Araujo, Mariela (Shell International Exploration and Production Inc.) | Chen, Chaohui (Shell International Exploration and Production Inc.) | Gao, Guohua (Shell International Exploration and Production Inc.) | Jennings, Jim (Shell International Exploration and Production Inc.) | Ramirez, Benjamin (Shell International Exploration and Production Inc.) | Xu, Zhihua (ExxonMobil) | Yeh, Tzu-hao (Shell International Exploration and Production Inc.) | Alpak, Faruk Omer (Shell International Exploration and Production Inc.) | Gelderblom, Paul (Shell International Exploration and Production Inc.)
Increased access to computational resources has allowed reservoir engineers to include assisted history matching (AHM) and uncertainty quantification (UQ) techniques as standard steps of reservoir management workflows. Several advanced methods have become available and are being used in routine activities without a proper understanding of their performance and quality. This paper provides recommendations on the efficiency and quality of different methods for applications to production forecasting, supporting the reservoir-management decision-making process.
Results from five advanced methods and two traditional methods were benchmarked in the study. The advanced methods include a nested sampling method MultiNest, the integrated global search Distributed Gauss-Newton (DGN) optimizer with Randomized Maximum Likelihood (RML), the integrated local search DGN optimizer with a Gaussian Mixture Model (GMM), and two advanced Bayesian inference-based methods from commercial simulation packages. Two traditional methods were also included for some test problems: the Markov-Chain Monte Carlo method (MCMC) is known to produce accurate results although it is too expensive for most practical problems, and a DoE-proxy based method widely used and available in some form in most commercial simulation packages.
The methods were tested on three different cases of increasing complexity: a 1D simple model based on an analytical function with one uncertain parameter, a simple injector-producer well pair in the SPE01 model with eight uncertain parameters, and an unconventional reservoir model with one well and 24 uncertain parameters. A collection of benchmark metrics was considered to compare the results, but the most useful included the total number of simulation runs, sample size, objective function distributions, cumulative oil production forecast distributions, and marginal posterior parameter distributions.
MultiNest and MCMC were found to produce the most accurate results, but MCMC is too costly for practical problems. MultiNest is also costly, but it is much more efficient than MCMC and it may be affordable for some practical applications. The proxy-based method is the lowest-cost solution. However, its accuracy is unacceptably poor.
DGN-RML and DGN-GMM seem to have the best compromise between accuracy and efficiency, and the best of these two is DGN-GMM. These two methods may produce some poor-quality samples that should be rejected for the final uncertainty quantification.
The results from the benchmark study are somewhat surprising and provide awareness to the reservoir engineering community on the quality and efficiency of the advanced and most traditional methods used for AHM and UQ. Our recommendation is to use DGN-GMM instead of the traditional proxy-based methods for most practical problems, and to consider using the more expensive MultiNest when the cost of running the reservoir models is moderate and high-quality solutions are desired.
Gao, Guohua (Shell Global Solutions, US Inc.) | Vink, Jeroen C. (Shell Global Solutions International B.V.) | Chen, Chaohui (Shell International Exploration & Production Inc.) | Araujo, Mariela (Shell International Exploration & Production Inc.) | Ramirez, Benjamin (Shell International Exploration & Production Inc.) | Jennings, Jim W. (Shell International Exploration & Production Inc.) | Khamra, Yaakoub El (Shell Global Solutions, US Inc.) | Ita, Joel (Shell Global Solutions, US Inc.)
Uncertainty quantification of production forecasts is crucially important for business planning of hydrocarbon field developments. This is still a very challenging task, especially when subsurface uncertainties must be conditioned to production data. Many different approaches have been proposed, each with their strengths and weaknesses. In this work, we develop a robust uncertainty quantification workflow by seamless integration of a distributed Gauss-Newton (DGN) optimization method with Gaussian Mixture Model (GMM) and parallelized sampling algorithms. Results are compared with those obtained from other approaches.
Multiple local maximum-a-posteriori (MAP) estimates are located with the local-search DGN optimization method. A GMM is constructed to approximate the posterior probability density function, by fitting simulation results generated during the DGN minimization process. The traditional acceptance-rejection (AR) algorithm is parallelized and applied to improve the quality of GMM samples by rejecting unqualified samples. AR-GMM samples are independent, identically-distributed (i.i.d.) samples that can be directly used for uncertainty quantification of model parameters and production forecasts.
