A general algebraic multiscale framework is presented for fractured porous media, which enables the treatment of fractures with multiple length scales and wide ranges of conductivity contrasts. To this end, fully integrated local basis functions for both matrix and fractures are constructed. These basis functions are employed to construct the multiscale coarse system for both matrix and fractures, and then interpolate the coarse solutions back to the fine-scale reference system. Combined with a second stage fine-scale solver, here, ILU(0), our development leads to an iterative multiscale strategy for heterogeneous fractured media, allowing for error reduction to any arbitrary level, while honoring mass conservation after any multiscale finite volume stage. In order to maintain generality, it is shown that when each fracture network is modeled using a single coarse grid cell, our formulation automatically reduces to that proposed by
Large-scale reservoir simulation is still a big challenge due to the difficulty of solving linear systems resulted from the Newton methods. For black oil simulation, more than 90% of running time is spent on the solution of linear systems. The problem is getting worse when developing parallel reservoir simulators using parallel distributed systems with tens of thousands of CPUs. Efficient linear solvers and preconditioners are critical to the development of parallel reservoir simulators.
Here we introduce our work on developing parallel preconditioners for highly heterogeneous reservoir simulations. A family of new Constrained Pressure Residual (CPR)-like preconditioners and advanced matrix pre-processing techniques are developed, including two new three-stage preconditioners and one four-stage preconditioners. A pressure system is solved by an algebraic multi-grid method, and the entire linear system is solved by the restricted additive Schwarz (RAS) method (one of the domain decomposition methods). To overcome a convective issue in reservoir simulation, a parallel potential-based matrix reordering method is employed to stabilize our preconditioners. Matrix decoupling methods, such as an alternative block factorization (ABF) strategy and a Quasi-IMPES (implicit pressure explicit saturation) strategy, are also applied. With the restricted additive Schwarz and algebraic multi-grid methods, our preconditioners have good scalability for parallel computers.
Our preconditioners have been applied to oil-water and black oil benchmark simulations. For the SPE 10 project, which is a big challenge for a linear solver because of highly heterogeneous permeability and porosity, our preconditioners with the GMRES linear solver are stable and efficient. When using 64 CPUs, the number of iterations of our linear solvers is less than 40. When applying our method to a standard black oil simulation with 100 millions of grid blocks, the number of iterations of our linear solvers is only two using 3,072 CPU cores. Our numerical experiments show that our preconditioners and linear solvers are stable with a large number of CPUs and are efficient for highly heterogeneous simulations.
Inference of spatially distributed reservoir properties from production data in scattered wells poses an under-constrained inverse problem that has nonunique solutions. One major contributor to problem ill-posedness is over-parameterization of spatially distributed reservoir properties. We recently introduced sparse representations of unknown reservoir properties for history matching by exploiting the correlation in their spatial distribution. In this approach, during history matching, instead of estimating reservoir properties for each model grid cell, the sparse representation of the reservoir properties are estimated from production data. The resulting history-matching problem can be solved using recent developments in sparse signal processing, widely known as
Ma, Eddie (Kuwait Oil Company) | Gheorghiu, Sorin (Schlumberger) | Banagale, Merlon (Kuwait Oil Company) | Dashti, Laila (Kuwait Oil Company) | Bond, Deryck (Kuwait Oil Company) | Ibrahim, Muhammad (Schlumberger) | Ali, Farida (Kuwait Oil Company) | Gurpinar, Omer (Schlumberger)
The Greater Burgan field in Kuwait is the largest clastic oil field in the world. Its sheer size, complex geology, intricate surface facility network, 5, 000 well-completions and 68-years of production history represent formidable challenges in reservoir simulation. In the last two decades, many flow simulation models, part-field and full-field, were developed as reservoir management tools to study depletion plan strategies and reservoir recovery. The new 2013 Burgan flow simulation was a major undertaking in terms of effort and financial cost. The model size, innovative technology, supporting resources, integrated workflow and meticulous planning applied to this project were unprecedented.
