Solutes In Porous Media Research Articles

The Form of Waiting Time Distributions of Continuous Time Random Walk in Dead-End Pores

Anomalous dispersion of solute in porous media can be explained by the power-law distribution of waiting time of solute particles. In this paper, we simulate the diffusion of nonreactive tracer in dead-end pores to explore the waiting time distributions. The distributions of waiting time in different dead-end pores show similar power-law decline at early time and transit to an exponential decline in the end. The transition time between these two decline modes increases with the lengths of dead-end pores. It is well known that power-law distributions of waiting time may lead to anomalous (non-Fickian) dispersion. Therefore, anomalous dispersion is highly dependent on the sizes of immobile zones. According to the power-law decline, we can directly get the power index from the structure of dead-end pores, which can be used to judge the anomalous degree of solute transport in advance.

Qualitative and Quantitative Evaluation of Permeability Changes during EOR Polymer Flooding Using Micromodels

Polymer solutions are used in chemical EOR processes to achieve incremental oil recoveries through obtaining favorable mobility ratios. In the process, the in-situ viscosity is a key parameter for the polymer flood design, as well as the changes in permeability due to the retention or adsorption (e.g.: plugging). Understanding the major causes of the plugging effects allows predicting injectivity problems as well as optimizing project design. The objective of this work is to use glass-silicon-glass micromodels in combination with tracer particles—attached to the flooded fluids—to qualitatively and quantitatively describe the extent of permeability changes after polymer injection. Laboratory work is performed in order to determine the physical properties of the polymer solutions when they flow through porous media, such as the presence of permeability reduction/plugging of the micromodel. A statistical analysis of the distribution and extent of plugged areas is performed and a study of the pressure response during various injection stages will complement the study. A biopolymer (Scleroglucan) was tested and compared to a commonly used polymer, giving a direct insight into their pros and cons. Five different concentrations of polymers were tested and put into relation with their quantitative and qualitative amount of sort of called retention. The amount of adsorption was determined experimentally in one case in order to draw the significance. By exploiting the potential of GSG-micromodels in combination with tracer particles, it was possible to visualize the reduction of flow paths and its increase during various injections for the first time. Expanding the working principle proposed in this work could provide further understanding of the behavior of any polymers. The results obtained and workflow presented in this work allow for additional understanding of polymer solutions behavior in flooding applications. Furthermore, the definition of optimized workflows to assess any kind of solutions in porous media and permeability changes is supported.

A polymer flooding mechanism for mature oil fields: Laboratory measurements and field results interpretation

Although polymer flooding has been widely studied in the literature on oil recovery, many questions remain about the technique's action mechanisms. It is widely accepted that the process is based on increasing viscosity of the aqueous phase (injected fluid). The polymer can also act secondarily by reducing the relative permeability to water. Fields that are markedly heterogeneous, highly depleted or that contain fluids with strong salinity are not good candidates for the process. To overcome these restrictions, the most recent applications have used more resistant polymers, with higher molar masses, injected in larger quantities, to preserve the level of viscosity under these adverse conditions. However, a pilot test in a field under adverse conditions showed surprising results and unexpected pressure abnormality, even with small quantities of partially hydrolyzed polyacrylamide (HPAM) with relatively low molar mass (6 × 106 g/mol). This paper reports the investigation of the prevailing mechanism responsible for these unexpected results, by means of laboratory tests of flow in porous media and rheological analysis of fluids and hydrodynamic diameter of polymer molecules. The Salt effect (NaCl) on polymer solutions was researched by rheology and particle size analysis at 25 and 50 °C. The results showed that mechanical/hydrodynamic retention prevails over polymer adsorption at the rock, with the former being caused by the increase in the polymeric hydrodynamic volume in fluid dilution processes. Besides this, increased temperature was found to favor greater hydrodynamic volume under tested conditions, while salinity had little influence. The results also shed new light on the flow behavior of polymer solutions in porous media, particularly in polymer injection for enhanced oil recovery from mature fields.

