In this study we explore the use of multilevel derivative-free optimization for history matching, with model properties described using PCA-based parameterization techniques. The parameterizations applied in this work are optimization-based PCA (O-PCA) and convolutional neural network-based PCA (CNN-PCA). The latter, which derives from recent developments in deep learning, is able to represent accurately models characterized by multipoint spatial statistics. Mesh adaptive direct search (MADS), a pattern search method that parallelizes naturally, is applied for the optimizations required to generate posterior (history matched) models. The use of PCA-based parameterization reduces considerably the number of variables that must be determined during history matching (since the dimension of the parameterization is much smaller than the number of grid blocks in the model), but the optimization problem can still be computationally demanding. The multilevel strategy introduced here addresses this issue by reducing the number of simulations that must be performed at each MADS iteration. Specifically, the PCA coefficients (which are the optimization variables after parameterization) are determined in groups, at multiple levels, rather than all at once. Numerical results are presented for 2D cases, involving channelized systems (with binary and bimodal permeability distributions) and a deltaic-fan system, using O-PCA and CNN-PCA parameterizations. O-PCA is effective when sufficient conditioning (hard) data are available, but it can lead to geomodels that are inconsistent with the training image when these data are scarce or nonexistent. CNN-PCA, by contrast, can provide accurate geomodels that contain realistic features even in the absence of hard data. History matching results demonstrate that substantial uncertainty reduction is achieved in all cases considered, and that the multilevel strategy is effective in reducing the number of simulations required. It is important to note that the parameterizations discussed here can be used with a wide range of history matching procedures (including ensemble methods), and that other derivative-free optimization methods can be readily applied within the multilevel framework.
Accurate and robust well modeling is essential for performing reservoir simulations of practical interest. The Multi-Segment well (MSWell) model is able to describe the well topology and accurately represent the multiphase multicomponent flow and transport behavior in the wellbore. The fully coupled method (FC) has been developed and widely applied on coupled reservoir and MSWell modeling due to its unconditional stability and consistent implementation. A local well solver can be applied to provide a better nonlinear precondition for MSWell variables in order to accelerate the nonlinear convergence of the FC method.
However, solving the coupled MSWell and reservoir model in a fully implicit scheme can still present limitations on some practical applications. First, the well or surface facility solver can be separate from the existing reservoir simulator, making it challenging to employ the fully implicit method. Second, complex linear and nonlinear solvers need to be designed to pair the specific wells and reservoir models. These solvers have to account for the different flow characteristics and discretization domains between reservoir and MSWell. A sequential coupling scheme can become preferable in such situations.
Sequential fully Implicit method (SFI) splits the fully coupled reservoir and MSWell equations into two parts and solves them sequentially. In spite of accomplishing an implicit coupling in a sequential scheme, SFI suffers the slow outer loop convergence rate especially when reservoir is strongly coupled with the wells, which is very often the case. The slow convergence is caused by the linear convergence rate of the fix point iteration used in the SFI. Here, we developed a sequential implicit Newton's method (SIN) for coupled MSWells. SIN incorporates a Newton update at the end of each sequential step to achieve a quadratic convergence of outer iterations, while require a limited extra computational cost. Numerical results show that SIN attains comparable nonlinear Newton iterations with the FC in the coupled heterogeneous reservoir and complex MSWell problems.
Pan, Huanquan (Stanford University) | Imai, Motonao (Japan Oil, Gas and Metals National Corporation/Waseda University) | Connolly, Michael (Stanford University) | Tchelepi, Hamdi (Stanford University)
Robust and efficient multiphase flash calculations are crucial in compositional and thermal simulations for complex fluid systems in which three or four phase may co-exist. Solution of the Richford-Rice (RR) equations is an important operation in the multiphase flash. The Newton method generally does not converge during solution of the RR equations unless very good initial values are provided. In this paper, the solution of the RR equations is formulated as a minimization of a convex function problem. For the first time, we use a trust-region (TR) method to solve the RR equations through minimization of the convex function. The Hessian matrix of the convex function is always positive-definite, and the TR-based solver guarantees convergence. The key to successful implementation is to determine the relaxation parameter in the Newton update. We select this relaxation parameter to meet the boundary of the objective function and to ensure an adequate step length. We tested the RR solver for three and four phase RR problems in the construction of phase diagrams. The test cases are representative of complex fluid systems encountered in enhanced oil recovery, including injection of CO2 into low temperature reservoirs and steam injection into heavy oil reservoirs at elevated temperatures. We performed tens of millions of multiphase flash computations, the results of which reveal our RR solver to be robust, efficient, insensitive to initial values, and capable of handling negative phase amounts. We also evaluated the effect of the initial values on convergence and recommend methods to estimate the initial values in our RR solver. In summary, our RR solver greatly improves the multiphase flash calculations and strengthens the coupling of phase equilibrium calculations to the governing equations in multiphase compositional and thermal simulation.
