Three-dimensional modeling of loose layers based on stratum development law

2.4.1 Recognition of SDPs

By analyzing the distribution of strata in boreholes, there are five SDPs (Figure 3) [46]. The effective recognition of SDPs based on the stratigraphic development information of adjacent boreholes is a vital prerequisite for loose layer modeling.

The strategy recognizing the SDPs of any pair of adjacent measured boreholes is as follows: First, the distribution of each stratum is determined by comparing the thickness value of the same stratum in the two boreholes and classifying the strata type (missing in the right borehole, missing in the left borehole, missing in both boreholes, or existing in both boreholes). Then, the adjacent strata of the same type are merged to generate different types of stratigraphic intervals. Finally, recognize the SDP according to the type and the number of strata in each stratigraphic interval.

The specific steps of recognition of SDPS are as follows:

1. Defining stratigraphic distribution types. Obtain a pair of adjacent measured boreholes, obtain the thickness of each stratum in each borehole, and quantitatively judge the stratigraphic distribution types between boreholes according to the following formula (Figure 7) and save them into SL = { sl i |i = 1,2 , … , N } .

where a · S i is the thickness of S i in borehole a on the left, b · S i is the thickness of S i in borehole b on the right, and sl i is the stratigraphic distribution type between the two boreholes.

1. Generating stratigraphic interval ds u . First, the stratigraphic distribution type sl i is read from SL in turn to form a string; then, the substrings with consistent adjacent characters in the string are extracted to form a substring set; and finally, each substring set is seen as a stratigraphic interval ds u and added to list DS = { ds u |u = 1,2 , … , DN } , ds u = ( S start , S end , flag , count ) , where u means the index of ds u , DN means the number of ds u s, S start is the stratum corresponding to the first character of substring, S end is the stratum corresponding to the first character of substring, count is the number of characters in the substring, and flag is any character in the substring.

2. Recognizing SDPs at different ds u Values. According to flag and count in ds u , the SDP is determined according to the following formula. Figure 8 shows the division of ds u s in Figure 7 and determines the SDPs.

where SO represents that the strata in ds u exist continuously between two adjacent boreholes, SP represents that the strata in ds u miss continuously in the left one of the two adjacent boreholes, SQ represents that the strata in ds u miss discontinuously in the left one of the two adjacent boreholes, SR represents that the strata in ds u miss continuously in the right one of the two adjacent boreholes, ST represents that the strata in ds u miss discontinuously in the right one of the two adjacent boreholes, and SU represents that the strata in ds u miss continuously between two adjacent boreholes.

Figure 7

Example of determining stratigraphic distribution type (for visual expression, the missing stratum is represented by light gray). A represents that the stratum misses in borehole b on the right. B represents that the stratum misses in borehole a on the left. C represents that the stratum misses in boreholes a and b. D represents that the stratum exists in boreholes a and b.

Figure 8

Example of differentiating SDPS: There are eight strata in boreholes a and b, which can be divided into seven stratigraphic intervals.

2.4.2 Determining the location of POVBs

Between adjacent measured boreholes, the number of POVBs and strata pinch-out positions are different according to different SDPs. The rules for determining the number of POVBs and strata pinch-out positions under different SDPs are shown in Table 2.

Table 2

Determination of the number and location of POVBs under different SDPs

SDP	Number of POVBs	Location of POVBs	Examples	Description
SO or SU	0 (in the DS, all ds u s whose SDPs are SO or SU)	—	—	—
SR or SP	ds u . count (in the DS, only one ds u whose SDP is SR or SP)	①Equidistant insertion between adjacent boreholes	(a)	A ds u exists whose SDP is SP with three missing strata
ST or SQ	1 (in the DS, only one ds u whose SDP is ST or SQ)		(b)	A ds u exists whose SDP is ST with one missing stratum
		②According to the development sequence, the stratum determines the pinch-out point on each POVB from far to near
Multiple SDPs	Max( ds u . count ) (in the DS, except that SDP is all ds u s of SO and SU, when any two ds u s do not exist continuously)		(c)	One ds u whose SDP is SP and two strata are miss, one ds u whose SDP is SQ and one stratum is missing. Due to discontinuity, two ds u s do not need to be merged, so the number of POVBs is the maximum missing strata number
	Max( ds u . count ) (in the DS, except that SDP is all ds u s of SO and SU, when two or more ds u s exist continuously, the continuous ds u s are merged to form a new DS)		(d)	Two ds u s whose SDPs are SQ, one d s u whose SDP is ST and one ds u whose SDP is SR. Merge continuous ds u s and take the maximum number of strata missing as the number of POVBs

