Streamline Simulation Research Articles

Optimized Cyclic Water Injection Strategy for Oil Recovery in Low-Permeability Reservoirs

With the worldwide decline in conventional oil production, tremendous unconventional resources, such as low-permeability reservoirs, are becoming increasingly important. Cyclic water injection (CWI) as an oil recovery method has attracted increasing attention in the present environment of low oil prices. However, the optimal CWI strategy is difficult to determine for a mature oilfield due to the involvement of multiple wells with multiple operational parameters. Thus, our main focus in this paper is to present a novel and systematic approach to optimize CWI strategies by studying a typical low-permeability, namely, reservoir G21. To this end, a comprehensive method that combines the advantages of streamline simulation and fuzzy comprehensive evaluation (FCE) was proposed to identify water channeling in the reservoir. Second, the reliability of the method was verified using tracer tests. Finally, a new hybrid optimization algorithm, the simulated annealing-genetic algorithm (SAGA), coupled with a reservoir simulator was developed to determine an optimal CWI strategy for the low-permeability reservoir. The results show that the CWI technique is viable as a primary means in the present environment of low oil prices to improve the waterflood performance in low-permeability reservoirs. The oil recovery of the most efficient strategy increases by 6.8% compared to conventional waterflooding. The asymmetric CWI scheme is more efficient than the symmetric CWI scheme for the low-permeability reservoir.

Applying improved analytical methods for modelling flood displacement fronts in bounded reservoirs (Quitman field, east Texas)

We apply an improved potential flow model based on conformal mapping to study the sweep pattern of waterfloods in bounded reservoirs. Solutions for streamlines and flood advancement obtained with the new model are validated using an independent but more intricate numerical streamline simulation method. Subsequently, the use of the benchmarked model is demonstrated in a review of the flow patterns in the Quitman field, a tabular reservoir comprising an 18 ft thick payzone of Harris sand interfingering with shales of the Eagle Ford formation. The reservoir is bound by impervious faults modelled by the method of images in earlier studies of the 1970's. In the present study, Riemann's mapping theorem in combination with the Schottky-Klein prime function is applied to find complex potentials that describe the flow in the bounded reservoir. Such an approach can model waterflooding in marginal fields like the Quitman oil field more accurately than previous potential flow methods and can visualize the sweep pattern and compute time-of-flight contours in a simpler and faster fashion than numerical streamline simulators.

Modeling semi-steady state near-well flow performance for horizontal wells in anisotropic reservoirs

This paper presents a novel methodology to model semi-steady state horizontal well flow performance in an anisotropic reservoir taking into account flow in the near-well region for an arbitrary well trajectory. It is based on an analytical productivity model describing coupled axial reservoir flow and radial well inflow. In order to apply this model in an anisotropic reservoir, the permeability field relative to the radial direction perpendicular to the well trajectory and the axial direction along the well trajectory must first be determined. A classical space transformation is used in concert with rotational transforms to obtain a virtual isotropic model. The transformation preserves the volumes and pressures. It is not a novel concept, but different from previous approaches in the sense that it is only applied in the near-well domain to formulate an equally isotropic media. As a result, the use of this virtual isotropic model requires the Dietz shape factor for an ellipse, transformed from the original cylindrical near-well domain. The Dietz shape factors are determined numerically in this research. The semi-steady state well/near-well model is implemented in a numerical simulator incorporating formation anisotropy and wellbore hydraulics. The specific productivity index along the well trajectory is generated using the virtual configuration. Numerical results for different anisotropy ratios and also incorporating frictional losses in the well are presented. Furthermore, the well/near-well model is applied in coupling with streamline reservoir model for a water flooding case. This appears to be the first coupling of a well hydraulics model and a streamline simulator. It presents the application of the well/near-well model in integrated reservoir simulation in an efficient and accurate manner. The results demonstrate that the coupling approach with a streamline reservoir model and the well/near-well is of great potential for advanced well simulation efficiently.

