We have previously introduced a transformation based upon the Fast Marching Methods (FMM) to describe the multi-dimensional diffusivity equation with heterogeneity as an effective one dimensional diffusivity equation in a streamtube. In the current study we develop and validate new asymptotic analytic approximations to this problem, which provide for a number of novel applications including rapid numerical simulation, reservoir and well characterization, sensitivity-based inversion using production data, and dynamic upscaling and downscaling. The novel semi-analytic asymptotic pressure approximation for the solution of an equivalent 1-D diffusivity equation is able to approximate the 3-D solution with heterogeneity. Earlier approaches have relied upon the numerical solution of the 1-D equation, and provide all the flexibility expected of a numerical approach. However, analytic solutions provide for the derivation of explicit relationships between the geometry of a propagating pressure "front" within a reservoir and pressure and rate measured at wells. In the current study, we extend the analytic treatment beyond simple fixed rate draw-down and test the predictions against analytic and numerical synthetic cases.
In this paper we provide a systemic validation of our semi-analytic solution technique and extend its utility to more realistic cases, including large changes in reservoir properties, pressure transient analysis with wellbore storage, and rate transient analysis in bounded reservoirs with fixed rate or fixed BHP production. This technique provides us with the ability to describe pressure propagation from fractured wells into the surrounding formations, which provides for a better drainage volume characterization, which is beneficial for both well spacing calculation and multi-stage fracture spacing optimization in unconventional reservoirs. It not only provides for the direct calculation of various welltest, rate transient and well performance concepts such as depth of investigation, welltest derivative, flow regimes and well productivity, but it can also predict pressure and flux distribution maps at any time of interest. Our study verifies that the new approach yields results very close to those generated by commercial simulators, indicating its promising application to rapid field production data analysis. As with other analytic approaches, the derived asymptotic solutions satisfy superposition in space and time, which allows for further application to cases with multiple wells and varying flow rates.
We show the validation of novel semi-analytic asymptotic pressure solutions to the diffusivity equation, which extend the calculation of rate and pressure transient from homogeneous reservoirs with regular well geometries to a series of reservoir problems with hydraulic fractured wells and reservoir heterogeneity. The treatment we present in this paper is faster than numerical finite difference simulation and allows for the development of fundamental relationships between reservoir performance and reservoir and well characteristics.
Molecular diffusion is a proven oil recovery mechanism in fractured reservoirs. Neglecting diffusion during the simulation of gas injections can lead to underestimated oil recoveries. This is particularly important in fractured reservoirs with low permeabilities. This paper presents the implementation of a diffusion model in a reservoir simulator. The implementation is verified with discrete fracture and dual-porosity model cases. The adopted diffusion model is based on irreversible thermodynamics. It uses full matrices of diffusion coefficients and chemical potential gradients as the driving force. To evaluate how much diffusion is dominant during fluid flow in fractured reservoirs, a form of the Péclet number is proposed as the ratio of characteristic times of diffusion to gravity drainage for a given reservoir and fluid properties. The incremental oil recovery from gas injection simulation results confirms the flow regime predicted by the Péclet number. This paper also examines the performance of simulations that involve diffusion by using various solution schemes, including explicit, fully implicit, and partially implicit methods. Optimal performance is achieved with the partially implicit method in which diffusion coefficients are updated at each timestep, while driving forces are updated inside Newton iterations. The simulation results also show that constant diffusivities might not provide a good representative for diffusion coefficients during gasflooding. They can cause oil recovery overestimation or underestimation. The authors demonstrate a technique to forgo diffusion calculations in the regions with convection-dominated flow regimes to help reduce computational time of the simulations involving diffusion. The speedup obtained for gas injection cases with a wide range of Péclet numbers is also examined.