A reduced-order modeling framework is developed and applied to simulate coupled flow-geomechanics problems. The reduced-order model is constructed using POD-TPWL, in which proper orthogonal decomposition (POD), which enables representation of the solution unknowns in a low-dimensional subspace, is combined with tra jectory piecewise linearization (TPWL), where solutions with new sets of well controls are represented via linearization around previously simulated (training) solutions. The over-determined system of equations is pro jected into the lowdimensional subspace using a least-squares Petrov-Galerkin procedure, which has been shown to maintain numerical stability in POD-TPWL models. The states and derivative matrices required by POD-TPWL, generated by an extended version of Stanford's Automatic-Differentiation-based General Purpose Research Simulator, are provided in an offline (pre-processing or training) step. Offline computational requirements correspond to the equivalent of 5-8 full-order simulations, depending on the number of training runs used. Runtime (online) speedups of O(100) or more are typically achieved for new POD-TPWL test-case simulations. The POD-TPWL model is tested extensively for a 2D coupled problem involving oil-water flow and geomechanics. It is shown that POD-TPWL provides predictions of reasonable accuracy, relative to full-order simulations, for well-rate quantities, global pressure and saturation fields, global maximum and minimum principal stress fields, and the Mohr-Coulomb rock failure criterion, for the cases considered. A systematic study of POD-TPWL error is conducted using various training procedures for different levels of perturbation between test and training cases. The use of randomness in the well bottom-hole pressure profiles used in training is shown to be beneficial in terms of POD-TPWL solution accuracy. The procedure is also successfully applied to a prototype 3D example case.
Understanding the effect of injected water salinity is becoming crucial, as it has been shown to have a strong impact on the efficiency of oil recovery process. Various experiments have concluded that carbonate wettability is altered when the water ionic-composition is changed. In this work, a numerical investigation of an oil blob mobilized by water is conducted inside a single pore. The presented model studies the synergy effect of multiphase flow and water salinity at the pore level.
To model the multiphase flow at the pore-scale, the full hydrodynamic Navier-Stokes equations are solved using direct numerical simulation (DNS). The effect of brine ionic-composition is examined through the electric double layer effect. Experimental zeta potential values, published in the literature, of crude oil/water and water/carbonate interfaces have been employed in the model, which capture the water salinity effect.
The simulation results show that the water wetting film surrounding the oil-droplet collapses to an adsorbed nanometer water layer when high salinity water is used. As a result, a large pressure gradient is required to mobilize the oil inside the pore and overcome the attractive surface forces between the oil/water and water/carbonate interfaces. For low-salinity injected water, the carbonate surface becomes more water-wet. The wetting film surrounding the oil blob becomes stable due to the repulsive electric double layer force. Therefore, less energy is required to mobilize the oil blob inside the pore compared to high water salinity. The effect of solid roughness and injected water flow rate are also studied, which show to have a strong impact on the oil displacement efficiency.
The novelty of the numerical method lies in efficiently capturing the nanoscale effect of the electric double layer in pore-scale multiphase flow at the microscale. The simulation results provide fundamental insights on the efficiency of low-salinity waterflooding at the pore level.
The critical gas saturation in permeable sands was studied as a function of depletion rate and the presence of an aqueous phase as the major experimental variables. Voidage-replacement ratios (VRR = injected volume/produced volume) less than 1 were used to obtain pressure depletion with active water injection. Three different live crude oils were considered. Two of the oils are viscous Alaskan crudes with dead-oil viscosities of 87.7 and 600 cp, whereas the third is a light crude oil with a dead-oil viscosity of 9.1 cp. The critical gas saturation for all tests ranged from 4 to 16%. These values for critical gas saturation are consistent with the finding that the gas phase displayed characteristics similar to those of a foamy oil. For a given oil and depletion rate, the critical gas saturation was somewhat larger for VRR = 0 than it was for VRR = 0.7. The oil recovery correlates with the critical gas saturation (i.e., for a given VRR, tests exhibit greater oil recovery when the critical gas saturation is elevated). For the conditions tested, there was not a strong correlation of critical gas saturation over more than two orders of magnitude of the rate of pressure depletion, for a given VRR. Such behavior might be consistent with theoretical studies reported elsewhere that suggest that the critical gas saturation is independent of the pressure-depletion rate when the rate of depletion is small.
This paper presents an analysis of the interactions between stimulation design and two important geomechanical effects: the variation of least principal stress (Shmin) between lithological layers and the stress shadow effect that arises from simultaneously propagating adjacent hydraulic fractures. To demonstrate these interactions, hydraulic fracture propagation is modeled with a 5-layer geomechanical model representing an actual case study. The model consists of a profile of Shmin measurements made within, below and above the producing interval. The stress variations between layers leads to an overall upward fracture propagation and proppant largely above the producing interval. This is due to interactions between the pressure distribution within the fracture and the stress contrast in the multiple layers. A sensitivity study is done to investigate the complex 3-D couplings between geomechanical constraints and well completion design parameters such as landing zone, cluster spacing, perforation diameter, flow rate and proppant concentration. The simulation results demonstrate the importance of a well characterized stress stratigraphy for prediction of hydraulic fracture characteristics and optimization of operational parameters.
