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The purpose of this page is to review the mathematics of fluid flow. We limit our review to essential aspects of partial differential equations, vector analysis, numerical methods, matrices, and linear algebra. These topics are discussed in the context of two fluid flow applications: analysis of the convection/dispersion equation and diagonalization of the permeability tensor. For more details about the mathematics presented here, consult the literature. Partial differential equations (PDEs) are frequently encountered in petroleum engineering.
Fast and reliable reservoir simulation is a key for the successful decision making in integrated reservoir studies. Large and complex multiphase reservoir models usually require expensive computational infrastructure. Physics-based model order reduction (MOR) methods have been introduced and applied (POD-DEIM, POD-TPWL), especially for mitigating the computational cost of black oil models in workflows that require multiple calls of the reservoir simulator. However, only a limited number of methods have looked deeper at the effectiveness of these techniques to multiphase and compositional simulation where expensive phase equilibrium calculations are added to the level of complexities associated with obtaining robust solutions. In this work, we develop the physics-based MOR techniques for rapid compositional simulations that accelerate calibrating of system of equations and phase equilibrium.
The combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) has been used extensively in two-phase flow systems. POD reduces the size of the system to be solved, and DEIM contributes to approximate the nonlinear terms for faster computation. These snapshot-based methods work in a two-step process. In the training case one can obtain the snapshots, which are the solutions at each time step, to derive the POD basis and DEIM basis. Then the test case is utilized to validate if the reduced model works for the different well control schedule cases, and to compare the speed of the simulation run time.
Results show that the POD-DEIM technique enables us to approximate the conventional model with high levels of accuracy up to more than 99%. And it also enables a faster simulation owing to the reduced order system.
In this study, we show the robustness of POD-DEIM method to reduce the computational cost for multi-phase, multi-component 3D reservoir model.
Landa-Marbán, David (Norwegian Research Centre) | Bødtker, Gunhild (Norwegian Research Centre) | Vik, Bartek Florczyk (Norwegian Research Centre) | Pettersson, Per (Norwegian Research Centre) | Pop, Iuliu Sorin (University of Hasselt) | Kumar, Kundan (University of Bergen) | Radu, Florin Adrian (University of Bergen)
In this paper, we study a Darcy-scale mathematical model for biofilm formation in porous media. The pores in the core are divided into three phases: water, oil, and biofilm. The water and oil flow are modeled by a generalized version of Darcy’s law, and the substrate is transported by mechanical dispersion, diffusion, and convection in the water phase. Initially, there is biofilm on the pore walls. The biofilm consumes substrate for production of biomass and modifies the pore space, which changes the rock permeability. The model includes detachment of biomass caused by water flux and death of bacteria, and it is implemented in the MATLAB Reservoir Simulation Toolbox (MRST). We discuss the capability of the numerical simulator to capture results from laboratory experiments. We perform a novel sensitivity analysis based on sparse-grid interpolation and multiwavelet expansion to identify the critical model parameters. Numerical experiments using diverse injection strategies are performed to study the impact of different porosity/permeability relationships in a core saturated with water and oil.
History matching is the most time-consuming phase in any reservoir-simulation study. As a means of accelerating reservoir simulations, a 2018 study proposed an approach in which a reservoir is treated as a combination of multiple interconnected compartments that, under a range of uncertainty, can capture the reservoir’s response during a recovery process. In this work, the authors extend that approach to represent a reservoir in a multiscale form consisting of multiple interconnected segments. To identify segments of the reservoir, spatial, temporal, and spatiotemporal unsupervised data-mining clustering techniques are used. Then, a novel nonlocal formulation for flow in porous media is presented in which the reservoir is represented by an adjacency matrix describing the neighbor and non-neighbor connections of comprising compartments.
Relative permeability and capillary pressure are the key parameters of the multiphase flow in a reservoir. To ensure an accurate determination of these functions in the areas of interest, the core flooding and centrifuge experiments on the relevant core samples need to be interpreted meticulously. In this work, relative permeability and capillary pressure functions are determined synchronously by history matching of multiple experiments simultaneously in order to increase the precision of results based on additional constraints coming from extra measurements. To take into account the underlying physics without making crude assumptions, the Special Core Analysis (SCAL) experiments are chosen to be simulated instead of using well know simplified analytical or semianalytical solutions. Corresponding numerical models are implemented with MRST (Lie, 2019) library. The history matching approach is based on the adjoint gradient method for the constrained optimization problem. Relative permeability and capillary pressure curves, which are the objectives of history matching, within current implementation can have a variety of representations as Corey, LET, B-Splines and NURBS. For the purpose of analyzing the influence of correlations on the history matching results in this study, the interpretation process with assumed analytical correlations is compared to history matching based on generic NURBS representation of relevant functions.