Oil shales are sedimentary rocks containing organic matter in form of kerogen which account for more than 5 trillion barrels of oil in place according to
Wang, Bin (University of Louisiana at Lafayette) | Du, Juan (CNOOC, Tianjin) | Feng, Yin (University of Louisiana at Lafayette) | Wang, Yihui (University of Louisiana at Lafayette) | Wang, Sijie (University of Louisiana at Lafayette) | Yang, Ruiyue (China University of Petroleum)
Reactive transport phenomenon, such as CO2 sequestration and Microbial EOR, has been of interest in streamline-based simulations. Tracing streamlines launched from a wellbore is important especially for time-sensitive transport behaviors. However, gridblocks containing wells are usually too large compared to wellbore radius. Field-scale simulations with local-grid-refinement (LGR) models are often consume huge computational time. An embedded grid-free approach is developed to account for the transport along streamlines in the vicinity of wellbore.
An embedded semi-analytical approach to integrate near wellbore transport behaviors into field-scale streamline simulation is developed, which consists of two-stage of development: tracing streamlines in a local gridblock (containing wells) and coupling with neighboring grids. The velocity field in a local gridblock is produced based on boundary element method, and then streamlines are numerically traced with TOF along each streamline based on the velocity field. This local streamline system is then coupled with Pollock-algorithm-based system at the interface between the local gridblock and its neighbors. Finally, the coupling result is verified by matching boundary conditions and transport equations are solved along streamlines.
The presented algorithm for solving near-wellbore streamlines is verified by both commercial finite element simulator and Pollock-algorithm-based 3D streamline simulator. Simulation results (including velocity field, time-of-flight (TOF), streamline pattern and concentration pattern) produced by different approaches are analyzed. Results show that the presented method can accurately perform the near-wellbore streamline simulation in a time-effective manner. The algorithm can easily extend to one grid containing multiple wells to account for the effect of interactions between wells on the overall flow pattern. Two and three-dimensional synthetic field-scale cases are investigated considering advection-reaction transport and multiple-well modeling. The streamline pattern and distribution of TOF within the gridblock containing wells are highly dependent on well location and number of wells. Assuming streamlines are evenly launched from the gridblock boundary and ignoring transport in the local gridblock containing wells are not always reasonable and may lead to overestimating the concentration front (up to 21.3% error).
Two algorithms are introduced in this work. One is referred to as Virtual-Boundary Element Method that is used to generate near wellbore streamlines. And the other one is an embedded semi-analytical approach to solve and integrate near wellbore transport behaviors into field-scale streamline simulation. This study provides a simple and grid-free solution, but is capable of capturing the flow field accurately near the wellbore with significant accuracy and computational efficiency. The method is promising for streamline-based reservoir simulation with time-sensitive transport, and other simulations which require an accurate assessment of interactions between wells in one particular gridblock.
Steam processes involving injection of a solvent with steam, such as CSS (Cyclic Steam Stimulation), SAGD (Steam-Assisted Gravity Drainage), and steam injection in isothermal mode are currently receiving a great deal of attention in Alberta. These combination processes are designed to reduce energy consumption and the emission of greenhouse gases over steam alone. Field results have been mixed and the original intent remains elusive. This paper addresses questions regarding the mechanism of solvent dissolution of bitumen and the expected improvement in oil recovery, if any, when a solvent is injected with steam.
The unique aspect of this work is that instantaneous phase equilibrium is not assumed, as is customary in numerical simulators for solvent-steam applications. Nor is partial equilibrium assumed based on an empirical factor or concept. Rather, equilibrium is based on an analytical model of the dissolution and mobilization of a drop of bitumen inside a pore by solvent and heat. The results of this part of the study show that solvent requires at least three times as long to mobilize bitumen as by heat conduction. Such a delay is implicit in the nature of diffusion and dispersion of a solvent. Several boundary conditions are tested for a spherical drop.
A new thermal compositional simulator with the single drop model was developed for this study and several thermal processes for solvent injection were investigated for non-equilibrium phase behaviour. The results were compared to the case of instantaneous equilibrium, confirming the reason for the lack of success with solvent injection.
The results and extensions of this work will be of great interest in heavy oil production because they serve to explain the expected performance and frequent lack of success in solvent-steam injection. Use of the developed simulator would make it possible to determine whether solvent injection is a good choice in a given situation.
