ECMOR XVII

Waterflooding has been commonly used for secondary oil recovery. However, it is well known that the efficiency of oil recovery decreases when the mobility ratio is large, or the reservoir is highly heterogeneous. In these scenarios, the polymer flooding technique arises as an efficient alternative to increase the production curves. The injection of a high viscosity polymer solution reduces the mobility ratio, improving the displacement and sweep efficiency. On the other hand, mechanical retention and adsorption phenomena give rise to formation damage close to the injection wells resulting in injectivity loss. In this context, our main goal is to construct a new computational model based on domain decomposition methods capable of coupling the phenomena in different spatial and time scales during the polymer flooding. From the mathematical point of view, we consider the polymer solution a pseudo-plastic flow with the hydrodynamic model given by a non-linear Darcy’s Law where the injected fluid viscosity depends on the shear rate as suggested by the Carreau’s Law. Furthermore, the polymer movement is quantified making use of a convection-diffusion-reaction transport equation where the non-linear reactive part is due to mechanical retention and adsorption. The studied model takes formation damage into account considering that porosity and permeability depend on the retained polymer concentrations mechanically retained or adsorbed. From the computational point of view, the non-linear mathematical model is discretized making use of the finite element method together with a staggered algorithm and the Newton-Raphson method. The kinetic law for mechanical retention is post-processed by the Runge-Kutta method. It is important to highlight that polymer may accumulate in the neighborhood of the injection well on a fast time scale causing injectivity loss. Contrary to the rest of the reservoir, where large time steps and a coarse spatial mesh can be used, on the neighborhood of the injection wells small time steps and a fine spatial mesh are sometimes required. In this context, we propose the application of domain decomposition techniques to couple the near-well/reservoir domains with accuracy and lower computational cost. To this end, we apply a multi-time step domain decomposition method to couple retention and adsorption near well phenomena with polymer transport in the reservoir. Finally, we propose some numerical simulations to show the efficiency of the domain decomposition as well as to quantify injectivity during polymer flooding.

Maximizing oil recovery is a challenging task for the oil industry worldwide, mainly in the presence of dynamic technical and economical constraints. To achieve this target, a number of enhanced oil recovery technologies are being applied, and one of the most successful and used methods is water alternating gas injection (WAG). The estimation of the optimal operating parameters of the WAG process is a complex problem which requires considerable number of time-consuming runs. Therefore, developing a faster alternative tool without scarifying the precision of the numerical simulators becomes essential. Proxy models that are user-friendly mathematical models based on machine learning and pattern recognition, have a noticeable ability to deal with highly complex problems, such as the outcomes of the numerical simulators in reasonable time.

The present work aims at establishing various dynamic proxy models for optimizing a constrained WAG project applied to real field data from “Gullfaks” in the North Sea. Two types of artificial neural network (ANN), namely multi-layer perceptron (MLP) and radial basis function neural network (RBFNN) were taught for predicting all the needed parameters for the formulated optimization problem. Levenberg–Marquardt (LM) algorithm was applied for optimizing the MLP model, while genetic algorithm (GA) and ant colony optimization (ACO) were applied for the proper selection of the RBFNN control parameters. Furthermore, the best proxy model found was coupled with GA and ACO for resolving the WAG optimization problems.

The results showed that the established proxies are robust, practical and effective in mimicking the performance of numerical reservoir model. In addition, the results demonstrated the effectiveness of GA and ACO in optimizing the parameters of WAG process for the real field data used in this study. The findings of this investigation contribute to the knowledge of the mathematics of oil recovery in various perspectives, namely the establishment of cheap and accurate time-dependent proxy models for real cases, the optimization of WAG process in the presence of various types of constraints and also the robustness of nature-inspired algorithms for resolving the optimization problems related to enhanced oil recovery.

