Estimation of Flow Functions During Drainage Using Genetic Algorithm

This paper presents a procedure to estimate relative permeability curves together with gridblock porosities and permeabilities by automatic history matching three-phase flow production data using the LBFGS algorithm for optimization and the adjoint method for generating sensitivity coefficients. Both power law and B-splines are considered for the representation of relative permeability curves. Power law is a simple representation; however, B-splines have the advantage of being able to accurately represent any set of relative permeability curves. In the B-spline representation, we provide a simple procedure based on a transformation of parameters (control points) defining the B-splines to ensure that monotonic curves are obtained. Without this transformation, the automatic history-matching process fails. If endpoint saturations are included as parameters, we show how to modify the adjoint procedure to account for the fact that initial conditions are sensitive to endpoint saturations. While the history-matching process is inherently non-unique, we show that reasonable estimates of relative permeability curves and porosity/ permeability fields can be obtained for a synthetic reservoir example. We also show the uncertainty quantification of the model parameters and reservoir performance prediction using the Randomized Maximum Likelihood (RML) method. Introduction Relative permeabilities are important parameters in reservoir performance calculations. Usually the relative permeability curves are obtained from labs through core flood tests. They can also be incorporated into the history matching procedure as model parameters. ARCHER and WONG (1973), and YANG and WATSON (1991) were among the early researchers to consider the estimation of relative permeability curves by history matching laboratory core flood data. A detailed literature review is given in REYNOLDS et al. (2004). REYNOLDS et al. (2004) also discussed the estimation of three-phase relative permeabilities by history matching roduction data. The current work is a natural extension of REYNOLDS et al. (2004). In this paper, we provide an alternative method for relative permeability estimation based on B-spline approximation. Compared to power law representation, Bsplines ave the advantage of being able to accurately represent any set of relative permeability curves. In the B-spline representation, we provide a simple procedure based on logtransformation of parameters (control points) defining the Bsplines to ensure the B-spline curves are monotonic or convex as used in CHEN et al. (2005). Since a reservoir simulator requires monotonic relative permeability curves, without this transformation, the automatic history-matching process may generate non-monotonic relative permeability curves and result in failed reservoir simulation run. A prior model for porosity and absolute permeability and relative permeability parameters is assumed to provide regularization, i.e., Bayesian estimation is applied to generate estimates. Model parameters, which are estimated by automatic history matching of production data, consist of gridblock porosties, gridblock absolute logpermeabilities and the parameters defining the relative permeability curves. We do not assume the endpoint saturations (i.e. irreducible water saturation, critical gas saturation and residual oil saturation) to be known and consider them as independent model parameters in history matching procedure.

The history matching, a key step in the reservoir numerical simulation, is performed to obtain a set of appropriate reservoir parameters to match the modeling results with the oilfield historical production data. The history matching problem is solved conventionally with Gauss-Newton, L-M, BFGS, and LBFGS methods, which require calculation of the gradient of the objective function, and is very slow in solving large-scale problems. In recent years, modern optimization algorithms, such as the genetic algorithm, particle swarm optimization, and differential evolution algorithm, have been applied in the reservoir history matching. In this paper, the hybrid particle swarm optimization and gravitational search algorithm (PSOGSA) is used to carry out the research on the automatic reservoir history matching. First, the basic concepts of the particle swarm optimization (PSO) and the gravity search algorithm (GSA) were introduced. Then, the concept and algorithm of PSOGSA were described. Moreover, the actual oil field data were used to establish a geological model and an objective function, and the PSOGSA was used to carry out the automatic history matching. The PSOGSA was compared with other algorithms. Finally, the effects of parameters on the PSOGSA were analyzed. The results show that the PSOGSA has a good performance in the automatic reservoir history matching, increases the diversity of particles, improves the global search capability, and accelerates the convergence. The PSOGSA is superior to the PSO or the GSA.

An automatic history-matching algorithm has been developed to determine relative permeabilities and capillary pressure curves simultaneously from the production history of a displacement test of oil and water through a core. Relative permeability and capillary pressure curves are represented by power functions. Those representations contain some parameters which are found by minimizing an objective function. The objective function is formed by the sum of the square of the differences between experimentally measured and numerically simulated production data. The numerical simulator is an IMPES finite-difference program which models the one dimensional two-phase flow. The minimization is performed by a non-linear regression algorithm: the Quasi-Newton Approximation for the Least-Squares Problem (Broyden-Fletcher-Goldfarb-Shanno formulae). The automatic history Bitching algorithm is applied to analyze the influence of capillary pressure on the determination of relative permeability curves. The algorithm is tested with simulated data and with actual experimental data. Database consisting of simulated experiments with random errors are tested to demonstrate the feasibility of the method. Besides, they provide exact relative permeability curves for comparison purposes. In effect, results obtained by using simulated data show the convergence and uniqueness of the algorithm. But the main observation is that relative permeabilities estimates can be in error when capillary pressure terms are neglected. Several sets of actual experimental data at inlet constant pressure are also tested. Convergent solutions are always found. Three different objective functions are applied. They are formed by oil rates, total fluid rates or a combination of both. The solution is dependent on the chosen objective function. Best adjustments are found by applying a combination of oil and total fluid rates, and including capillary pressure.