The proposed method is first validated with 1-D nonlinear synthetic problems having multiple MAP points. The AR-GMM samples are better than the original GMM samples. Then, it is tested with a synthetic history-matching problem using the SPE-1 reservoir model with 8 uncertain parameters. The proposed method generates conditional samples that are better than or equivalent to those generated by other methods, e.g., Markov chain Monte Carlo (MCMC) and global search DGN combined with the Randomized Maximum Likelihood (RML) approach, but have a much lower computational cost (by a factor of 5 to 100). Finally, it is applied to a real field reservoir model with synthetic data, having 235 uncertain parameters. A GMM with 27 Gaussian components is constructed to approximate the actual posterior PDF. 105 AR-GMM samples are accepted from the 1000 original GMM samples, and are used to quantify uncertainty of production forecasts. The proposed method is further validated by the fact that production forecasts for all AR-GMM samples are quite consistent with the production data observed after the history matching period.
The newly proposed approach for history matching and uncertainty quantification is quite efficient and robust. The DGN optimization method can efficiently identify multiple local MAP points in parallel. The GMM yields proposal candidates with sufficiently high acceptance ratios for the AR algorithm. Parallelization makes the AR algorithm much more efficient, which further enhances the efficiency of the integrated workflow.
Alfred, Dicman (Marathon Oil Corporation) | Ramirez, Benjamin (Marathon Oil Corporation) | Rodriguez, Jesus (Marathon Oil Corporation) | Hlava, Kimberly (Marathon Oil Corporation) | Williams, Darren (Marathon Oil Corporation)
Fine-scale, reservoir models are necessary for locating high performance zones in unconventional reservoirs. The reservoir in the Woodford play is a naturally fractured, heterogeneous mudrock. The most common technique for characterizing the natural fracture system in such a reservoir is use of a DFN (discrete fracture network) model. However, such models are data intensive, time consuming to build and may provide non-unique solutions. We propose a workflow that estimates the contribution from fractures through a continuum model using well performance and static attributes derived from well observations and seismic data. The utility of the workflow is demonstrated by a blind test against a local sector model history match that shows excellent agreement with production history.
The integrated workflow includes a purpose-built stratigraphic framework for the Woodford. The Woodford was divided into five intervals using facies changes, biostratigraphic information, and log signatures. Core descriptions and x-ray diffraction data indicate total organic carbon and clay can be used to discriminate distinct depositional facies and intrinsic reservoir properties within the Woodford. Correlations that relate total organic carbon and clay volumes to seismic information, such as acoustic impedance and bulk density, enable the construction of petro-elastic models. Using petro-elastic models, a petrophysical seismic inversion is used to derive density, total organic carbon, and clay volumes. Structural seismic attributes such as dip, curvature, and coherence are also incorporated into the model. Finally, correlations obtained from multivariate statistics of well performance data and static model properties are applied to create spatial distributions of natural fracture drivers including: effective permeability and shape factor distributions across the field.
Advances in horizontal drilling and multistage hydraulic fracturing have made production from low permeability reservoirs possible and economical. Careful measurements of core permeability in shale reservoirs indicate that matrix permeability is in the nano-Darcy range. These permeabilities are too low to support economic flow rates that have been achieved in such reservoirs. Both empirically and theoretically, it can be concluded that production success is due to the multistage hydraulic fracture stimulation, which creates a large number of interconnected micro- and macro-fractures near the horizontal wellbore to support economic production rates. Using this hypothesis, we can simulate the reservoir performance of unconventional shale reservoirs both with single-porosity and dual-porosity mathematical models. In fact, the model parameters can be adjusted to provide an explanation for the decline of the rate exponent in the hyperbolic decline analysis, as we have reported in an earlier paper.
In this paper we present the details of the reservoir modeling philosophy and gridding methodology applied to an abnormally high-pressure, unconventional shale reservoir. Results for single- and dual-porosity models will be presented and compared with the decline curve analysis (DCA) results. Furthermore, the effect of gas condensation in the pores is discussed both from the flow and thermodynamics points of view. It is concluded that much uncertainty exists about the exact nature of flow and production mechanisms in low-permeability shale reservoirs; nonetheless, one can predict future performance with acceptable engineering accuracy and reliability using intrinsically different models.