Lie, K. -A. (SINTEF) | Kedia, K. (ExxonMobil Upstream Research Company) | Skaflestad, B. (SINTEF) | Wang, X. (ExxonMobil Upstream Research Co.) | Yang, Y. (ExxonMobil Upstream Research Co.) | Wu, X. -H. (ExxonMobil Upstream Research Co.) | Hoda, N. (ExxonMobil Upstream Research Co.)
This paper presents a general framework for constructing effective reduced-order models from an existing high-fidelity reservoir model, irrespective of grid topology. We employ a flexible hierarchical grid coarsening strategy that is designed to preserve geologic features and structures in the underlying model such as environments of deposition and faults. The strategy supports selecting and combining coarsening methods that are targeted to the flow patterns in different parts of the reservoir. This includes, but is not limited to, explicit user-imposed boundaries, using efficient field-wide flow indicators, topological and geometric partitioning and methods for amalgamating and splitting clusters of cells.
Collectively, these schemes enable an automatic strategy that separates a model into flow-dependent compartments that are respectively close to, far away from, or in between regions of sharp flow transients such as wells. These compartments may then be coarsened using different tailored techniques and target grid resolutions providing much more flexibility compared to traditional coarsening methods. We demonstrate that various techniques for flow-based transmissibility upscaling can be deployed on the resulting coarsened model to compute effective model properties. The hierarchical construction strategy allows efficient exploration of the geologic features of a reservoir that most impact flow patterns and well communication. The coarsened models are shown to be rank and trend accurate, enabling a more exhaustive sensitivity analysis if needed. We study the accuracy of the reduced-order model with a particular emphasis on the upscaled model's ability to capture effects of multiple phases in simulation runs compared to the full high-fidelity model.
A new methodology for the joint optimization of economic project life and time-varying well controls is introduced. The procedure enables the maximization of net present value (NPV) subject to satisfaction of a specified modified internal rate of return. Use of this framework allows an operator to avoid situations where NPV continues to increase in time, but the late-time cash flows are negligible (in terms of an appropriate financial metric) relative to the capital invested in the project. The optimization is formulated as a nested procedure in which economic project life is optimized in the outer loop, and the associated well settings (time-varying bottomhole pressures in the cases considered) are optimized in the inner loop. The inner-loop optimization is accomplished by use of an adjointgradient-based approach, while the outer-loop optimization entails an interpolation technique. We demonstrate the successful application of this framework for production optimization for two-and three-dimensional reservoir models under waterflood. The tradeoff between maximized NPV and rate of return is assessed, as is the impact of discount rate on optimal operations. We believe this to be the first production optimization formulation that explicitly incorporates both NPV and rate of return. As such, this approach may represent an alternative to existing treatments that entail the bi-objective optimization of long-and short-term NPV.
Wang, Cong (Colorado School of Mines) | Xiong, Yi (Colorado School of Mines) | Huang, Zhaoqin (China University of Petroleum) | Winterfeld, Philip (Colorado School of Mines) | Ding, Didier (IFPEN) | Wu, Yu-Shu (Colorado School of Mines)
Gas flow in shales is complicated by the highly heterogeneous and hierarchical rock structures (i.e., ranging from organic nanopores, inorganic nanopores, less permeable micro-fractures, more permeable macro-fractures, to hydraulic fractures). The dominant fluid flow mechanism varies in these different flow regimes, and properties of these rock structures are sensitive to stress changes with different levels. Although traditional single-porosity and double-porosity models can simulate certain time range of reservoir performance with acceptable accuracy, they are not generally applicable for the prediction of long-term performance and have limitations to improve our understandings of enhanced hydrocarbon recovery. In this paper, we present a multi-domain, multi-physics model, aiming to accurately simulate the fluid flow in shale gas reservoirs with more physics-based formulations.