A KDE‐Based Random Walk Method for Modeling Reactive Transport With Complex Kinetics in Porous Media

AbstractIn recent years, a large body of the literature has been devoted to study reactive transport of solutes in porous media based on pure Lagrangian formulations. Such approaches have also been extended to accommodate second‐order bimolecular reactions, in which the reaction rate is proportional to the concentrations of the reactants. Rather, in some cases, chemical reactions involving two reactants follow more complicated rate laws. Some examples are (1) reaction rate laws written in terms of powers of concentrations, (2) redox reactions incorporating a limiting term (e.g., Michaelis‐Menten), or (3) any reaction where the activity coefficients vary with the concentration of the reactants, just to name a few. We provide a methodology to account for complex kinetic bimolecular reactions in a fully Lagrangian framework where each particle represents a fraction of the total mass of a specific solute. The method, built as an extension to the second‐order case, is based on the concept of optimal Kernel Density Estimator, which allows the concentrations to be written in terms of particle locations, hence transferring the concept of reaction rate to that of particle location distribution. By doing so, we can update the probability of particles reacting without the need to fully reconstruct the concentration maps. The performance and convergence of the method is tested for several illustrative examples that simulate the Advection‐Dispersion‐Reaction Equation in a 1‐D homogeneous column. Finally, a 2‐D application example is presented evaluating the need of fully describing non‐bilinear chemical kinetics in a randomly heterogeneous porous medium.

Effect of solute immobilization on the stability problem within the fractional model in the solute analog of the Horton-Rogers-Lapwood problem.

The paper is devoted to the linear stability analysis within the solute analogue of the Horton-Rogers-Lapwood (HRL) problem. The solid nanoparticles are treated as solute within the continuous approach. Therefore, we consider the infinite horizontal porous layer saturated with a mixture (carrier fluid and solute). Solute transport in porous media is very often complicated by solute immobilization on a solid matrix of porous media. Solute immobilization (solute sorption) is taken into account within the fractal model of the MIM approach. According to this model a solute in porous media immobilizes within random time intervals and the distribution of such random variable does not have a finite mean value, which has a good agreement with some experiments. The solute concentration difference between the layer boundaries is assumed as constant. We consider two cases of horizontal external filtration flux: constant and time-modulated. For the constant flux the system of equations that determines the frequency of neutral oscillations and the critical value of the Rayleigh-Darcy number is derived. Neutral curves of the critical parameters on the governing parameters are plotted. Stability maps are obtained numerically in a wide range of parameters of the system. We have found that taking immobilization into account leads to an increase in the critical value of the Rayleigh-Darcy number with an increase in the intensity of the external filtration flux. The case of weak time-dependent external flux is investigated analytically. We have shown that the modulated external flux leads to an increase in the critical value of the Rayleigh-Darcy number and a decrease in the critical wave number. For moderate time-dependent filtration flux the differential equation with Caputo fractional derivatives has been obtained for the description of the behavior near the convection instability threshold. This equation is analyzed numerically by the Floquet method; the parametric excitation of convection is observed.

Polymer Flow Through Porous Media: Numerical Prediction of the Contribution of Slip to the Apparent Viscosity

The flow of polymer solutions in porous media is often described using Darcy’s law with an apparent viscosity capturing the observed thinning or thickening effects. While the macroscale form is well accepted, the fundamentals of the pore-scale mechanisms, their link with the apparent viscosity, and their relative influence are still a matter of debate. Besides the complex effects associated with the rheology of the bulk fluid, the flow is also deeply influenced by the mechanisms occurring close to the solid/liquid interface, where polymer molecules can arrange and interact in a complex manner. In this paper, we focus on a repulsive mechanism, where polymer molecules are pushed away from the interface, yielding a so-called depletion layer in the vicinity of the wall. This depletion layer acts as a lubricating film that may be represented by an effective slip boundary condition. Here, our goal is to provide a simple mean to evaluate the contribution of this slip effect to the apparent viscosity. To do so, we solve the pore-scale flow numerically in idealized porous media with a slip length evaluated analytically in a tube. Besides its simplicity, the advantage of our approach is also that it captures relatively well the apparent viscosity obtained from core-flood experiments, using only a limited number of inputs. Therefore, it may be useful in many applications to rapidly estimate the influence of the depletion layer effect over the macroscale flow and its relative contribution compared to other phenomena, such as non-Newtonian effects.

