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Aiming at the problem that it is difficult to balance the accuracy and velocity of the fixed-step solution of standard parabolic equation in the study of radio wave propagation in a wide range of complex environments, a variable-step solution method of standard parabolic equation is proposed. Firstly, after deducing the relationship between the error of SSFT solution and step size, frequency and other factors; And the basic selection range of step size for SPE variable step size solution is given. Secondly, the action mechanism of different environmental factors and the requirement of changing trend for step size are expounded through simulation, and the complex environment of SPE application is classified according to the requirement of error. Finally, the method is used to simulate the characteristics of the typical complex environment. The simulation results show that the method can save up to 71.4% of the time compared with the fixed-step method of parabolic equation under the condition of ensuring the calculation accuracy. The reliability and efficiency of the proposed method are verified, and the calculation efficiency of the parabolic equation radio wave prediction can be greatly improved. Therefore, the parabolic equation method with variable step size can ensure the accuracy of calculation, reduce the memory and time required for calculation, greatly improves the efficiency of calculation, and has practical significance in the application of real-time prediction of electromagnetic wave propagation in a wide range of complex environments.

In this paper, a variable step size method for the PE (Parabolic Equation) is proposed to solve the problem of low computational efficiency in the study of radio wave propagation at a large range of complex environments. Firstly, the relationship between the error and the step size, the frequency and other factors of the SSFT (Split-Step Fourier Transform) solution in the standard atmosphere is deduced, and then the basic selection range of the step size is given for the variable step size method. Secondly, the action mechanism of different environmental factors and the requirement of changing trend for step size are expounded through simulation, and the complex environment of PE application is classified according to the requirement of error. Finally, the electric wave characteristics of typical complex environment are advanced by the method. The simulation results show that compared with the small step method, the variable step size method of the PE can reduce the horizontal sampling point by 55.9% and the calculation time by 69.6% when the accuracy is close to the small step method, and the variable step size method is more accurate than the fixed large size step method. Therefore, the variable step size on the PE can not only ensure the accuracy of calculation, but also reduce the memory and time required, which greatly improves the reliability and efficiency. Moreover, it is suitable for solving the problem of the radio wave propagation in complex environment.

Aiming at the problems of high water cuts in the later stages of multi-throughput and the inability to effectively develop the remaining oil between wells in ordinary heavy oil reservoirs. The development of block flood control and viscosity reduction complex flooding was proposed, in which the flood control agent can achieve the blocking and deep regulation of high permeability channels. Dissection, effectively expanding the volume of subsequent viscosity reducing agents. Through a large number of preferred evaluations, a polymer with viscosity-increasing properties for certain mineral formation water salinity, low viscosity at injection temperature, and high viscosity at the bottom hole temperature (65 ° C) was selected, and other properties of the polymer were tested. The results show that the polymer is hydrophobically associative and has temperature-sensitive properties. As the temperature increases, the polymer viscosity first increases and then decreases. At the reservoir temperature, the maximum viscosity of 3000 mg/L polymer reaches 919 mPa.s; The polymer also has excellent anti-aging properties and shear resistance. After storage at the reservoir temperature (65 °C) under anaerobic conditions for 90 days, the viscosity retention rate reaches 97%; In the shear force of the reservoir (7.5 s−1) conditions, the viscosity of the polymer will not decrease due to multiple shears. The displacement effect of 1# polymer was verified by the dual-tube high-low-permeability mold displacement experiment. The results show that the polymer can effectively block the high-permeability layer and effectively use the low-permeability layer.

Additional file 1: Figure S1. PCA was applied to the Leaf4, Leaf7, and petal data sets to identify trends in shape and size variations among MAGIC lines. The outline of a leaf was represented by the Cartesian coordinates of its 25 points, and similarly, a petal was represented by 25 points in the same way. The GIF showed the data set for each PC expressed in standard deviations from the mean position of the point within the collection of leaves and petals. Figure S2. Range of PC values obtained for the leaf and petal allometric model. The mean of each MAGIC line is represented by a blue diamond, and the bars display the range of all values observed for this line. PC units are standard deviations. Figure S3. Variations along each PC for Leaf4, Leaf7 and Petal allometry within the MAGIC lines. Each histogram represents the distribution of MAGIC lines along one of the PCs from the leaf and petal allometry model. Figure S4. QTL scan of the PCAs for the fourth leaf allometry among MAGIC lines. Figure S5. QTL scan of the PCAs for the seventh leaf allometry among MAGIC lines. Figure S6. QTL scan of the PCAs for petal allometry among MAGIC lines. Figure S7. QTL scan of the PCAs for the fourth leaf and petal allometry covariation among MAGIC lines. Figure S8. QTL scan of the PCAs for the seventh leaf and petal allometry covariation among MAGIC lines. Figure S9. QTL scan of the life-history traits among MAGIC lines. Figure S10. The genetic correlation between life-history traits and the allometry model. The vertical lines represent the QTLs overlapping with the peak SNP within 1 Mb. The different QTLs with the same accession allele conferring maximum effects are shown in red. Figure S11. The promoter sequence alignment and expression analysis of candidate genes. The candidate gene ERECTA (AT2G26330) in the LF7_2.1 locus, obtained the minimum Leaf7.PC2 value in the Bur-0 accession, had specific variation in the Bur-0 promoter sequence and had the lowest expression level in Bur-0. The second candidate gene AGO4 (AT2G27040) in the LF7_2.1, PE_2.5 and LF7PE_3.2 loci had specific variation in the promoter in the Ler-0 accession, with the maximum Leaf7.PC2, Petal.PC2 value and minimum Leaf7-Petal.PC3 value and obtained the highest expression level in Ler-0.

Aiming at the low computational efficiency of the fixed-step solution of the standard parabolic equation (SPE) in the study of complex environmental radio wave propagation, a variable step size technology for the standard parabolic equation is proposed. Based on the relationship between the factors such as step size and frequency with the calculation error in the SSFT solution, the basic criteria for the SPE variable step size solution are proposed; secondly, the complex environment of SPE application is graded, and the corresponding step size is selected for different complex environment levels under the condition of satisfying the error requirement. Finally, the method is used to simulate the radio wave characteristics of typical complex environment. The calculation results verify the reliability and efficiency of the proposed method. It shows that the proposed method can greatly improve the computational efficiency of parabolic equations under the condition of ensuring the accuracy of calculation.