An idealized model has been developed for the purpose of studying the characteristic behavior of a fractured nanopore medium, which contains five regions: organic nanopores, inorganic nanopores, local micro-fractures, global natural fractures, and hydraulic fractures. Fluid flow governing equations in this model vary according to the different dominant fluid flow mechanisms in different regions. For example, the apparent permeability, which is the intrinsic permeability multiplied by a correction factor, is used to account for the gas slippage through nanopores of shale matrix; while the organic and inorganic nanopores in this matrix have different capacities for gas adsorption. On the other hand, for fluids flow in natural fractures and hydraulic fractures with high velocity, the non-Darcy flow model is used to capture the strong inertia when is comparable to viscous force.
Numerical studies with practical interests are discussed. Several synthetic, but realistic test cases are simulated. Input parameters in these cases are evaluated using either the laboratory or theoretical work. Our results demonstrate that this model is able to capture the typical production behavior of unconventional reservoirs: a great initial peak, the sharp decline in the first few months, followed by a long flat production tail. A series of sensitivity analyses, which address the organic matter content, organic matter connectivity, natural fracture density, and hydraulic fracture spacing, will also be conducted.
Reservoir simulation plays an important role in the petroleum industry. Today, there is a specific demand to run ensembles of megaand even giga-cell models. The iterative solution of the large systems of nonlinear governing equations, which describe the multiphase mass transfer in the subsurface, takes the most of the simulation time. The linearization part of the solution process occupies a significant fraction of that time, especially in compositional models. Moreover, the implementation of the linearization step usually embodies the most substantial, complex, and specific part of the computational loop in modern simulators, defining which physical mechanisms and assumptions are employed. This significantly complicates the implementation of simulation codes for heterogeneous computing hardware, which promises significant improvements in simulation time. In this work, we use the recently proposed Operator-Based Linearization (OBL) approach to develop a general purpose reservoir simulation code aiming to substantially decrease the simulation time. OBL offers a simplified linearization method, enhancing the computational performance of simulation and providing an opportunity of a painless porting to heterogeneous computing architectures. To distinct the contribution of both factors, we developed two versions of the compositional simulation prototype code: for traditional CPU and GPU-accelerated hardware architectures. While the former allowed us to speed the linearization stage up by an order of magnitude in comparison with the conventional approach, based on Automatic Differentiation (AD), the latter improved it further by another order of magnitude. The developed prototype realizes the potential of the OBL approach and GPU computing architecture, proving significant improvement in general purpose simulation performance.
Field-scale simulations of complex processes, often suffer from long simulation times. One of the main reasons is that the Newton-Raphson (NR) process used to solve each simulation time step requires many iterations and small time-step sizes to converge. Since the selection of solution variables impacts the nonlinearity of the equations, it is attractive to have a practical method to rapidly explore the use of alternative primary variables in general-purpose reservoir simulators.
Many reservoir simulators use pressure, saturations, and temperature in each gridblock as primary solution variables, which are referred to as natural variables. There is also a class of reservoir simulators that uses pressure, total component masses (or moles), and internal energy in each gridblock as primary variables. These simulators are referred to as mass-variable based reservoir simulators. For a given choice of primary variables, most simulators have dedicated, highly optimized procedures to compute the required derivatives and chain rules required to build the Jacobian matrix. Hence, it is usually not possible to switch between mass and natural variables. In this work, however, we establish a link at the numerical solution level between naturaland mass-variable formulations and design a novel (nonlinear) block-local method that transforms mass-variable shifts (computed at each NR iteration) into equivalent natural variable shifts.
We demonstrate on a number of simulation models of various complexity that, by use of the proposed approach, a mass-variable based flow simulator can still effectively use natural variables, where the change of variables can be made locally per gridblock. Results indicate that in some models the total number of NR iterations, linear solver (LS) iterations, and backups are reduced when using natural variables, instead of mass variables. However, the improvement is fairly modest and not generally observed. Our findings also signify that, depending on the specific characteristics of the simulation problem at hand, mass-variable based simulators may perform comparably or outperform natural-variable based simulators.
The proposed variable switching method can be used effectively to evaluate the impact of using different primary solution variables on problem nonlinearity and solver efficiency. With this method, the impact of interchanging primary solution variables on problem nonlinearity can be rapidly evaluated.