As the Burgan field has matured over time, the reservoir pressure has declined in certain areas, with associated reduced productivity. The reduction of wells' productivity, combined with the increasing water production, has necessitated improved oil recovery (IOR) initiatives in order to meet the Kuwait Oil Company (KOC) corporate vision-2030, sustaining oil production and ensuring high recovery from Burgan reservoirs. This paper describes the development of a dynamic model to design pressure maintenance projects for optimal reservoir management and IOR strategies. It was built on a history match model which has a 68-years of history matching on three levels, Global (Field), Regional (Reservoirs / Gathering Centers) and wells. These three levels depict the concerted history matching effort in accordance with the recurrent data quality. Details of geologic and dynamic modeling have been documented and presented in previous Burgan SPE papers and are not repeated in this paper.
The primary objectives of the Burgan prediction model are meeting the production target profiles with optimal field development plans (FDP) and to maximize oil recovery. There are two pressure maintenance projects, Wara Pressure Maintenance Project (WPMP) and Burgan Sand Upper (BGSU-PMP), included in the prediction model. In this paper, WPMP is discussed in detail as the waterflood project is now entering operation stage after 10 years of planning and construction. BGSU-PMP is part of the Burgan FDP but is not focused within the scope of this paper.
Sub-surface modeling in the giant Greater Burgan field complex is not just enormous, it is also arduous and challenging. The accomplishment by the team was momentous despite a less-than-expected result. Nonetheless, lessons learnt offered valuable information for future improvement. It has been a long and difficult journey from geological model to dynamic model over the last five years. Yet, in pursuing IOR and EOR, the journey has just begun.
We study unconditionally stable sequential methods for the all-way coupled thermoporomechanical problems. We first propose two sequential methods: the undrained-adiabatic split that combines the undrained split in poromechanics with the adiabatic split in thermomechanics, and the extended fixed-stress split. We perform new stability and convergence analysis for the undrained-adiabatic and extended fixed-stress split methods, introducing a new extended norm for nonlinear stability analysis, which can cover all-way coupled thermoporomechanical problems. In this study we show that the two methods are unconditionally stable (i.e., contractive and B-stable), when we take implicit time stepping. We also perform spectral analysis in order to investigate convergence of the two methods when linearizing the coupled problem. The spectral analysis will be useful for designing reliable pre-conditioners of the monolithic method. The spectral analysis shows that the two sequential methods are convergent and that the extended-fixed stress split is more accurate than the undrained-adiabatic split for strong coupling. We show numerical examples, which support the a-priori stability and convergence estimates.
Faults and complex wells are two important types of internal boundaries to resolve in reservoir simulation. Faults are physical boundaries which may form local barriers or conduits to fluid flow. In structured-grid simulation, fault surfaces are typically represented as zig-zag cell edges where the depths may be shifted across the fault face. The better representation of fault traces using unstructured gridding has been the subject of research in the petroleum literature for over two decades. The use of long horizontal and multi-branch complex wells for production from tight and heterogeneous reservoirs is also common practice nowadays. These wells can be densely populated which make classical local grid refinement (LGR) methods difficult to apply. It is highly desirable to represent the perforation inflow and the near-wellbore flow more accurately in full-field simulation.
The paper extends the Voronoi gridding method (
Following an introduction of unstructured-grid methods in reservoir simulation, the gridding algorithm is discussed in details. This is followed by simulation examples, which includes a full-field compositional simulation example of a faulted gas-condensate reservoir completed with many deviated and horizontal wells. An in-house parallel reservoir simulator is used to run the models. Simulation results using both the structured corner-pointed-geometry (CPG) grid and the unstructured-grid method are compared. The advantages of unstructured approach in complex field-scale simulation are demonstrated.