Yetkin, Can (Nitec LLC) | Ramirez, Benjamin (Colorado School of Mines) | Al-kobaisi, Mohammed (Colorado School of Mines) | Kazemi, Hossein (Colorado School of Mines) | Ozkan, Erdal (Colorado School of Mines)
Permeability anisotropy, via full permeability tensor, has not been accurately accounted for in reservoir models because of the implementation complexity. Specifically, numerical implementation of the off-diagonal components of the permeability tensor is inconvenient and cumbersome. This paper shows how directional permeability, calculated from a full permeability tensor, can be used as a simple replacement both in coding numerical models and in day-to-day engineering analysis. For the former, we have implemented the directional permeability for single-phase flow in a 9-point finite-difference formulation, which is easy to code. This formulation, however, is easily applicable to any control-volume formulation including the perpendicular bisector (PEBI) grid. The implementation of this technique has produced excellent numerical results in numerical simulation of multi-phase flow displacement. For routine engineering applications, we have also applied this technique to generate pressure responses for a four-well interference test in a highly anisotropic system. The analysis of the test results by conventional type-curve matching produced the correct reservoir geometric-average permeability and a very good approximation for the direction of the maximum permeability, which is a testimony to the credibility of the formulation. The use of this formulation should be very useful in determining major natural fracture trends in reservoirs undergoing water or gas injection, and in modeling fracture trends in other fractured reservoir situations, such as tight sands.
Al-kobaisi, Mohammed (Colorado School of Mines) | Kazemi, Hossein (Colorado School of Mines) | Ramirez, Benjamin (Marathon Oil Co.) | Ozkan, Erdal (Colorado School of Mines) | Atan, Safian (Marathon Oil Co.)
This paper continues the work presented in Ramirez et al. (2009). In Part I, we discussed the viability of the use of simple transfer functions to accurately account for fluid exchange as the result of capillary, gravity, and diffusion mass transfer for immiscible flow between fracture and matrix in dual-porosity numerical models. Here, we show additional information on several relevant topics, which include (1) flow of a low-concentration water-soluble surfactant in the fracture and the extent to which the surfactant is transported into the matrix; (2) an adjustment to the transfer function to account for the early slow mass transfer into the matrix before the invading fluid establishes full connectivity with the matrix; and (3) an analytical approximation to the differential equation of mass transfer from the fracture to the matrix and a method of solution to predict oil-drainage performance.
Numerical experiments were performed involving single-porosity, fine-grid simulation of immiscible oil recovery from a typical matrix block by water, gas, or surfactant-augmented water in an adjacent fracture. Results emphasize the viability of the transfer-function formulations and their accuracy in quantifying the interaction of capillary and gravity forces to produce oil depending on the wettability of the matrix. For miscible flow, the fracture/matrix mass transfer is less complicated because the interfacial tension (IFT) between solvent and oil is zero; nevertheless, the gravity contrast between solvent in the fracture and oil in the matrix creates convective mass transfer and drainage of the oil.
Characterization and quantification of fractures in naturally fractured reservoirs is a very difficult task; nonetheless, when natural fractures contribute significantly to fluid movement and hydrocarbon drainage in the reservoir, a dual-porosity approach is adopted to quantify reservoir performance. The dual-porosity concept can be perceived and quantified in several ways, as shown in Fig. 1.
The dual-porosity concept was conceived on the premise that a very highly conductive fracture medium was formed as an interconnected network of secondary porosity within a pre-existing porous rock of primary porosity. A third medium of lower-conductivity fractures (i.e., microfractures) can be added to the flow system in some important applications. Regardless of the formulation, the flow in the high-conductivity fracture network takes place at high velocities from one grid cell to another irrespective of the flowing phase. In two- or three-phase flow, there is usually a local exchange of fluids between the fractures and the adjacent matrix at comparatively low velocities. Contrast in fluid velocities in the two flow systems is a very important issue in naturally fractured reservoirs because, in multiphase flow, typically water or gas can move rapidly in the fractures and surround the matrix blocks partially or totally. Once a matrix block is surrounded partially or totally by a particular fluid, then transfer of fluid phases and components takes place between the fracture and matrix. Deciphering the recovery mechanisms and describing the pertinent equations of mass transfer constitute the heart of this paper--both Part I (Ramirez et al. 2009) and Part II. Similar issues extend to any variants of the dual-porosity concept, such as the triple-porosity, irrespective of the idealization concept.