The global oil and gas industry will experience intensive well decommissioning activities, with tens of thousands of well plug and abandonment (P&A) operations forecast to be executed worldwide, over the next few decades. In the North Sea alone, 2,624 wells are expected to be decommissioned over the next decade (
Transient flow modelling of the well P&A system allows new key performance indicators (KPIs) to be developed, e.g. evolution of hydrocarbon saturations within the P&A well over time and the time hydrocarbons reach the surface. These KPIs are not provided by steady-state P&A models. Results presented in this paper demonstrate the value derived from applications of the developed framework to: Optimally allocate resources by identifying fit-for-purpose P&A designs (e.g. the number, location and length of wellbore barriers, and the value of remedial operations on annular flow barriers). Support risk-based decision-making via investigation and comparison of how multiple P&A design options perform for given well/reservoir conditions. Identify critical modelling parameters and optimally allocate data gathering resources to reduce uncertainty. The "likelihood of occurrence" of each cement defect type is one such critical, but very uncertain, model input parameter.
Optimally allocate resources by identifying fit-for-purpose P&A designs (e.g. the number, location and length of wellbore barriers, and the value of remedial operations on annular flow barriers).
Support risk-based decision-making via investigation and comparison of how multiple P&A design options perform for given well/reservoir conditions.
Identify critical modelling parameters and optimally allocate data gathering resources to reduce uncertainty. The "likelihood of occurrence" of each cement defect type is one such critical, but very uncertain, model input parameter.
Ranaee, Ehsan (Department of Energy, Politecnico di Milano) | Guédon, Gaël Raymond (Department of Energy, Politecnico di Milano) | Moghadasi, Leili (Eni SpA) | Inzoli, Fabio (Department of Energy, Politecnico di Milano) | Riva, Monica (Department of Civil and Environmental Engineering, Politecnico di Milano) | Maddinelli, Giuseppe (Eni SpA) | Bartosek, Martin (Eni SpA) | Guadagnini, Alberto (Department of Civil and Environmental Engineering, Politecnico di Milano)
We aim at developing a viable workflow for the characterization of reservoir responses under Water Alternating Gas (WAG) conditions for enhanced oil recovery. We do so through a numerical Monte Carlo (MC) framework and by relying on (
We consider uncertainty in (
In the case of a homogeneous reservoir, we note that reservoir simulation responses are strongly sensitive to the degree of convexity of the two-phase relative permeability curves. In the case of heterogeneous reservoir settings, results are almost similarly sensitive to porosity, characteristics of the relative permeability model, and the degree of heterogeneity of the reservoir. In the case of well-connected (randomly) heterogeneous fields, the importance of the porosity is stronger than in the heterogeneous setting lacking well connected regions.
Characterization of reservoir model attributes relying on pore-scale simulation approaches in the presence of uncertainty can provide a robust term of comparison which can be integrated within a classical reservoir simulation approach relying on relative permeability data stemming from core-flooding experiments. Our results document that uncertainties in the evaluation of (
In general, a probabilistic framework for a modeling process involves two uncertainty spaces: model parameters and state variables (or predictions). The two uncertainty spaces in reservoir simulation are connected by the governing equations of flow and transport in porous media in the form of a reservoir simulator. In a forward problem (or a predictive run), the reservoir simulator directly maps the uncertainty space of the model parameters to the uncertainty space of the state variables. Conversely, an inverse problem (or history matching) aims to improve the descriptions of the model parameters by using the measurements of state variables. However, we cannot solve the inverse problem directly in practice. Numerous algorithms, including Kriging-based inversion and the ensemble Kalman filter (EnKF) and its many variants, simplify the system by using a linear assumption.
The purpose of this paper is to improve the integration of measurement errors in the history-matching algorithms that rely on the linear assumption. The statistical moment equation (SME) approach with the Kriging-based inversion algorithm is used to illustrate several practical examples. In the Motivation section, an example of pressure conditioning has a measurement that contains no additional information because of its significant measurement error. This example highlights the inadequacy of the current method that underestimates the conditional uncertainty for both model parameters and predictions. Accordingly, we derive a new formula that recognizes the absence of additional information and preserves the unconditional uncertainty. We believe this to be the consistent behavior to integrate measurement errors.