2.4.3 Calculating the stratum thickness in POVBs

In this article, the process of calculating the thickness of each stratum in POVBs mainly involves the following steps: (1) generating stratigraphic thickness matrix PT = { pt i , j |i = 1,2 , … , N;j = 1,2 , … , V } , where i is the index of the stratum, j is the index of the POVB, and pt i , j is the thickness of stratum S i in POVB j (Figure 9); (2) calculating the thickness of each stratum in each POVB and saving in PT based on the thickness and ds u of that in the measured boreholes on both sides; and (3) completing the ground elevation of each POVB and the top or bottom surface elevation of each stratum in this POVB according to the data structure of measured boreholes.

Figure 9

Stratigraphic thickness matrix of the POVBs between measured boreholes a and b.

The methods of calculating the thickness of each stratum in the POVBs are explained in detail:

1. Get a POVB j between boreholes a and b , then pick up a stratum S i in it.

2. According to Section 2.4.2, determine which ds u the S i is located in and then calculate its thickness. There are two situations:

   If S i is located in the ds u whose SDP is SO, the thickness of S i is calculated using the following formula and saved into pt i , j .

   (3) pt i , j = a · S i + j V + 1 × ( a · S i − b · S i ) .

   If S i is located in the ds u whose SDP is not SC, the thickness of S i is calculated using the following formula and saved into pt i , j .

   (4) pt i , j = 1 − j V − k ( a . S i ) + 1 * a · S i , if a · S i > 0 and j ≤ V − k ( a · S i ) + 1 0 , if a · S i > 0 and j > V − k ( a · S i ) + 1 j V − k ( b · S i ) + 1 , if b · S i > 0 and j ≥ k ( b · S i ) 0 , if b · S i > 0 and j < k ( b · S i ) ,

   where k ( a · S i ) is the mark value of the S i in the ds u in the borehole a and k ( b · S i ) is the mark value of the S i in the ds u in the borehole b . The method used to determine the mark value is to obtain S i , in which a · S i > 0 ( b · S i > 0 ) in the borehole a · (b), and incrementally marking S i from 1, which will be marked as k ( a · S i ) ( k ( b · S i ) ). For example, we assume that there is a ds u = ( S 2 , S 5 , various flags but no D , 4 ) between boreholes a and b ; if a · S 2 > 0 , a · S 4 > 0 and a · S 5 > 0 , then k ( a · S 2 ) = 1 , k ( a · S 4 ) = 2 and k ( a · S 5 ) = 3 , and if b · S 3 > 0 , then k ( b · S 3 ) = 1 .

3. Increase the identifier i and cycle step (2) to complete the calculation of all stratigraphic thickness information in POVB j .

4. Increase the identifier j and cycle steps (1)–(3) to complete the calculation of all POVBs.

3.1.1 Study area and dataset

The first study area is located in the Qinhuai District, Nanjing, China. The modeling area is approximately 57.11 km², and scattered exposed bedrock areas are distributed (Figure 12a). There are 341 measured boreholes in this case, and the 3D models of some measured boreholes are shown in Figure 12b. In this case, there are seven stratigraphic units, and the strata are denoted as S 1 , … , S 6 and S 7 from top to bottom.

Figure 12

Modeling area and measured borehole model: (a) modeling area with exposed bedrock and 341 measured boreholes and (b) 3D measured boreholes with seven stratigraphic units.

3.1.2 Modeling performance

The modeling parameters are presented in Table 3, and the modeling results are shown in Figure 13. The model quality inspection methods proposed in Section 2.6 are used to test the modeling performance. The boreholes W120, W201, W227, and W39 (see Figure 12a) were selected as the verification boreholes, and the stratigraphic thickness error table of the two modeling methods is shown in Table 4. The mean and the standard deviation of the method of this paper are 1.23 and 1.10, respectively and all are less than the value of the comparison method, which are 3.24 and 3.23, respectively. Therefore, compared with the horizons-to-solids method using elevation interpolation, the method of this article, which interpolates stratigraphic surfaces by stacking, can fit the strata thicknesses more accurately and improve the accuracy of the model.