Streamline simulation of water-oil displacement in a heterogeneous fractured reservoir using different transfer functions

Main parts of oil and gas reserves are stored in fractured reservoirs. Simulation of multiphase flow in fractured reservoirs requires a large amount of calculations due to the complexity, reservoir scale and heterogeneity of the rock properties. The accuracy and speed of the streamline method for simulating hydrocarbon reservoirs at field scale make it more applicable than conventional Eulerian simulators using finite difference and finite element techniques. Conventional simulators for fractured reservoirs consume a great deal of time and expense and require powerful CPUs like supercomputers. This makes the development of a fast, powerful and precise simulation method of great importance. The present study was undertaken to develop a computational code as a streamline simulator to study waterflooding in a two-dimensional fractured reservoir with heterogeneous permeability using the Dual Porosity-Single Permeability (DPSP) model. In this simulator, the pressure equation is solved implicitly over an Eulerian grid and then the streamlines are generated using Pollock's semi-analytical method and are traced. At this point, the Time-Of-Flight (TOF) is developed and the saturation equations are mapped and solved explicitly along the streamlines. Next, the results are transferred back onto the Eulerian grid and the calculations are repeated until the simulation end time. In fractured reservoirs, the interaction between the matrix and fracture is included in the transfer functions. Transfer functions model fluid flow and production mechanisms between the matrix and fracture. They introduce source/sink equations between the matrix and fracture and they are distributed throughout the media. In the current study, a problem is simulated using streamline method and several important transfer functions. A new linear transfer function with a constant coefficient is introduced that is based on differences in water saturation between the matrix and fracture. The simulation results were then compared and a commercial software is applied to solve the same problem. The results of the streamline simulator were compared with those of the commercial software and showed appropriate accuracy for the newly introduced transfer function. The accuracy and efficiency of the streamline simulator for simulation of two-phase flow in fractured reservoirs using different transfer functions are confirmed and the results are verified.

Erratum to: Streamline simulation of water-oil displacement in a heterogeneous fractured reservoir using different transfer functions

Erratum to: Streamline simulation of water-oil displacement in a heterogeneous fractured reservoir using different transfer functions

Rules for Flight Paths and Time of Flight for Flows in Porous Media with Heterogeneous Permeability and Porosity

Porous media like hydrocarbon reservoirs may be composed of a wide variety of rocks with different porosity and permeability. Our study shows in algorithms and in synthetic numerical simulations that the flow pattern of any particular porous medium, assuming constant fluid properties and standardized boundary and initial conditions, is not affected by any spatial porosity changes but will vary only according to spatial permeability changes. In contrast, the time of flight along the streamline will be affected by both the permeability and porosity, albeit in opposite directions. A theoretical framework is presented with evidence from flow visualizations. A series of strategically chosen streamline simulations, including systematic spatial variations of porosity and permeability, visualizes the respective effects on the flight path and time of flight. Two practical rules are formulated. Rule 1 states that an increase in permeability decreases the time of flight, whereas an increase in porosity increases the time of flight. Rule 2 states that the permeability uniquely controls the flight path of fluid flow in porous media; local porosity variations do not affect the streamline path. The two rules are essential for understanding fluid transport mechanisms, and their rigorous validation therefore is merited.

Application of Streamline Simulation to Gas Displacement Processes

Performance evaluation of miscible and near-miscible gas injection processes is available through conventional finite difference (FD) compositional simulation. Streamline methods have also been developed in which fluid is transported along the streamlines instead of using the finite difference grid. In streamline-based simulation, a 3D flow problem is decoupled into a set of 1D problems solved along streamlines. This reduces simulation time relative to FD simulation, and suppresses the numerical dispersion errors that are present in FD simulations. Larger time steps and higher spatial resolution can be achieved in these simulations. Thus, streamline-based reservoir simulation can be orders of magnitude faster than the conventional finite difference methods. Streamline methods are traditionally only applied to incompressible flow processes. In this paper, the method is adopted and assessed for application to compressible flow processes. A detailed comparison is given between the results of conventional FD simulation and the streamline approach for gas displacement processes. Finally, some guidelines are given on how the streamline method can potentially be used to good effect for gas displacement processes.