Quantification of non-Newtonian fluid dynamics of a wormlike micelle solution in porous media with magnetic resonance

Nuclear magnetic resonance (NMR) pulsed gradient stimulated echo (PGStE) techniques were used to observe anomalous transport phenomena for flow of a non-Newtonian wormlike micelle solution through a model porous media. Understanding the flow behavior of wormlike micelle solutions in porous media is important due to the growing interest of these solutions in enhanced oil recovery. NMR velocity imaging was unable to discern differences in the flow field between shear-thickening wormlike micelle solution and water due to spatial resolution limitations. However, the probability of displacement, i.e. the propagator, was skewed towards slower velocities and long tails at high displacements for the micelle solution and incorporation of a fractional dynamics approach using the moments of the probability distribution showed a deviation from asymptotic Gaussian statistics.

SYNERGISM STUDY OF MIXTURES OF IONIC AND NONIONIC SURFACTANTS IN ENHANCED OIL RECOVERY ADSORPTION

Adsorption of surfactants from aqueous solutions in porous media is fundamental in enhanced oil recovery (EOR) because surfactant loss reduces the effectiveness of the chemical solution injected, making the process economically unfeasible. This paper analyzes surfactants’ synergistic effects in adsorption and enhanced oil recovery. This study used nonionic (nonylphenol 20 ethylene oxide – R200) and anionic surfactants (saponified coconut oil – OCS and base soap - SB) with the two mixtures studied (R200/OCS and R200/SB). The mixture R200/SB, due to its higher viscosity because of the presence of SB, saturated the ionic surfactant (SB) faster, reaching a higher oil recovery (70.0%). The addition of SB surfactant in the R200 solution led to an increase in oil recovery (70.0%) when compared to R200 alone (64.5%).

Application of binding theory for seepage of viscoelastic fluid in a variable diameter capillary

The polymer solution for polymer flooding is a viscoelastic fluid. There exist both shear flow and elongational flow when the polymer solution flows in a porous medium, where an additional dissipation is involved. The additional dissipation caused by elongational deformation is often ignored while studying the flow of the fluid in a porous medium. For a complex polymer solution, the generated elongational pressure drop cannot be ignored. In a capillary of fixed diameter, the polymer solution is only impacted by the shear force, and its rheological property is pseudoplastic. Therefore the variable diameter capillary and the converging-diverging flow model with different cross sections are required to describe the flow characteristics of the polymer solution in porous media more accurately. When the polymer solution flows through the port, we have the elongational flow and the polymer molecules undergo elongational deformation elastically. By using the mechanical energy balance principle and the minimum energy principle, a mathematical model of non-Newtonian fluid inlet flow was established by Binding. On the basis of the Binding theory, with the application of the theory of viscoelastic fluid flow in the circular capillary and the contraction – expansion tube, the relations between the viscoelastic fluid flow rate and the pressure drop are obtained.

2-d mathematical and numerical modeling of fluid flow inside and outside packing in catalytic packed bed reactor

Generally, the momentum equation of fluid flow in porous media was solved by neglecting the terms of diffusion and convection such as Ergun, Darcy, Brinkman and Forchheimer models. Their model primarily applied for laminar flow. It is true that these model are limited to condition whether the models can be applied. Analytical solution for the model type above is available only for simple one-dimensional cases. For two or three-dimentional problem, numerical solution is the only solution. This work advances the flow model in porous media and provide two-dimentional flow field solution in porous media, which includes the diffusion and convection terms. The momentum lost due to flow and porous material interaction is modeled using the available Brinkman-Forchheimer equation. The numerical method to be used is finite volume method. This method is suitable for the characteristic of fluid flow in porous media which is averaged by a volume base. The effect of the solid and fluid interaction in porous media is the basic principle of the flow model in morous media. The Brinkman-Forchheimer consider the momentum lost term to be determined by a quadratic function of the velocity component. The momentum and the continuity equation are solved for two-dimentional cylindrical coordinat . the result were validated with the experimental data. The velocity of the porous media was treated to be radially oscillated. The result of velocity profile inside packing show a good agreement in their trend with the Stephenson and Steward experimental data. The local superficial velocity attains its global maximum and minimum at distances near 0.201 and 0.57 particle diameter, dp. velocity profile below packing was simulated. The result were validated with Schwartz and Smith experimental data. The result also show an excellent agreement with those experimental data.Keywords : finite volume method, porous media, flow distribution, velocity profile

Statement of Removal

ABSTRACTPolymer flooding is an attractive branch of chemical enhanced oil recovery techniques. Regrettably, chemical and mechanical degradation reduces the effectiveness of polymer flooding. This paper explores the effects of silica nanoparticles on flow behavior of polymer solution in porous media by employing both rheological experiments and numerical simulations. Experimental results indicate that dispersed silica nanoparticles in polymer solution are responsible for polymer adsorption reduction and the solution viscosity improvement. Simulation results revealed that if the injectant has some dispersed silica nanoparticles, the cumulative heavy oil recovery and breakthrough time will be bettered compared to the conventional polymer flood.