Enhanced-oil-recovery (EOR) processes involve complex flow, transport, and thermodynamic interactions; as a result, compositional simulation is necessary for accurate representation of the physics. Flow simulation of compositional systems with high-resolution reservoir models is computationally intensive because of the large number of unknowns and the strong nonlinear interactions. Thus, there is a great need for upscaling methods of compositional processes. The complex multiscale interactions between the phase behavior and the heterogeneities lie at the core of the difficulty in constructing consistent upscaling procedures.
We use a mass-conservative formulation and introduce upscaled phase-molar-mobility functions for coarse-scale modeling of multiphase flow. These upscaled flow functions account for the subgrid effects caused by the absolute permeability and relative permeability variations, as well as the effects of compressibility. Upscaling of the phase behavior is performed as follows. We assume that instantaneous thermodynamic equilibrium is valid on the fine scale, and we derive coarse-scale equations in which the phase behavior may not necessarily be at equilibrium. The upscaled thermodynamic functions, which represent differences in the component fugacities, are used to account for the nonequilibrium effects on the coarse scale. We demonstrate that the upscaled phase-behavior functions transform the equilibrium phase space on the fine scale to a region of similar shape, but with tilted tie-lines on the coarse space. The numerical framework uses K-values that depend on the orientation of the tie-lines in the new nonequilibrium phase space and the sign of upscaled thermodynamic functions.
The proposed methodology is applied to challenging gas-injection problems with large numbers of components and highly heterogeneous permeability fields. The K-value-based coarse-scale operator produces results that are in good agreement with the fine-scale solutions for the quantities of interest, including the component overall compositions and saturation distributions.
Permanent downhole gauges (PDGs) provide a continuous record of pressure, temperature, and, sometimes, flow rate during well production. The continuous record provides rich information about the reservoir and makes PDG data a valuable source for reservoir analysis (e.g., pressure-rate deconvolution for reservoir-model identification). It has been shown in previous work that the convolution-kernel (CK) -based data-mining approach is a promising tool to interpret flow-rate and pressure data from PDGs. The CK method denoises and deconvolves the pressure signal successfully without an explicit-breakpoint detection. However, the bottlenecks of computational efficiency and the incomplete recovery of reservoir behaviors limit the application of the method to interpret real-PDG data.
In this paper, three different machine-learning techniques were applied to flow-rate/pressure interpretation. We formulated the machine-learning techniques into a linear regression (LR) on parameters that connect the nonlinear flow-rate features with pressure targets. Such a formulation leads to a closed-form solution, which speeds up the computation dramatically. The machine-learning algorithms that were formulated using LR were shown to have the same learning quality as the CK method, and they outperformed it with much less computational effort. Next, the kernel method was applied to address the issue of the incomplete recovery of reservoir behaviors, because it efficiently expanded the dimension of the feature space without an explicit representation of the features, but it led to overfitting. Finally, kernel ridge regression (KRR) used the expanded features given by the kernel function to capture the more detailed reservoir behaviors, while controlling the prediction error using ridge regression (RR). It was shown that KRR recovers the full reservoir behaviors successfully (e.g., wellbore-storage effect, skin effect, infinite-acting radial flow, and boundary effect).
Some potential uses of temperature data from PDGs are also discussed in this paper. Machine learning was shown to be able to model the temperature and pressure data recorded by PDGs, even if the actual physical model is complex. This originates from the fact that, by using features as an approximation of model characteristics, machine learning does not require a perfect knowledge of the physical model. The modeling of pressure using temperature data was extended to two promising applications: pressure-history reconstruction using temperature data, and the cointerpretation of temperature and pressure data when flow-rate data are not available.
Orr, Franklin M. (Stanford University)
Franklin M. Orr Jr., Stanford University Summary Recent progress in carbon capture, utilization, and storage (CCUS) is reviewed. Considerable experience has now been built up in enhanced-oil-recovery (EOR) operations, which have been under way since the 1970s. Storage in deep saline aquifers has also been achieved at scale. Introduction The challenge of making deep reductions in greenhouse gas (GHG) emissions in this century is a daunting one given the scale of the use of energy by humans and our current dependence on fossil fuels, which provide essential energy services at low cost to modern societies. Meeting the challenge of reducing GHG emissions will require a fully diversified portfolio of approaches, such as much more energy-efficient end-use technologies (e.g., cars, home and business heating and air conditioning, lighting); electrification of energy services coupled with reduced GHG emissions from electric power generation; fuel switching in transportation and electric power generation; deployment of additional renewable power generation; land-use changes toward lower-emission agriculture; emission reductions of short-term forcers such as black carbon, CH Integrated assessments of the various pathways indicate that portfolios that include significant deployment of CCUS have lower estimated costs than those without CCUS (Clarke et al. 2014; Krey et al. 2014). In 2005, the Intergovernmental Panel on Climate Change (IPCC) issued a detailed special report (SRCCS) on many aspects of carbon capture and storage (CCS) (Metz et al. 2005). Wilcox (2012) provided detailed descriptions of specific capture technologies and their energy requirements, as did Boot-Handford et al. (2014), who gave additional commentary on pipeline transportation issues, subsurface storage issues, and a European policy perspective.