An Embedded Grid-Free Approach for Near-Wellbore Streamline Simulation

SummaryReactive-transport phenomena, such as carbon dioxide sequestration and microbial enhanced oil recovery (EOR), have been of interest in streamline-based simulation (SLS). Tracing streamlines launched from a wellbore is important, especially for time-sensitive transport behaviors. However, discretized gridblocks are usually too large compared with the wellbore radius. Field-scale simulations with local-grid-refinement (LGR) models often consume huge computational time. An embedded grid-free approach to integrate near-wellbore transport behaviors into streamline simulations is developed, comprising two stages of development: tracing streamlines in a wellblock (a gridblock containing wells) and coupling streamlines with neighboring grids. The velocity field in a wellblock is produced using a gridless virtual-boundary-element method (VBEM), where streamlines are numerically traced with the fourth-order Runge-Kutta (RK4) method (Butcher 2008). The local streamline system is then connected with the global streamline system, which is produced with the Pollock (1988) algorithm. Finally, the reactive-transport equation will be solved along these streamlines.The algorithm presented for solving near-wellbore streamlines is verified by both a commercial finite-element simulator and a Pollock-algorithm-based 3D streamline simulator. A series of computational cases of reactive-transport simulation is studied to demonstrate the applicability, accuracy, and efficiency of the proposed method. Velocity field, time-of-flight (TOF), streamline pattern, and concentration distribution produced by different approaches are analyzed. Results show that the presented method can accurately perform near-wellbore streamline simulations in a time-effective manner. The algorithm can be directly applied to one grid containing multiple wells or to off-center wells. Furthermore, assuming streamlines are evenly launched from the gridblock boundary or ignoring transport in the wellblock is not always reasonable, and may lead to a significant error.This study provides a simple and grid-free solution, but is capable of capturing the flow field near the wellbore with significant accuracy and computational efficiency. The method is promising for reservoir SLS with time-sensitive transport, and other simulations requiring an accurate assessment of interactions between wells in one particular gridblock.

Complex analytical solutions for flow in hydraulically fractured hydrocarbon reservoirs with and without natural fractures

• Streamlines in naturally fractured reservoir modeled by new analytical formulation. • Natural fractures may affect time-of-flight contours and drainage contours. • Flow is supported in natural fracture direction and limited normal to the fractures. • Prototype slab as natural analog for flow in hydraulically fractured reservoirs. Reservoir drainage towards producer wells in a hydraulically and naturally fractured reservoir is visualized by using an analytical streamline simulator that plots streamlines, time-of-flight contours and drainage contours based on complex potentials. A new analytical expression is derived to model the flow through natural fractures with enhanced hydraulic conductivity. Synthetic examples show that in an otherwise homogeneous reservoir even a small number of natural fractures may severely affect streamline patterns and distort the drainage contours. Multiple parallel natural fractures result in a drainage region that is narrower in the direction normal to the natural fractures while the drainage reach is larger in the natural fracture direction. Reservoirs with numerous natural fractures are shown to be characterized by more tortuous drainage patterns than reservoirs without natural fractures. Finally, the analytical flow model for naturally fractured reservoirs is applied to a natural analog of flow into hydraulic fractures. The tendency of the injected fluid to stay confined to the fracture network as opposed to matrix flow is entirely controlled by the hydraulic conductivity contrast between the fracture network and the matrix.

A new streamline model for near-well flow validated with radial flow experiments

Streamline simulation is a powerful tool that can be used for full field forecasting, history matching, flood optimization, and displacement visualization. This paper presents the development and the application of a new semi-analytical streamline simulation method in the near-wellbore region in polar/cylindrical coordinate systems. The main objective of this paper is to study the effects of the permeability heterogeneity and well completion details in the near-wellbore region. These effects dictate the streamline geometries, which in turn influence well productivity. Previous streamline applications used a constant flow rate for each stream tube. In this paper, streamline simulation is performed under the assumption of constant pressure boundaries, which is a novel and non-trivial extension of streamline simulation. Solutions are constructed by treating each stream tube as a flow unit by invoking analytical solutions for such geometries. In addition, visualization experiments are conducted to investigate the effect of the heterogeneity. Two-dimensional waterflooding visualization experiments in radial porous media are performed with constant pressure boundaries. The streamline simulator is applied to history match the relative permeabilities using these experiments, thereby validating the new near-well streamline method.