Nonlinear time series model is a time series model applied to data that has the nonlinear pattern. One of the nonlinear time series models is Self-Exciting Threshold Autoregressive (SETAR). The SETAR model is a time series model that data modeling is done by dividing data into multiple regimes, whereas each regime following an autoregressive (AR) model. The division of the regime based on the score of the delay and threshold of the data itself. The number of SETAR model parameters not only resulted from the best model search process but also resulted in a SETAR model that is not yet optimum. Based on these findings, this study used Genetic Algorithm (GA) to produce the best and optimum SETAR model. In this research, using SETAR simulation data modeling and return data of Bank Rakyat Indonesia (BRI) were performed. The method used to model the data is Grid Search (GS) and Genetic Algorithm (GA). The result of analysis of SETAR simulation data shows that GA method gives better modeling result than GS method. The GA motive AIC value for the amount of 200 data is -3.976178 which is smaller than the AIC GS method of 1.361723. For the amount of data of 500 AIC values, GA method is also smaller than AIC GS method. In BRI stock return data, GA method also gives better modeling result compared to GS. It is marked by the GA AIC method value of -11147.66 less than -11146.26 which is the AIC method of GS. Thus, the result of analysis of SETAR model simulation data and BRI stock return shows that GA method gives better modeling result compared to GS method based on generated AIC value.

Numerical analysis of circular straight mufflers equipped with three chambers at high-order-modes

IntroductionIn recent years, SMPMSM is quietly famous compared with other conventional motors in industrial applications due to higher efficiency, higher reliability, low cost, and smooth torque [1]–[3]. In general, several objective optimizations need to be consider, which required an optimization technique for an electrical motor. This paper deals with an optimal design of a surface-mounted permanent magnet synchronous machine (SMPMSM) with an exact analytical subdomain model by using a genetic algorithm method. To analyze the characteristic of permanent magnet (PM) motors, the classical optimization method, such as the finite element method (FEM), is intensively used. However, FEM has several time problems that require a longer computational time to evaluate the performance of PM motors. This problem can be overcome by using a genetic algorithm (GA) method combined with a subdomain model (SD), which developed an improved performance of SMPMSM, for instance, total harmonic distortion (THDv) and phase back-EMF. In this design, a three-phase 12-slot/8-pole PM motor is established with an exact SD model with radial magnetization patterns. Then, the GA ensemble with SD model to search the optimality of SMPMSM machine design. In the final analysis, the optimal new design of SMPMSM demonstrated by comparing with the initial design that is investigated by FEM. The result of induced back-EMF, and magnetic flux density of optimal design is compared with the initial design to show the advantages of GA optimization method.Analytical Model, Genetic Algorithm and Parameter of optimizationThe purpose of this analysis is to investigate the optimization design of a three-phase 12-slot/8-pole SMPMSM with a radial magnetization pattern. This optimization problem has been solved by obtains the minimum THDv as a fitness function. In order to process the optimization design of SMPMSM, several parameters need to be optimized i.e magnet pole-pitch, αp and slot-opening angle, boa. The roulette method is used to select the objective function of optimization with 50 maximum generations per iteration. The cross and mutation probability need to be set which requires linear and uniform recombination which is 0.89 and 0.025 respectively. The parameters of 12-slot/8-pole SMPM motor as follows: radius of the rotor, Rr = 23mm, the radius of PM surface Rm = 26mm, the inner radius of the stator Rsb = 42.5mm, axial length, la = 50mm, air-gap length, lg = 1mm, remanence of the magnet, Br, 1.12T, relative coil permeability, µr = 1.05 with the number of pole pairs, Np is 8 and number of stator slots, Ns is 12.The result, Discussion and ConclusionThe phase back electromotive force has a significant relationship with flux-linkage because phase back electromotive force is derived from the rate of change of flux-linkage. In Fig. 1. show the waveform of phase back electromotive force against rotor position of initial and optimal design in radial magnetization patterns. The initial design of phase back electromotive force waveform shows more trapezoidal compare with the optimal design which has a sinusoidal waveform. The comparison of total harmonic distortion between the initial and optimal design of SMPMSM are demonstrated in Fig. 2 below. The result from Fig.2. shows the highest total harmonic distortion of 12-slot/8-pole motor is an initial design which is 11.34% compared with the optimal design where 1.3%. The magnitude of harmonic in phase back electromotive force |En| of initial design shows 22.24 V compare to optimal design which is 17.4 V. This shows a good agreement that the optimal motor design gives lowest total harmonic distortion compare with the initial design. The optimizes of SMPMSM motor with the following parameters are given in Table 1. **

