# Constructing of 3D Fluvial Reservoir Model Based on 2D Training Images
## 3 Comparative Analysis
### Connectivity  
Connectivity can reflect the degree of variation in the probability of connectivity of regionalized variables at a certain distance in a certain direction. In this paper, the idea of connectivity evaluation is borrowed from [PERSON] et al. [21]. [PERSON] used the training images produced by 2D ground-penetrating radar to simulate the generation of a 3D subsurface aquifer structure. One of the evaluation methods is to decompose the simulated 3D body into 2D vertical slices and calculate the MPH (multiple-point histogram). MPH indicates the frequency of spatial patterns in the grid and aims to reproduce the spatial features in the training image comparatively; the smaller the distance, the more consistent the spatial structure, and the better the simulation result; the larger the distance, the more inconsistent the structure, and the poorer the simulation result. The MPH of the training image is used as a reference, and the deviation of the simulation result MPH from it is quantified, thus providing information about the quality of the simulation result. Therefore, the 3D models generated by the above three algorithms were evaluated in terms of connectivity with the original training images in the x-, y-, and z-directions. The connectivity is calculated as:  
\[K(h;n)=E\{I(u)I(u+h)\ldots I(u+nh)\}=Prob\{I(u)=1,I(u+h)=1,\ldots,I(u+nh)=1\} \tag{1}\]  
where h is a unit vector in any given direction and K(h; n) denotes the probability of observing n consecutive points. I(u) is the property at point u. When I(u) = 1, it represents that point u is a sand body; when I(u) = 0, it represents that point u is mud. The steps of connectivity evaluation are: (1) First, slice the above three 3D models and find the connectivity curves of the sandstone in the models. Each slice contains the connectivity curves in horizontal and vertical directions. (2) Select 3 groups of 10 slices for each 3D model with equal spacing in the x-, y-, and z-directions. Find the connectivity of the sandstone in each direction in the three groups of models and find its mean value. (3) Compare the changes in the connectivity of the sandstone in the three directions in each group of models with the training images. (1) The algorithm in this paper is used to generate the y-direction connectivity of the 3D model as an example (Figure 8). Ten slices with positions of y = 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 are selected in the y-direction of the 3D model, and the horizontal and vertical connectivity of each slice is found with equal spacing, as shown in Figures 8 and 9. The connectivity curves of the 10 slices in the y-direction basically match with the training image, confirming that the 3D model generated from the 2D slices can obtain higher-order  
Figure 7: Generated model with 3D TI as a function of variance in three directions. statistics while obtaining spatial correlation and location distribution features similar to those deposited in the training images. (2) The connectivity of the x-, y-, and z-directions of the fluvial facies model generated by the method in this paper is compared with the methods of Snesim and DS, and the connectivity functions in each direction are found in the same way as in (1). The locations of equal intervals were selected: x = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. The slice sizes in the y-direction were the same as in (1), z = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and the probability curves of sandstone connectivity in the three directions were compared as follows:  
Figure 10 shows the comparison of the connectivity of the sandstone in the x-direction. Figure 10a shows the connectivity curves with different lag distances in the horizontal direction, and it can be seen that the method in this paper is in suitable agreement with the Snesim connectivity function, while the DS method generates the model with the largest difference in connectivity with the training images. Figure 11 shows the comparison of the connectivity of sandstone in the y-direction.