The effect of temperature dependent relative permeability on heavy oil recovery during hot water injection process using streamline-based simulation

Extraction of heavy viscous oil from oilfields requires thermal enhanced oil recovery techniques to reduce the oil viscosity and improve its mobility. Usually, in these problems the effects of temperature on the relative permeability are ignored. In this paper, hot water injection into oil reservoirs (3D heterogeneous porous media) with temperature dependent viscosity and relative permeability functions is simulated numerically using an improved streamline approach to enhance computational performance. To this end, pressure and conduction equations are solved simultaneously on a 3D Eulerian grid, and saturation and heat convection equations are solved along multiple 1D streamlines. Results of the old and the modified thermal streamline simulator are compared in terms of the norm of error, and a significant improvement in accuracy of results is observed. Finally, the new numerical simulator employed to investigate effects of temperature dependent relative permeability on oil recovery of four sample heavy oilfields. It has been shown that these effects have a remarkable impact on the efficiency of hot water injection process and large errors can be caused, ignoring relative permeability changes with temperature in realistic reservoirs.

Streamline Simulation of Barium Sulfate Precipitation Occurring Within the Reservoir Coupled With Analyses of Observed Produced-Water-Chemistry Data To Aid Scale Management

SummaryIn waterflooded reservoirs under active scale management, produced-water samples are routinely collected and analyzed, yielding information on the evolving variations in chemical composition. These produced-water chemical-composition data contain clues as to the fluid/fluid and fluid/rock interactions occurring in the subsurface, and are used to inform scale-management programs designed to minimize damage and enable improved recovery.In this interdisciplinary paper, the analyses of produced-water compositional data from the Miller Field are presented to investigate possible geochemical reactions taking place within the reservoir. The 1D and 2D theoretical model has been developed to test the modeling of barium sulfate precipitation implemented in the streamline simulator FrontSim. A completely 3D streamline simulation study for the Miller Field is presented to evaluate brine flow and mixing processes occurring in the reservoir by use of an available history-matched streamline reservoir-simulation model integrated with produced-water chemical data. Conservative natural tracers were added to the modeled injection water (IW), and then the displacement of IW and the behaviors of the produced water in two given production wells were studied further. In addition, the connectivity between producers and injectors was investigated on the basis of the comparison of production behavior calculated by the reservoir model with produced-water chemical data. Finally, a simplified model of barite-scale precipitation was included in the streamline simulation, and the calculation results with and without considering barite precipitation were compared with the observed produced-water chemical data. The streamline simulation model assumes scale deposition is possible everywhere in the formation, whereas, in reality, the near-production-well zones were generally protected by squeezed scale inhibitor, and, thus, the discrepancies between modeled and observed barium concentrations at these two given wells diagnose the effectiveness of the chemical treatments to prevent scale formation.

Modeling Reservoir Circulation and Economic Performance of the Neal Hot Springs Geothermal Power Plant (Oregon, U.S.): An Integrated Case Study

We review the key factors that facilitated the successful development of a geothermal reservoir at Neal Hot Springs (NHS), Oregon, by a private company. The NHS geothermal field development plan, financing tools, and fluid circulation model provide a template for the development time, financing and well rates required to sustain geothermal power generation in similar setups elsewhere. The early economic performance and life-cycle analysis are specified using company data, applying a financial evaluation tool developed by the National Renewable Energy Laboratory (NREL). The Neal Hot Springs geothermal project was enabled by a combination of financing sources: U.S. federal loan guarantee and cash grant, state tax credit and private equity. Flow patterns in the NHS geothermal reservoir are visualized using a novel streamline simulation tool applying known well rates and benchmarking the residence time of re-injected brine based on tracer tests performed by the company. Temperature of produced brine has stayed remarkably constant over the first years of NHS plant operations, which we ascribe to advection of heat by an upwelling far-field flow beneath the plant’s production wells.

Technology Focus: History Matching and Forecasting (April 2017)