We consider a two-phase Darcy flow in a fractured porous medium consisting in a matrix flow coupled with a tangential flow in the fractures, described as a network of planar surfaces. This flow model is coupled with the mechanical deformation of the matrix assuming that the fractures are open and filled by the fluids, as well as small deformations and a linear elastic constitutive law. In this work, the model is derived and discretized using the gradient discretization method which covers a large class of conforming and non conforming discretizations. This framework allows a generic convergence analysis of the coupled model using a combination of discrete functional tools. The convergence of the discrete solution to a weak solution of the model is proved using a priori and compactness estimates. This is, to our knowledge, the first convergence result for this type of models taking into account two-phase flows and the nonlinear poro-mechanical coupling including the cubic nonlinear dependence of the fracture conductivity on the fracture aperture. Previous related works consider a linear approximation obtained for a single-phase flow by freezing the fracture conductivity. Numerical experiments are presented to illustrate this result using a Two-Point Flux Approximation cell centered finite volume scheme for the flow and a P2 finite element method for the mechanics. Iterative coupling algorithms are investigated to solve the coupled discrete nonlinear systems at each time step of the simulation.

The significant world oil reserves related to fractured karst reservoirs in Brazilian pre-salt fields adds new frontiers to the (1) development of numerical methods for upscale giant fields with multiscale heterogeneities, (2) history matching and production strategy optimization under critical uncertainties and (3) forecast of the future reservoir performance. However, there is a lack of benchmark models with a heterogeneous dynamic behavior typical from fractured karst reservoirs, to develop and validate novel numerical methods. This work presents a simulation benchmark model, available as public domain data, which represents a fractured carbonate karst reservoir and add a great opportunity to test new methodologies for reservoir development and management using numerical simulation.

The work structure is divided in three steps: (1) development of a reference model, a fine grid model with high level of geologic details, treated as the real field, (2) development of a simulation model under uncertainties considering an initial stage of the field development phase, and, (3) elaboration of a benchmark proposal for studies related to the oil field development and production strategy selection. Based on the available information from well logs, several uncertainty attributes were considered in structural framework, facies and petrophysical properties. Dynamic, economic and technical uncertainties were also considered. The reference model is a giant field divided by two stratigraphic zones - the upper zone characterized by stromatolites and the lower one by coquinas. Moreover, the model is characterized by two regions with karst features near the horizons surfaces and a cluster of fractures near faults. Volcanic rocks and high permeable trends near faults are included as non-mapped uncertainties in the simulation model, as the information from well logs at the initial stage of field development does not intercept this geologic attribute. This approach will lead to several challenges on reservoir development and management.

As this benchmark is representative of a giant field, it is divided in four sectors. Sector 1 has already a production strategy defined, aiming studies regarding field management. The strategy considers WAG (water alternate gas/CO2) as recovery mechanism and the presence of 13 wells in a first wave (6 producers and 7 injectors), and other 4 wells can be added in a second wave. Field development studies can be applied in the other sectors.

This Benchmark provides a great opportunity for develop and test novel numerical methods in giant reservoirs with geologic and dynamic pre-salt trends.

Recent developments based on nanotechnology have shown the immense potential of application in EOR & IOR operations, which is supported by successful results on the lab and field-scales. However, the poor understanding and the shortage of a robust framework for nanoparticle transport-and-retention modelling in porous media is a downside for its properly spread in the Ο &G industry. In this work, we propose a novel modelling framework that allows to represent jointly mechanical and chemical mechanisms for nanoparticle retention and remobilisation in porous media. This model is formulated under a phenomenological approach that considers a strong physical basis of these processes on the macroscale. Retention and remobilisation dynamics are modelled under a non-equilibrium approximation using an α-order kinetic which depends on equilibrium condition. The mathematical formulation was programmed using the open-source package Chebfun, as a function of dimensionless variables to make up-scaling to higher scales more feasible. The impact of dimensionless variables in nanoparticle transport and retention was studied by a sensibility analysis which allowed to identify their effect on nanoparticle transport and retention. In this sense, some simplifications are proposed for the model according to the dimensionless variables. In order to validate this framework and its implementation, a set of lab tests was designed and carried out using silica-nanoparticle-based nanofluid in sand packs. Some concentration jumps were used to catch its effect on nanoparticle retention and remobilisation. Experimental data show a good agreement with simulation data under each operation condition and parameter fitting. Additionally, the model is capable of predicting the profile of nanoparticle concentration and its evolution on time. Changes in that profile can be predicted if operating conditions change, allowing their optimisation. Finally, this modelling framework is implemented in the multi-physic and multi-component tool DFTmp Simulator to simulate specific EOR & IOR application on the field-scale. Using the fitting parameters obtained previously, an application of IOR is simulated considering a multiphase system and other phenomena simultaneously.