Balogun, Adetayo S. (Shell E&P) | Kazemi, Hossein (Colorado School of Mines) | Ozkan, Erdal (Colorado School of Mines) | Al-kobaisi, Mohammed (Colorado School of Mines) | Ramirez, Benjamin (Colorado School of Mines)
Accurate calculation of multiphase fluid transfer between the fracture and matrix in naturally fractured reservoirs is a very crucial issue. In this paper, we will present the viability of the use of a simple transfer function to accurately account for fluid exchange resulting from capillary and gravity forces between fracture and matrix in dual-porosity and dual-permeability numerical models. With this approach, fracture- and matrix-flow calculations can be decoupled and solved sequentially, improving the speed and ease of computation. In fact, the transfer-function equations can be used easily to calculate the expected oil recovery from a matrix block of any dimension without the use of a simulator or oil-recovery correlations.
The study was accomplished by conducting a 3-D fine-grid simulation of a typical matrix block and comparing the results with those obtained through the use of a single-node simple transfer function for a water-oil system. This study was similar to a previous study (Alkandari 2002) we had conducted for a 1D gas-oil system.
The transfer functions of this paper are specifically for the sugar-cube idealization of a matrix block, which can be extended to simulation of a match-stick idealization in reservoir modeling. The basic data required are: matrix capillary-pressure curves, densities of the flowing fluids, and matrix block dimensions.
Naturally fractured reservoirs contain a significant amount of the known petroleum hydrocarbons worldwide and, hence, are an important source of energy fuels. However, the oil recovery from these reservoirs has been rather low. For example, the Circle Ridge Field in Wind River Reservation, Wyoming, has been producing for 50 years, but the oil recovery is less than 15% (Golder Associates 2004). This low level of oil recovery points to the need for accurate reservoir characterization, realistic geological modeling, and accurate flow simulation of naturally fractured reservoirs to determine the locations of bypassed oil.
Reservoir simulation is the most practical method of studying flow problems in porous media when dealing with heterogeneity and the simultaneous flow of different fluids. In modeling fractured systems, a dual-porosity or dual-permeability concept typically is used to idealize the reservoir on the global scale. In the dual-porosity concept, fluids transfer between the matrix and fractures in the grid-cells while flowing through the fracture network to the wellbore. Furthermore, the bulk of the fluids are stored in the matrix. On the other hand, in the dual-permeability concept, fluids flow through the fracture network and between matrix blocks.
In both the dual-porosity and dual-permeability formulations, the fractures and matrices are linked by transfer functions. The transfer functions account for fluid exchanges between both media. To understand the details of this fluid exchange, an elaborate method is used in this study to model flow in a single matrix block with fractures as boundaries. Our goal is to develop a technique to produce accurate results for use in large-scale modeling work.
Ramirez, Benjamin (Marathon Oil Co.) | Kazemi, Hossein (Colorado School of Mines) | Al-kobaisi, Mohammed (Colorado School of Mines) | Ozkan, Erdal (Colorado School of Mines) | Atan, Safian (Marathon Oil Co.)
Accurate calculation of multiphase-fluid transfer between the fracture and matrix in naturally fractured reservoirs is a crucial issue. In this paper, we will present the viability of the use of simple transfer functions to account accurately for fluid exchange resulting from capillary, gravity, and diffusion mass transfer for immiscible flow between fracture and matrix in dual-porosity numerical models. The transfer functions are designed for sugar-cube or match-stick idealizations of matrix blocks.
The study relies on numerical experiments involving fine-grid simulation of oil recovery from a typical matrix block by water or gas in an adjacent fracture. The fine-grid results for water/oil and gas/oil systems were compared with results obtained with transfer functions. In both water and gas injection, the simulations emphasize the interaction of capillary and gravity forces to produce oil, depending on the wettability of the matrix.
In gas injection, the thermodynamic phase equilibrium, aided by gravity/capillary interaction and, to a lesser extent, by molecular diffusion, is a major contributor to interphase mass transfer. For miscible flow, the fracture/matrix mass transfer is less complicated because there are no capillary forces associated with solvent and oil; nevertheless, gravity contrast between solvent in the fracture and oil in the matrix creates convective mass transfer and drainage of oil.
Using the transfer functions presented in this paper, fracture- and matrix-flow calculations can be decoupled and solved sequentially--reducing the complexity of the computation. Furthermore, the transfer-function equations can be used independently to calculate oil recovery from a matrix block.