Technology Focus The initial work on computer techniques for assisted history matching date back to the 1960s. However, it was a long journey between the development of early methods and operational use. Initially, these methods were referred to as “automatic history matching,” giving the wrong impression that it was something we could delegate to a computer. Fortunately, industry and academia soon realized that “assisted” was a more accurate term than “automatic.” Nowadays, there is an impressive amount of literature and a large number of assisted-history-matching methods. The diversity is so vast that it is challenging to divide these methods into categories. For example, there are several flavors of methods based on sensitivity and gradient-optimization algorithms. There are also methods grounded on stochastic optimization, evolutionary algorithms, design of experiments, proxy modeling, streamline simulation, and Kalman filters. This is just to mention a few, and the list is still growing. Regardless of the specific method of your choice, I believe the main development in history matching is the recognition that it can be formulated as a Bayesian inference problem. Bayes’ rule provides an elegant framework to formalize the process of learning from data to update our beliefs. The beauty is in the fact that Bayes’ rule gave to history matching the correct meaning. History matching is no longer a searching process to find the best model. Instead, history matching is a process of mitigating uncertainty in light of new information. The papers summarized in this feature and the ones indicated in the additional-reading list are excellent examples of recent developments and applications of assisted-history-matching techniques. All are aligned with the modern Bayesian interpretation. I hope you enjoy the reading. Recommended additional reading at OnePetro: www.onepetro.org. SPE 179549 Streamline-Based Rapid History Matching of Bottomhole Pressure and Three-Phase Production Data by Dongjae Kam, Texas A&M University, et al. SPE 182684 Generation of a Proposal Distribution for Efficient MCMC Characterization of Uncertainty in Reservoir Description and Forecasting by Xin Li, The University of Tulsa, et al. SPE 182693 A Robust Iterative Ensemble-Smoother Method for Efficient History Matching and Uncertainty Quantification by Xiang Ma, ExxonMobil, et al.

Statement of Removal

ABSTRACTAlthough nowadays finite difference simulators have many advantages for reservoir engineers to predict reservoirs in three dimensions, they have some disadvantages, such as spending much run time for complex fields with several thousand grids, which – by using streamline simulation – the run time was decreased considerably. In this paper, first a brief summary of streamline simulation and its uniqueness applications were discussed and then an Iranian oil reservoir as a case study was simulated by using FrontSim streamline simulator. Then, besides testing the results of Sayyafzadeh et al. (2010) to find the location of infill wells, the most suitable locations for infill drilling were selected. After that, vertical and horizontal infill wells were investigated and overall 34 scenarios were tested by using the streamline simulation method and FrontSim software. Finally the results of simulation and testing infill drilling scenario show significant increase in field oil production and field oil recovery factor, approximately 13%, and also a reduction in field water cut. In addition, distinguishing fractures or faults, well’s drainage area, and understanding how much injectors affect the field oil production are some of the other benefits of the streamline simulation method. The novelty of this paper was to study the effects of horizontal infill wells and to compare the results between horizontal and vertical infill drilling using streamline simulation.

Optimizing volumetric sweep efficiency in water flooding by streamline simulation

ABSTRACT Early shutting time in production wells due to water production performs an important role to determine production efficiency and useful life of the reservoir. In this study, in order to postpone the shut-in time of producing wells, increase oil displacement, and enhance production efficiency, production and injection wells capabilities with respect to their position in the reservoir were studied by using the concept of streamline. In the oil reservoirs, increasing injection flow rate does not necessarily enhance oil displacement and recovery. Therefore, suitable injection rates according to injection and production wells position have to be optimized. Also, production wells flow rate can affect sweep efficiency optimization extremely and increase the efficiency of injection wells. In this study, according to the position of production and injection wells and water production rates resulting from injection wells, four scenarios with different injection and production rates were investigated. This optimization has led to a reduced water production and water injection. Also, it increased the production efficiency and reservoir life.

Rules for Flight Paths and Time of Flight for Flows in Porous Media with Heterogeneous Permeability and Porosity

Porous media like hydrocarbon reservoirs may be composed of a wide variety of rocks with different porosity and permeability. Our study shows in algorithms and in synthetic numerical simulations that the flow pattern of any particular porous medium, assuming constant fluid properties and standardized boundary and initial conditions, is not affected by any spatial porosity changes but will vary only according to spatial permeability changes. In contrast, the time of flight along the streamline will be affected by both the permeability and porosity, albeit in opposite directions. A theoretical framework is presented with evidence from flow visualizations. A series of strategically chosen streamline simulations, including systematic spatial variations of porosity and permeability, visualizes the respective effects on the flight path and time of flight. Two practical rules are formulated. Rule 1 states that an increase in permeability decreases the time of flight, whereas an increase in porosity increases the time of flight. Rule 2 states that the permeability uniquely controls the flight path of fluid flow in porous media; local porosity variations do not affect the streamline path. The two rules are essential for understanding fluid transport mechanisms, and their rigorous validation therefore is merited.