Parameter identification is the process of constructing the mathematical model of a dynamic system based on measured dynamic responses. A non-classical search method namely genetic algorithm (GA) is employed in this study, which has several advantages over classical system identification techniques. Nevertheless, direct application of GA does not necessarily work well, particularly with regards to computational efficiency in fine-tuning when the solution approaches the optimal value. Thus, a local search (LS) method is introduced into the GA approach to improve this capability. Numerical example of a cantilever beam is presented to show the efficiency of the combined GA and LS method (GA-LS). The FE model of the cantilever beam is established with 10 beam elements. In this example, only the element properties associated with flapping motion are identified. The GA-LS method gives much better results than the GA method does. The mean error is reduced to 5.3% from 14.9% while to 14.5% from 47.2% for the maximum individual error. It is concluded that the GA-LS method is a more efficient search method than the GA method. The GA-LS method is then applied to identify the mass and stiffness parameters of a model helicopter blade using measured excitations and accelerations in this paper. Since the mass and stiffness of the model helicopter blade are not available, the only way to verify the identified parameters is to compare the ffequencies and mode shapes calculated from the identified parameters and the measured frequencies and modal shapes of the actual blade. The mean eiror of the frequencies is 0.9% and the maximum individual error 2.4%. The sequence of identified mode shapes is the same with measured mode shapes. Transactions on Engineering Sciences vol 43, © 2003 WIT Press, www.witpress.com, ISSN 1743-3533

In an optical Pancharatnam-Berry (PB) phase metasurface, each sub-wavelength dielectric structure of varied spatial orientation can be treated as a point source with the same amplitude yet varied relative phase. In this work, we introduce an optimized genetic algorithm (GA) method for the synthesis of one-dimensional (1D) PB phase-controlled dielectric metasurfaces by seeking for optimized phase profile solutions, which differs from previously reported amplitude-controlled GA method only applicable to generate transverse optical modes with plasmonic metasurfaces. The GA–optimized phase profiles can be readily used to construct dielectric metasurfaces with improved functionalities. The loop of phase-controlled GA consists of initialization, random mutation, screened evolution, and duplication. Here random mutation is realized by changing the phase of each unit cell, and this process should be efficient to obtain enough mutations to drive the whole GA process under supervision of appropriate mutation boundary. A well-chosen fitness function ensures the right direction of screened evolution, and the duplication process guarantees an equilibrated number of generated light patterns. Importantly, we optimize the GA loop by introducing a multi-step hierarchical mutation process to break local optimum limits. We demonstrate the validity of our optimized GA method by generating longitudinal optical modes (i. e., non-diffractive light sheets) with 1D PB phase dielectric metasurfaces having non-analytical counter-intuitive phase profiles. The produced large-area, long-distance light sheets could be used for realizing high-speed, low-noise light-sheet microscopy. Additionally, a simplified 3D light pattern generated by a 2D PB phase metasurface further reveals the potential of our optimized GA method for manipulating truly 3D light fields.