In reservoir simulation studies, one of the crucial factors affecting the accuracy and hence reliability of the results is the representation of well connections in the numerical reservoir grid. Although there have been numerous attempts to redefine the relationship between wellbore pressure, grid cell pressure and the corresponding fluid flowrate, the original Peaceman formulae are still the most prevalent simulation software option by far. The simplicity of their implementation overshadows their limited applicability to symmetric 2D scenarios of purely cylindrical radial flow, also built into the "3D projected Peaceman" formula.

One of the attempts to improve the inflow model was the Multi-Point Well Connection (MPWC) method (SPE 173302) which solves the local flow problem using the Boundary Element Method (BEM). In terms of its boundary conditions, pressures of the next-neighbour cells surrounding the well-connection cell appear in the final coupling formula, which makes the method difficult to implement and computationally less efficient.

A new method has been formulated to overcome the drawbacks of MPWC and still utilise the benefits of BEM. The proposed Free-Space Well Connection (FSWC) method converts the next-neighbour cells into infinitesimal layers of equivalent transmissibilities and applies free-space boundary conditions to their outer surfaces. All cell faces are adaptively refined into a required number of boundary elements and their pressures and fluxes are expressed by means of free-space Green’s functions representing well perforation sources/sinks. The method is applicable to cells and perforations of arbitrary geometry, including perforations outside the cell of interest, and to general cases of heterogeneous anisotropic rock permeability. Balancing all boundary pressures and fluxes yields the resulting well-connection transmissibility (or well index) and inter-cell transmissibility multipliers that emulate the flow asymmetry outside the well-connection cell.

Accuracy of the FSWC method has been verified against various analytical and numerical models. Even for the ideal case of a fully penetrating vertical well in the centre of a square reservoir, the FSWC-computed well index is closer to the analytical solution than that of Peaceman. Despite its broad applicability, superior accuracy and robustness, the method is fast and requires just a few CPU seconds to reach the desired precision. This is demonstrated by various examples with realistic well trajectories from full-field reservoir simulation runs.

We propose a proxy model to separate oil and water production total predicted liquid rate. This is essential to optimal waterflooding management. The proxy models studied here are widely used to estimate parameters in the field of petroleum engineering due to their low computational cost and do not require prior knowledge of reservoir properties. The approach uses production history and the producer-based capacitance and resistance (CRMP) model, together with the combination of two fractional flow models, Koval ( Cao, 2014 ) and Gentil ( Gentil, 2005 ). We will henceforth call Kogen this combined model.

The combined fractional flow model can be formulated as a constrained nonlinear curve fitting. The objective function to be minimized is a measure of the difference between calculated and observed water cut values (Wcut) or net present values (NPV). The constraint limits the difference in water cuts of the Koval and Gentil models at the time of transition between the two. The problem can be solved using gradient-based method the sequential quadratic programming (SQP) algorithm. In this study, the gradient is computation by finite differences. The parameters of the CRMP model are the connectivity between wells, time constant, and productivity index. These parameters can be found using a Nonlinear Least Squares (NLS) algorithm. With these parameters, it is possible to predict the liquid rate of the wells. The Koval and Gentil models are used to calculate the Wcut in each producer well over the concession period which in turn allows to determine the accumulated oil and water productions.

Two synthetic models, Brush Canyon Outcrop and Brugge model are used to validate the proposed strategy. Then we compare the solutions obtained with the three fractional flow models (Koval, Gentil, and Kogen) with results obtained directly from the simulator.

It has been observed that the proposed combined model, Kogen, consistently generated more accurate results. In addition, CRMP/Kogen proxy model has demonstrated its applicability, especially when the available data for model construction is limited, always producing satisfactory results for production forecasting with a low computational cost.

Controlled/Low Salinity Waterflooding (LSWF) is an augmented waterflood with well-reported improved displacement efficiency compared with conventional waterfloods. Physical mixing or dispersion of the injected low-salinity (LS) brine with the formation high-salinity (HS) brine substantially reduces the low-salinity effect. Numerical dispersion often misrepresents this mixing in conventional LSWF-simulations, causing errors in the results. Uncertainty in the reservoir description further makes the evaluated performance questionable. Existing studies have suggested optimal amounts for the injected LS-brine to sustain its displacement stability during inter- well flows with physical mixing, but with poor or no consideration of uncertainty. This work focuses on optimizing the injected LS-brine amount considering reported flow uncertainties while ensuring adequate correction of the erroneous influence of numerical dispersion on physical mixing. We investigate the impacts of flow uncertainties on the optimal LS slug-size. The sensitivity of the optimal slug-size to heterogeneity is examined under uncertainty. We evaluate how the interaction between physical mixing and geological heterogeneity influences slug integrity and performance.