Other examples are used to validate the new formula with both linear and nonlinear (i.e., the saturation equation) problems, with single and multiple measurements, and with different configurations of measurement errors. For broader applications, we also develop an equivalent formula for algorithms in the Monte Carlo simulation (MCS) approach, such as EnKF and ensemble smoother (ES).
The significant oil reserves related to karst reservoirs in a Brazilian presalt field add new frontiers to the development of upscaling procedures to reduce time for numerical simulations. This work aims to represent karst reservoirs in reservoir simulators using special connections between matrix medium and karst medium, each modeled in different grid domains of a single-porosity flow model. This representation intends to provide a good relationship between accuracy and simulation time.
The concept follows the embedded discrete-fracture model (EDFM) developed by Li and Lee (2008) and later extended by Moinfar et al. (2014); however, this work extends the approach for karst reservoirs [embedded discrete-karst model (EDKM)] by adding a representative volume through gridblocks to represent karst geometries and porosity. For the extension of the EDFM approach in a karst reservoir, we adapt the methodology to four stages: construction of a single-porosity model with two grid domains; geomodeling of karst and matrix properties, each for the corresponding grid domain; application of special connections through the conventional reservoir simulator to represent the transmissibility between the matrix and the karst medium; and calculation of transmissibility between the matrix and the karst medium.
For a proper verification, we applied the EDKM methodology in a carbonate reservoir with megakarst structures, which consists of nonwell-connected enlarged conduits and greater than 300 mm of aperture. The reference model was a refined grid with karst features explicitly combined with matrix facies, including coquinas interbedded with mudstones and shales. The gridblock of the reference model measures approximately 10 x 10 x 1 m. For the simulation model, the matrix-grid domain has a gridblock size of approximately 100 x 100 x 5 m. The karst-grid domain had the same block size as the refined grid. Flow in the individual karst-grid domain or matrix-grid domain is governed by Darcy’s equation, implicitly solved by a simulator. However, the transmissibility for the special connections between karst and matrix blocks is calculated as a function of open area to flow, matrix permeability, and block-center distance. The matrix properties were scaled up through conventional analytical methods. The results show that EDKM had a considerable performance regarding a dynamic matching response with the reference model, within a reduced simulation time, while maintaining a higher dynamic resolution in the karst-grid domain without using an unconstructed grid.
This work aims to contribute to the extension of the EDFM approach for karst reservoirs, which can be applied to commercial finite-difference reservoir simulators. This could be a solution to reduce simulation time without disregarding the explicit representation of karst features in structured grids.
Chen, Zeliang (Rice University) | Wang, Xinglin (Rice University) | Jian, Guoqing (Rice University) | Zhang, Leilei (Rice University) | Dong, Pengfei (Rice University) | Singer, Philip M. (Rice University) | Hirasaki, George J. (Rice University)
Unconventional resources are of great importance in the global energy supply. However, the ultralow permeability, which is an indicator of the producibility, makes the unconventional production challenging. Therefore, the permeability is one of the critical petrophysical properties for formation evaluation. There aremany existing approaches to determine permeability in the laboratory using core analysis. The methods can be divided into two categories: steady-state and unsteady-state approaches. The steady-state approach is a direct measurement using Darcy’s law. This approach has disadvantages because of the accuracy in the measurement of low flow rate and the long run time. The unsteady-state approach includes pulse decay, oscillating pressure, and Gas Research Institute methods. These approaches are complicated in terms of setups and interpretations. Both steady-state and unsteady-state approaches typically have a constraint on the maximum-differential pressure. We propose a novel unsteady-state method to determine the permeability by transient-pressure history matching. This approach involves simulation and experiments. On the experiment side, the ultralow-permeability core undergoes 1D CO2-flooding experiments, during which the transient pressure is monitored for history matching. On the simulation side, the transient-pressure history is simulated using the finite-volume method incorporating real-gas pseudopressure and table lookup to deal with the nonlinearity in fluid properties and singularity during phase transition. The free parameter permeability in the simulation is adjusted for history matching to determine the rock permeability. Our new unsteady-state approach is developed for fast and convenient permeability estimation for unconventional formation cores. This approach is a valuable addition to existing permeability measurement methods.