To improve the current industry standard one-parameter-at-a-time sensitivity analysis method, we propose a new sensitivity analysis framework that utilizes Plackett-Burman design and Random Forests (a well-known data mining method). The new framework significantly reduces the number of required simulation runs (i.e. samples), and at the same time compellingly improves the automatic history matching error. The proposed sensitivity analysis framework starts with generating samples/simulations using Plackett-Burman design. Here, each simulation is executed based on different combinations of the parameters' input values. Once the samples are ready, the parameters' input values and the target vector (i.e. the history matching error vector) are used to construct a Random Forests model. This model is used to rank the importance of each history matching parameter. Hence, parameters with low impact on the history matching error are discarded, and the remaining are used for Genetic Algorithm-based automatic history matching. The impact of an internal Random Forests parameter (number of decision trees) on the history matching error is also observed. The highly faulted reservoir with water injection is used for history matching and 10 parameters are defined uncertain consisting of fault transmissibilities, permeabilities, and connate water saturation. The aim is to match the bottom hole pressure and watercut for several oil producing wells. The one-parameter-at-a-time method requires 21 samples, and the selected top 4 parameters from this method are mainly fault transmissibilities. The resulting automatic history matching results gave a final error of 463.338. With the new framework, which requires only 12 samples instead of 21, the final error was 628.041, given that 100 trees are utilized. As we increased the number of trees to 500, the final error is significantly better (i.e. 36% improvement) than the results obtained from the one-parameter-at-a-time method. We can see that using the new framework, the majority of the top 4 parameters are related to permeabilities. Here, there is no significant change in terms of computation time as we increase the number of trees from 100 to 500; total computation time is less than 10 seconds. For this case study, not only this new framework requires significantly less number of initial simulation samples, it significantly improved the history matching error (as compared to the industry standard one-parameter-at-time sensitivity analysis method). Therefore, practicing engineers can utilize this framework to save time and simultaneously improve accuracy of the history matching parameter ranking. Parameter ranking computation with Random Forests takes less than 10 seconds.

Solving a real-world university examination timetabling is an intractable problem. Examination timetabling problems (ETP) can be defined as a problem of assigning a set of final examinations schedules to a given number of timeslots and rooms subject to a set of constraints. These constraints are usually divided into hard and soft constraints. Hard constraints must be satisfied under any circumstances. Whilst, soft constraints are not only essential but also should be satisfied as much as possible. Due to the complexity of the problem, it is usually impossible to have solutions that satisfy all soft constraints. The quality of the timetable is measured based on soft constraints violation. A weighted penalty/value is attached to each violation of the soft constraint and the objective is to minimize the total penalty value of these violations. Although there have been a numerous approaches in solving the ETP, but this paper only focus on Genetic Algorithm (GA) method based on the fact that it is robust and can fits into complex problem space. This paper gives a summary of the GA method on ETP and it aims in helping other researcher to shorten their time in understanding the application of GA in the selected problem.

The thickness of an aquifer (B) is a critical hydrogeologic parameter in understanding groundwater system, especially in a karst aquifer where the high heterogeneity in this parameter brings more uncertainty to the evaluation of groundwater resources. In this study, the genetic algorithm (GA) method was used to estimate B, as well as five other parameters, including K r (horizontal hydraulic conductivity), K z (vertical hydraulic conductivity), S (storativity), S y (specific yield), and α (delayed coefficient) of a karst aquifer based on a pumping test. The search bounds of the six parameters for the GA method were obtained from numerous hypothetic numerical tests. The feasibility of the GA method for parameter estimation was verified by the results obtained from the gradient method in terms of a pumping test conducted in the Platte River valley in Nebraska. Then, this approach was used to estimate the hydraulic parameters in a karst aquifer with unknown aquifer thickness based on a pumping recovery test conducted in the Houzhai karst basin, southwestern China. The results show that the K r , K z , and the anisotropy ratio of K r to K z are strongly affected by B. The differences between the estimated parameter values at two observation wells indicate the existence of high heterogeneity in the karst aquifer. The parameter B has the highest absolute normalized sensitivity among these variables, implying that the estimated B may be more reliable as compared to other parameters. This study demonstrates that the GA method is an alternative method of parameter estimation with the unknown aquifer thickness.

A fundamental problem in the automated history matching is the development of optimization algorithms to find optimal solution in multidimensional search spaces. These optimization algorithms can be of two different types: global optimizers, which are able to work with noisy multi-modal functions and local optimizers, which are not able to work with noisy multimodal functions. Global optimizers present a shortcoming related to very large number of function calls. In reservoir engineering, optimization methods with a large number of function calls is a drawback since a reservoir simulation can take days. Due to this drawback in the last years, techniques such as Least Squared, Thin Plane Spline, Kriging and Artificial Neural Networks (ANN) have been used as proxies to reservoir simulators. This article shows the results of the research developed with relation to the application of evolutionary algorithms to find global optima in the automated history matching. Several ANN are used as proxies to reservoir simulator. The ANN used are Generalized Regression Neural Network; Fuzzy System with Subtractive Clustering; Radial Basis Network (two architectures). The research is realized in a distributed parallel software that supports decisions of reservoir engineers, UNIPAR. In this software the results obtained from evolutionary algorithms are fine-tuned by using local optimizers based on the Hooke-Jeeves optimizationmethod. Two case studies with synthetic and real reservoirs are examined. Introduction An important research field of the Petroleum Engineering is the reservoir management whose challenge is to optimize the reservoir performance considering criteria related toenvironmental, political, economic, and technological aspects.The numerical simulation is a technique with extensiveapplication in the reservoir management. Reservoir models areobtained from geological models and used to simulate fluidflow [1], [2]. These models have to be constantly corrected toyield the observed production behavior. The procedure called History Matching constitutes in the adjusting of the reservoir model with base in the perturbation of parameters. The process is controlled by comparisons among production data, well test obtained from reservoir with measured data from model. Some important steps of the History Matching are:Parametrization - Choice of the parameters of the model which must be perturbed to reduce the difference between simulated and observed production behavior.Objective Function - The definition of the misfit function to quantify the difference between modeled and observed production behavior. There are studies where the objective function works with geology data [3], and data shape [4] together with production data.Sensitivity and uncertainty analysis - The sensitivity analysis measures the impact that the parameters have on the objective function. Parameters with impact of first order must be chosen. The uncertainty analysis measures the uncertainty ofthe parameters, for example the porosity and permeability in the vicinity of wells have small uncertainty, since that these values are obtained from well logs, values of porosity mathematically calculated have greater uncertainty. Stopping criteria - To define stopping criteria is a fundamental step since that if the stopping criteria are defined incorrectly the proper model is not found.