Transport and removal of a solvent in porous media in the presence of bitumen, a highly viscous solute

Removal of solvent from the gangue generated by an oil sands solvent extraction process is essentially a process of solution transport in a porous medium with a highly viscous solute (left over bitumen of a completed extraction process or residual bitumen) at low concentrations. The solvent usually has much higher vapor pressure and lower viscosity than the residual bitumen. The concentrations of the residual bitumen are typically in the range of 0.6–2.4wt% while those of the solvent (cyclohexane) is 8–12wt% at the start of the drying process to remove the solvent. Drying of gangue was carried out for 2h under ambient conditions using petri dishes with bed heights of 0.6, 1.0 and 1.4cm. A typical drying curve (i.e., cumulative volatiles mass loss versus time) shows two distinct drying stages in which the first stage is the fast evaporation of cyclohexane (99% by weight removal) followed by a stage in which slow evaporation of water is observed. The residual bitumen was observed, unexpectedly, to migrate to the top surface of the porous media with cyclohexane via capillary transport and accumulate there in the first stage of drying. The rate of removal of cyclohexane decreased with increasing residual bitumen content. Consequently, duration of the fast cyclohexane removal stage from the gangue sample containing 2.4wt% bitumen was 1.5 times longer than that of the 0.6wt% bitumen gangue sample, when the initial cyclohexane concentration was 12wt%; it was 1.2 times longer when the initial cyclohexane concentration was 8wt%. Our studies reveal that poor capillary transport of solution in porous media at higher bitumen concentrations to be the most likely cause of slower drying rather than vapor pressure lowering of bulk solution and pores blockage. The cause is similar to previous reports in literature with the solute being salt in salt-water solution.

Elastic instabilities during the flow of hydrolyzed polyacrylamide solution in porous media: effect of pore-shape and salt.

We experimentally investigate the flow of hydrolyzed polyacrylamide (HPAM) solution with and without salt in model porous media at high Weissenberg numbers (Wi > 1.0). The effect of pore shapes on the flow pattern and pressure drop is explored by using periodic arrays of circular and square pillars in aligned and staggered layouts. In the apparent shear-thinning regime, we observe stationary dead zones upstream of the pillars. In addition, we confirm that the size of stationary dead zones correlates with the level of shear-thinning, by varying the amount of salt in HPAM solution. At higher shear rates (or Wi), these dead zones are periodically washed away. We present the mechanism of this elastic instability and characterize it based on the pressure drop fluctuation spectral density.

Chemical potential of a two-component liquid in porous media: The case of unsaturated soil

The aim of this paper is to propose a method to determine the chemical potential of two miscible components of a solution in porous media, particularly in the case of an unsaturated soil. It is limited to the capillary state; the equilibrium of the liquid phase in the soil is determined by gas–liquid interfaces whose geometry obeys Laplace’s law. Deduction of the chemical potential of the two components from the chemical potential of pure water is based on the assumption that, for the same volume occupied by the fluid phase in the porous medium, the geometry of the fluid phase is independent of the composition of the solution.The expressions of the chemical potentials of the two components were established. They show the superposition of two effects: the effect of interfacial forces and the effect of the presence of the other component. A study of the water–alcohol solution in a clayey silty sand soil was then conducted to alcohol mole fractions between 0 and 0.15 and water content in the 5%–15% range. In this field, the chemical potential of the water was seen to be affected by the capillary effects below a water content of 10%. Above that, its variation as a function of the mole fraction of alcohol differed little from that of the potential of the water in a free solution. The chemical potential of the alcohol was little affected by the capillary effects.

