaining oil after a water flood. Currently, commercial and academic chemical flooding simulators use Hand's model and linear interpolation rules based on a salinity scan to model micro-emulsion phase behavior. Those simple models have significant limitations when the reservoir conditions are different than experimental conditions. Recently an equation-of-state (EoS) based on HLD-NAC was developed that gives improved phase behavior predictions away from the tuned experimental data. The new EoS for SP modeling was incorporated into our in-house compositional simulator, PennSim. In this research, we expand PennSim with an updated flash calculation module that includes alkali and geochemistry. The new simulator can more accurately model ASP flooding since phase behavior changes consistently with aqueous chemistry, oil saturation, alkali concentration, salinity, pressure, and other key variables.
ASP phase behavior depends on many factors including pressure, temperature, oil EACN, and geochemistry. PennSim solves the mass conservation equations of oil acidic components, water, and aqueous solute along with phase equilibrium relations and chemical reactions using an IMPEC approach. The reactions between oleic acids, aqueous and solid species, surfactant adsorption and mineral dissolution reactions are considered. A scalable capillary desaturation curve (CDC) with permeability and porosity is used for heterogeneous field cases. An algebraic multigrid linear solver is used to speed up simulation.
The functionalities of the developed code are demonstrated with multiple injection scenarios, such as the combination of an alkali and polymer slug with and without added synthetic surfactant. Simulations are made to model 1D core floods and 2D heterogeneous reservoirs with multiple wells. The simulation results show that the main difference between the new CDC and traditional CDC curve is the size of the developed oil bank, where the traditional CDC predicts a larger oil bank in the high permeability channel.