History matching is a very important activity during the development and management of petroleum reservoirs. Matched models are fundamental to ensure reliable future forecasts, and enhance the level of understanding of the field via geological and reservoir models. Automated history matching (Figure 1) uses a mathematical algorithm to help choose new parameter values so that models better match historical data. The algorithm may be deterministic such as a gradient based method or stochastic such as a genetic or neighbourhood algorithm (NA). The former finds optimal solutions rapidly but the overall search is limited. Locally optimum solutions are all that may be found and uncertainty analysis is not effective. Stochastic methods are quasi-global but can be quite expensive. In this work we consider an approach to speed up the convergence rate of the NA. A proxy model is used to direct the stochastic search using gradients more effectively than is usual for the NA (Arwini and Stephen 2010). We call the new method Neighbourhood Algorithm with Proxy derived Gradients (NAPG). This results in finding solutions with far fewer models. The approach is further improved by updating the proxy model as we progress with history matching and the initialization is optimized using experimental design methods. We apply the approach to the Schiehallion field where we also use time lapse (4D) seismic data as dynamic constraints to reduce uncertainty to get a more reliable model that we might use it afterwards for the forecasting.

The use of gradient-based optimization methods in automatic history matching requires that the reservoir simulator provides the necessary derivative information in the most efficient way. The implementation of these techniques in existing complex simulators is a difficult task, however. This paper describes the implementation of the forward and adjoint methods for derivative calculation in a full-featured adaptive implicit black-oil simulator. To show how the derivative calculation methods can be used even outside the framework of standard optimization algorithms, a new algorithm for automatic history matching, based on the truncated singular value decomposition, is proposed. Application to simple synthetic cases shows promising results.

Two-phase oil/water relative permeability measurements were conducted at ambient and high temperatures in two different rock-fluid systems; one using a clean Poly-Alpha-Olefin (PAO) oil and the other with Athabasca bitumen. The tests were performed in a clean sand-pack with the confining pressure of 800 psi, using deionized water as the aqueous phase. Both the JBN method and the history match approach were utilized to obtain the relative permeability from the results of isothermal oil displacement tests. The contact angle and IFT measurements were carried out to assess any possible wettability alteration and change in fluid/fluid interaction at higher temperatures. Results, Observations, Conclusions: The results of the clean system using the viscous PAO oil confirmed that the two-phase oil/water relative permeability in this ultra-clean system is practically insensitive to the temperature. The slight variation in oil endpoint relative permeability, especially at ambient condition, was attributed to variations in the packing of sand. It was found that the history matching derived two-phase relative permeability from the highest temperature test provides reasonably good history matches of the other displacements that were conducted at lower temperatures. In addition, it is shown that the JBN approach based relative permeability curves show larger variations, primarily due to insufficient volume of water injection at lower temperatures, which makes the practical residual oil saturation much higher than the true residual. In contrast with the ultra-clean system, the results obtained with bitumen showed much larger variations in relative permeability with temperature. Most of the reported studies involving history matching approach treat the low-temperature measurements as the base case and show that changes in relative permeability are needed to history-match the tests at higher temperatures. We have shown that the displacement done at the highest temperature provides a more reliable estimate of the relative permeability and, in some cases, this relative permeability can successfully history match tests done at lower temperatures. In view of the impracticality of injecting sufficient water to reach close to real residual oil saturation at low temperatures, it would be better to obtain relative permeability data at high temperatures for characterizing the two-phase flow behavior of viscous oil systems.