An Investigation of Polymer Adsorption in Porous Media Using Pore Network Modelling

Polymer injection in hydrocarbon reservoirs has been of great interest in petroleum engineering as an enhanced oil recovery method. In order to have a reliable injection plan, an accurate estimation of polymer behaviour is required to control the fluid flow in porous media. However, it is difficult to calculate the flow properties of polymer solutions—known as non-Newtonian fluids—in porous media, due to variable viscosity at different shear rates. Adsorption of polymer molecules onto the rock surface decreases the available cross-sectional area of polymer solution, therefore at a constant rate, the shear rate increases. As a result, it is important to investigate how polymer adsorption can affect the polymer solution flow behaviour in porous media. Moreover, prior to any polymer injection plan, the amount of adsorbed polymer should be calculated, due to the high cost of polymers. In the present study, pore network modelling was used to investigate the effect of polymer adsorption on flow behaviour of polymer solutions in porous media. To achieve this goal, a simple pore was constructed to adequately represent the real structure of the porous medium. It was then validated by simulation of Newtonian fluid flow in the network and comparison of the calculated flow properties with the experimental data. After validation, the pore network was used for simulation of non-Newtonian fluid flow in porous media. The experimental apparent viscosity of non-Newtonian fluid was used for verification of the procedure and the developed network. The apparent viscosity of non-Newtonian was calculated successfully using macroscopic properties of the rock sample and measured bulk properties of non-Newtonian fluid. The effect of polymer adsorption on calculation of non-Newtonian fluid apparent viscosity was then investigated. It was shown that the polymer adsorption process plays an important role in calculation of polymer solution apparent viscosity. The results reveal that polymer adsorption cannot be neglected, especially in near well-bore areas where the adsorption is high. Finally, the effect of wettability in calculating the total amount of adsorbed polymer volume was studied. The results suggest that neglecting the rock’s wettability can result in an overestimation or underestimation of the amount of adsorbed polymer.

Effective Properties of Pore Network Elements Derived from Reactive Transport in Individual Pores

Adsorption of solutes in porous media is typically represented as an equilibrium process. The adsorption process is determined by the concentration of the solute in the fluid next to the solid grain in individual pores. The concentration near the solid surface is different from the average concentration within the pore due to the development of concentration gradients within the pore space. Simulations with pore network models are usually based on the average concentration in individual pore throats. To advance the realism of such models, we develop a relationship between pore-scale adsorption coefficients and corresponding coarse-grained adsorption parameters. A numerical scheme is proposed: firstly the solute transport within a single pore is simulated undergoing equilibrium adsorption at the pore wall, and secondly flux-averaged concentration breakthrough curves are obtained. By fitting the breakthrough curves, the coarse-grained adsorption parameters are determined from the pore-scale hydraulic and adsorption parameters.

Study on Flow Parameters Measurement Experiment of Polymer Solution in Porous Media

Due to the complexity of porous media and viscoelasticity of polymer solution, the flow behavior of polymer solution in porous media is more complex than that in the rheometer, and the flow regular pattern of polymer solution in porous media directly affects the formulation of polymer flooding projects. This paper designs experiment on flow parameters measurement of polymer solution (partially hydrolyzed polyacrylamide) in porous media, and the regular pattern of influence on resistance factor, residual resistance factor and flow viscosity in porous media caused by core permeability, polymer concentration, polymer molecular weight and polymer injection rate was given. The followings are the results: even for the same batch core with similar parameters, due to reasons such as natural fractures which could lead to comparatively large difference of core parameters between the same batch of core, thus in order to ensure the reliability of experiment data, it is necessary to detect core quality before core flooding experiment. Core end effect has significant effect on the results of core flooding experiment, it suggests that we should eliminate the effect of core end effect when carry out core flooding experiment, otherwise it is difficult to obtain valid experiment data. There exists a critical polymer concentration, when polymer concentration is lower than critical concentration, resistance factor is basically equal to apparent viscosity; when polymer concentration is higher than critical concentration, resistance factor is far less than apparent viscosity and the difference value increases with the increase of polymer concentration. Flow viscosity of polymer solution in porous media is far less than apparent viscosity in the rheometer and when polymer concentration increases, the increase extent of flow viscosity is far less than the increase of apparent viscosity. Results of this research are of great importance for enhancing flooding effect and guiding oilfield production.