We propose an improved ‘effective salinities’ concept to evaluate appropriate effective salinities to characterize the desired representative physical mixing supressing the large numerical dispersion effects usually encountered in coarse-grid LSWF-simulations. This ensures reliable representation of physical dispersion in such grids. We consider different models with characterized levels of heterogeneity and essential variables that control the impact of mixing on LSWF performance based mainly on reported data. New indicators are defined to evaluate the displacement stability and performance of injected LS-brine thereby relating its technical and economic performance. Slug performance is evaluated at different injection times to examine the sensitivity of recovery to LS injection start-time. Performance uncertainty is assessed through a designed four-stage computationally-effective approach: Parameter-space sampling to design representative experiments; Proxy modelling; Proxy validation and verification; and Monte Carlo simulation to provide a wider representative sample for the parameter-space.

We can now reliably represent physical dispersion in LSWF-simulations of current commercial reservoir simulators. Recovery is observed to be relatively insensitive to LS injection start-time until breakthrough of preinjected HS-brine. This is important for LS injection designs as they need not commence immediately for secondary-mode. The potential favourable influence of the spatial distribution of heterogeneity is seen, with links to transverse dispersion. The evaluated optimal sizes from existing studies are observed to be, at best, only suitable as displacement stability thresholds for slug injection considering uncertainty. We find an optimal slug-size of at least 1.0 HCPV to reduce risk under uncertainty.

In this work, we propose a new time-stepping method for the simulation of transport in two-phase flows. Our method relies on constant initial saturation conditions and builds on the streamline-based discretization. For example, in sampling methods such as multi-level Monte Carlo, many probable scenarios of an uncertain permeability field have to be simulated with inexpensive models in order to quantify the uncertainty of phase saturations. However, since the statistical error converges slowly, large ensembles are needed and therefore, the computational cost per sample has to be small. We illustrate the performance of our new inexpensive, yet accurate time-stepping scheme in Buckley-Leverett type problems involving multi-Gaussian as well as more realistic channelized permeability fields.

Production optimization under uncertainty is complex and computationally demanding, a particularly challenging process for carbonate reservoirs subject to WAG injection, represented in large ensembles with high simulation runtimes. Search spaces of optimization are often large, where reservoir models are complex and the number of decision variables is high. The computational costs of ensemble-based production optimization can be decreased by reducing the size of the ensemble with representative models (RM). The validity of this method requires that the RM maintain representativeness throughout the optimization process, where the production strategy changes at each evaluation. Many techniques of RM selection use production forecasts of the ensemble for an initial production strategy, which raises questions about the robustness of the RM. This work investigates approaches to ensure the consistency of RM in ensemble-based long-term optimization. We use a metaheuristic optimization algorithm that finds sets of RM that represent the ensemble in the probability distribution of uncertain attributes and the variability of production, injection, and economic indicators ( Meira et al., 2020 ). Our case study is a benchmark light-oil fractured carbonate with features of Brazilian pre-salt reservoirs and many reservoir and operational uncertainties. We obtained production, injection and economic indicators using different approaches to provide valuable insight for RM selection. We inferred about RM fitness for production optimization based on their adequacy for uncertainty quantification for varying production strategies. Despite the effects of changing decision variables on RM representativity, our results suggest the possible use of RM for ensemble-based production optimizations with limitations related to the estimation of the probabilistic objective function due to mismatches in the probabilities of occurrence. Using production indicators obtained from a base production strategy decreased RM representativeness when compared to RM selection based on a more robust evaluation of reservoir performance using a wide-covering well pattern and no restrictions from production facilities. Finally, our results suggest valid RM selection using production forecasts for intermediate dates of the simulation period, an important contribution for ensembles with very high simulation runtimes. We also provide a broad theoretical background on the uncertain reservoir system and on approaches to obtain reduced ensembles and their applications.