The surrogate model of choice in this investigation uses Trajectory Piecewise Linearization as numerical complexity reduction technique and Proper Orthogonal Decomposition for dimension reduction. The stability of the model is assured through the use of Petrov-Galerkin left projector in finding the reduced space solution of the linearized problem. The TPWL/POD approximation is further refined by the addition of a kriging correction model. The high fidelity model is a two-phase, 3D, fully implicit simulator that uses mass fraction-based formulation. The approximate model is used to expedite a waterflooding optimization problem where design variables are BHP controls to maximize the lifecycle Net Present Value. The proposed optimization strategy is based on a trust region framework. It decomposes the original problem into a sequence of local problems performed on the surrogate model bounded by a trust region whose extent is adaptively managed by the strategy during the optimization process depending on surrogate accuracy. Should surrogate accuracy deteriorate as a result of changes in well controls during the optimization process TPWL/POD model must be retrained to incorporate new state snapshots corresponding to those controls. This work proposes TPWL/POD retraining criteria based on the trust region accuracy parameter and an error indicator that represents the average distance between stored snapshots and the corresponding simulation states. The proposed optimization strategy is applied to a 24000 cell reservoir based on SPE-10 problem with two injector and four producer wells considering four control cycles. Differences in fluid densities and fluid compressibilities are take into account increasing problem nonlinearity. Different trends are used for the correction model. A parameter study is conducted to fine tune the proposed correction criteria. Excellent results are obtained with NPV values exceeding those obtained by coupling the SQP optimizer directly to the simulator as well as the TPWL/POD approximation with no correction or retraining. The strategy proved to be very effective because of the reuse of previous computation through stored Jacobians and continuous refinement of the kriging correction model. Unnecessary retraining simulations were avoided by the criteria which can be used by other surrogate based strategies making use of TPWL/POD approximation.
The use of proxy models for optimisation of expensive functions has demonstrated its value since the 1990's in many industries. Within reservoir engineering, similar techniques have been used for over a decade for history matching, both in commercial tools and in-house software.
In addition to efficient history matching, proxy models have a distinct advantage when performing uncertainty quantification of probabilistic forecasts. Markov Chain Monte Carlo (MCMC) methods cannot realistically be applied directly with reservoir simulations, and even fast proxy models can fail dramatically to adequately represent the range of uncertainty if implemented without due care.
A pitfall of the use of proxy models is that they are considered ‘black box’ and their quality is difficult to measure. Engineers prefer to deal with deterministic simulation models which they can evaluate and understand.
The main pitfall of simple random walk MCMC techniques, which have begun to appear within reservoir engineering workflows, is a focus on theoretical properties which are not observed in practical implementations. This gives rise to potential gross errors, which are not generally appreciated by practitioners. Advances in recent years within the field of Bayesian statistics have significantly improved this situation, but have not yet been disseminated within the oil and gas industry.
This paper describes the limitations of random walk MCMC techniques which are currently used for reservoir prediction studies, and shows how Hamiltonian MCMC techniques, together with an efficient implementation of proxy models, can lead to a more reliable and validated probabilistic uncertainty quantification, whilst also generating a suitable ensemble of deterministic reservoir models. Scientific comparison studies are performed for both an analytical case and a realistic reservoir simulation case to demonstrate the validity of the approach.
The benefit of this methodology is to allow asset teams to effectively manage reservoir decisions using a robust and validated understanding of uncertainty. It lays the scientific foundations for the next generation of uncertainty tools and workflows.
Static high resolution three dimensional geological models are routinely constructed to provide an integrated description of a reservoir which includes seismic, well log, and core data, and which characterize the reservoir heterogeneity at multiple scales. These models also represent the structure and stratigraphy of the reservoir within the design of the modeling grid, which may include fault blocks, faults, pinch-outs, layering and cross-bedding. The growth of computational resources has remained rapid, and both geologic models and flow simulation models have increased in size. 50 million cell geologic models are routine, while simulation models are typically one or two orders of magnitude coarser. Hence upscaling of the geologic models for flow simulation remains part of the subsurface workflows.
The industry also faces new reservoir engineering challenges. Unconventional reservoirs (tight gas / shale oil / shale gas) have sufficiently low permeabilities that the time for pressure transients are no longer measured in hours or days, but instead are measured in decades or longer. The separation between transient testing and steady state reservoir management is no longer applicable. Similarly, our upscaling algorithms have relied upon steady state concepts of flow, which may no longer be applicable.
In the current study, a novel diffuse source transmissibility upscaling approach is described. It applies pressure transient concepts to the calculation of the effective transmissibility between reservoir simulation coarse cell pairs. Unlike the usual steady state upscaling algorithms, it is a completely local calculation and is not dependent upon knowledge of, or assumptions about, global reservoir flow patterns. It is well suited to performance prediction within unconventional reservoirs as it utilizes drainage volume concepts to the calculation of coarse cell average pressures, although its use is not restricted to unconventional reservoirs. The approach is tested using the conventional reservoir SPE10 waterflood data set. It is then validated at field scale using an onshore US tight gas reservoir model. The approach is shown to reduce simulation run times by up to two orders of magnitude without significant loss of accuracy in performance prediction.