How Viscoelastic-Polymer Flooding Enhances Displacement Efficiency

SummaryIncreasing flooding-solution viscosity with polymers provides a favorable mobility ratio compared with brine flooding and hence improves volumetric sweep efficiency. Flooding with a polymer solution exhibiting elastic properties has been reported to increase displacement efficiency, resulting in a sustained doubling of the recovery enhancement compared with the use of conventional viscous-polymer flooding (Wang et al. 2011). Flooding with viscoelastic-polymer solutions is claimed also to increase recovery more than expected from changes in capillary number alone (Wang et al. 2010). This increase in displacement efficiency by viscoelastic polymers is reported to occur because of changes in the steady-state-flow profile and enhancements in oil stripping and thread formation. However, within the industry there are doubts that a genuine effect is observed, or that improvements in displacement efficiency occur with field-applicable flow regimes (Vermolen et al. 2014).In this study, we demonstrate that flooding with viscoelastic-polymer solutions can indeed increase recovery more than expected from changes in capillary number. We show a mechanism of fluctuations in flow at low Reynolds number by which viscoelastic-polymer solutions provide improvements in displacement efficiency. The mechanism, known as elastic turbulence, is an effect previously unrecognized in this context. We demonstrate that the effect may be obtained at field-relevant flow rates. Furthermore, this underlying mechanism explains both the enhanced capillary-desaturation curves and the observation of apparent flow thickening (Delshad et al. 2008; Seright et al. 2011) for these viscoelastic solutions in porous media. The work contrasts experiments on flow and recovery by use of viscous and viscoelastic-polymer solutions. The circumstances under which viscoelasticity is beneficial are demonstrated. The findings are applicable to the design of formulations for enhanced oil recovery (EOR) by polymer flooding.A combination of coreflooding, micromodel flow, and rheometric studies is presented. The results include single-phase and multiphase floods in sandstone cores. Polymer solutions are viscoelastic [partially hydrolyzed polyacrylamide (HPAM)] or viscous (xanthan). The effects of molecular weight, flow rate, and concentration of the HPAMs are described. The data lead us to suggest a mechanism that may be used to explain the observations of improved displacement efficiency and why the improvement is not seen for all viscoelastic-polymer floods.

Characterizing the Role of Clay and Silica Nanoparticles in Enhanced Heavy Oil Recovery During Polymer Flooding

Polymer flooding as one of the most effective methods for enhancing oil recovery has a significant effect on tertiary production of reservoirs with poor vertical sweep efficiencies and those which are extremely heterogeneous. Water-soluble polymers are used to both increase the viscosity of water and decrease the mobility ratio of the displaced and displacing fluids. However, due to high amount of polymer adsorption, presence of divalent cations and mechanical degradation, polymer molecules may lose their properties. Hence it is necessary to further improve the displacement effectiveness of polymer flooding. Despite the fact that several ways have been suggested to enhance the performance of polymer flooding, application of nanoparticles in the improvement of rheological properties of the polymer solution used in enhanced oil recovery has not been reported previously and is still an ongoing subject. This paper investigates the effects of nanoparticles on flow behavior of polymer solution in porous media by employing both experimental studies and numerical simulations. The simulation model first is validated against well-controlled laboratory experiment to have reliable predictions of the full-field implementations. The results show that the amount of polymer adsorption and viscosity of polymer solution will improve when the clay or silica nanoparticles is present in the injectant, and accordingly, the cumulative oil recovery and breakthrough time will be bettered. The result of sensitivity analysis demonstrates that polymer molecules are more degradable during polymer flooding compared to nanoparticles-assisted polymer flooding. Based on the validated model, 3D simulations of nanoparticles-assisted polymer field pilot were performed and the results revealed that the cumulative oil recovery, water cut and breakthrough time will improve when the injectant has some dispersed nanoparticles.

Study on complex flow law near polymer flooding wellbores and adaptability evaluation of shaped charges

The flow law near polymer flooding wellbores under the condition of perforation is very complex and there are many types of shaped charges. In this study, a non-Darcy flow model near polymer flooding wellbores was given, and a flow experiment was carried out to measure the resistance coefficient, residual resistance coefficient, and flow viscosity of the polymer solution in porous media under the conditions of different permeability and different polymer molecular weight. According to the results of numerical simulation, this paper calculated the flow efficiency of different types of shaped charges and evaluated the adaptability of different charges under polymer flooding condition. The results of this study have an important significance for guiding the optimization of shaped charges and the optimization design of perforating parameters.