Advancement of sweep zones in waterflooding: conceptual insight based on flow visualizations of oil-withdrawal contours and waterflood time-of-flight contours using complex potentials

Sweep zones are traced in synthetic reservoir models of waterflood advancement based on potential functions. Time-of-flight contours, oil-withdrawal contours and streamlines corresponding to fluid withdrawal paths are visualized. The effects of differential well rates on waterflood sweep regions for a range of well architectures are systematically investigated using reservoirs that are continuous isotropic with and without impervious fault barriers. Complex potentials are capable of solving the drainage path for any constellation of producer and injection wells, accounting for any discontinuities that affect the flow path and productivity of the wells. Flood patterns are visualized for a series of doublets and 7-spot well patterns. Loss of planned drainage symmetry occurs when an undiscovered fault barrier obstructs and diverts the waterflood. Our method is assumed effective in illustrating the value of analytical streamline simulations for first-order assessment of sweep patterns in hydrocarbon field produced with waterflooding. The critical impact of injection rates and fault barriers on the shape of the waterflooding patterns is visualized in detail. The analytical streamline simulator allows tracing of the respective flow paths of displacing oil and water in the reservoir and visualizes both oil-withdrawal contours and waterflood time-of-flight contours. Generic rules are formulated to aid sweep maximization both prior to drilling and during the surveillance of producing wells.

Numerical simulation of two-phase flow in fractured porous media using streamline simulation and IMPES methods and comparing results with a commercial software

Streamline simulation is developed to simulate waterflooding in fractured reservoirs. Conventional reservoir simulation methods for fluid flow simulation in large and complex reservoirs are very costly and time consuming. In streamline method, transport equations are solved on one-dimensional streamlines to reduce the computation time with less memory for simulation. First, pressure equation is solved on an Eulerian grid and streamlines are traced. Defining the “time of flight”, saturation equations are mapped and solved on streamlines. Finally, the results are mapped back on Eulerian grid and the process is repeated until the simulation end time. The waterflooding process is considered in a fractured reservoir using the dual porosity model. Afterwards, a computational code is developed to solve the same problem by the IMPES method and the results of streamline simulation are compared to those of the IMPES and a commercial software. Finally, the accuracy and efficiency of streamline simulator for simulation of two-phase flow in fractured reservoirs has been proved.

Streamline-Based History Matching for Multicomponent Compositional Systems

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 174750, “Streamline-Based History Matching of Arrival Times and Bottomhole-Pressure Data for Multicomponent Compositional Systems,” by Shusei Tanaka, Dongjae Kam, Akhil Datta-Gupta, and Michael J. King, Texas A&M University, prepared for the 2015 SPE Annual Technical Conference and Exhibition, Houston, 28–30 September. The paper has not been peer reviewed. Streamline-based history-matching techniques have provided significant capabilities in integrating field-scale water-cut and tracer data into high-resolution geologic models. However, application of the streamline-based approach for simultaneous integration of water cut and bottomhole pressure (BHP) has been rather limited. In this paper, the authors introduce a novel semianalytic approach to compute the sensitivity of the BHP data with respect to gridblock properties. Introduction The streamline-based method has many advantages in terms of computational efficiency and applicability. The main advantage of the streamline-based method is that it is able to calculate parameter sensitivity with a single streamline simulation or post-processing of the grid-based finite-difference simulation results. The calculated sensitivity is comparable with the sensitivities computed from numerical perturbation or with adjoint-based sensitivities. It is possible to calculate the parameter sensitivity from a commercial finite-difference simulator by use of streamlines traced using the flux field. This allows accounting for detailed physics by means of finite-difference simulation while taking advantage of the power of the streamlines for sensitivity computations. Streamline and Parameter Sensitivity Streamline-based history matching starts with the tracing of streamlines from a given set of static and dynamic conditions. This process is outlined in detail in the complete paper. Amplitude, Travel-Time, and Generalized Travel-Time Inversion (GTTI) Methods The approach to minimizing the objective function through all the observed points and simulation results is defined as amplitude inversion. The travel-time inversion, instead, attempts to match single reference times such as water-breakthrough point or peak tracer response. The amplitude matching is general in terms of reduction of the objective function, because while the travel-time inversion reduces the objective function at only a single point, the amplitude matching can cover all the data points. However, the travel-time-based approach is often used because amplitude inversion is a highly nonlinear problem and presents difficulties in computational cost and in reducing the objective function when there is a large amount of wells and data points.