Figure 13

Modeling results: (a) 3D solids with measured boreholes; (b) local amplification of the 3D solids; (c) local amplification of the 3D solids in the bottom; (d) four main substrata in the 3D solids; (e) cross section passing through boreholes; and (f) cross section passing through both sides of the bedrock area.

Furthermore, two stratigraphic cross sections were generated (Figure 13e and f). The pinch-out mode of the strata on both sides of the exposed bedrock area is consistent with the development characteristics of the loose layers (Figure 2), and the locations of strata pinch-out follow the SDPs. Therefore, the modeling method of this article is applicable to the area where bedrock and loose layers are alternately distributed.

3.2.1 Study area and dataset

This case is located in the Hangkonggang District, Zhengzhou, China. The modeling area is approximately 426.5 km² and includes 69 measured boreholes (Figure 14a). Figure 14b shows the 3D models of some measured boreholes. There is no exposed bedrock area, and the terrain is flat. There are 35 stratigraphic units in this case, and the strata are denoted as S 1 ,…, S 34 , and S 35 from top to bottom.

Figure 14

Modeling area and measured borehole model: (a) modeling area with 69 measured boreholes, (b) measured boreholes data for modeling, and (c) 3D measured boreholes with 35 stratigraphic units.

3.2.2 Modeling performance

The modeling parameters are presented in Table 5, and the modeling results are shown in Figure 15 (Z-axis exaggeration 20 times for clear visualization). This case selects boreholes ZG36, G128, G92, and G25–44 (Figure 13a) as the verification boreholes and then chooses seven key strata to make the stratigraphic thickness error table shown in Table 6. The mean and the standard deviation of the method of this paper are 1.30 and 0.70, respectively, and all are less than the value of the comparison method, which are 2.06 and 1.03, respectively.

Figure 15

Modeling results: (a) 3D solids, (b) and (c) local amplifications of the 3D solids, (d) solids separation diagram, and (e) cross section passing through boreholes.

In addition, because the modeling area does not contain bedrock, Method II is not carried out completely. However, Figure 15d shows that the pinch-out patterns are still valid when there are many stratigraphic units in the boreholes.

4.1.1 The number and distribution of the measured boreholes

The measured boreholes are the data basis for constructing the 3D loose-layer model, and their quantity and distribution are the primary factors for determining the quality of the model. In the modeling area, a large number of measured boreholes with a uniform distribution can reveal the loose layer’s distribution characteristics reasonably and ensure the accuracy of the model. In contrast, too few or unevenly distributed measured boreholes will make it difficult to accurately define the pinch-out positions of missing strata and ensure the model accuracy in the sparse area of boreholes. In China, the general principles of the measured borehole layout can refer to the national standard code for engineering geological drilling (DZ/T 0017-91).

In addition, at least one or more measured boreholes are required in the area between exposed bedrocks to control the spatial distribution of loose layers. If the area between the exposed bedrock lacks measured boreholes, the loose layer in this area will not be modeled correctly.

4.1.2 Screening of the measured boreholes

If boreholes that are too shallow are involved in constructing the 3D loose-layer model, the illusion of a missing local stratum will appear. Therefore, it is necessary to screen the shallow boreholes to ensure that the borehole depth is basically the same and to meet the requirements of 3D modeling. According to the bottom stratum ID and the depth of each borehole, the shallow borehole can be automatically screened.

4.1.3 Discrete spacing d

The number of DVBs directly determines the smoothness of the model. If the discrete spacing d is set too large, the number of DVBs is small. For the model established in this case, the strata surfaces are rough, the boundary edges and corners are sharp, and it has a poor visualization effect. If the discrete spacing d is set to small, the number of DVBs will be large. In this case, the strata surfaces will be smooth, and the boundary transition will be gentle.

However, it is meaningless to pursue the model’s accuracy further when the fluctuation of the local level is indistinguishable to the human eye, which will reduce the efficiency of model construction, increase the amount of model data, and impact the execution efficiency of model display and 3D spatial analysis applications. Therefore, the appropriate discrete spacing can achieve a balance among the visualization effect, modeling efficiency, and the amount of model data.

4.1.4 DEM accuracy

During the construction of a 3D loose-layer model, the ground elevation of virtual boreholes needs to be obtained from the DEM to ensure that the surface of the topmost stratum is consistent with the DEM. Therefore, the DEM accuracy has an important influence on the modeling accuracy of the 3D loose-layer model.

First, the DEM used in this article can be generated by interpolating the ground elevation of the measured boreholes. In this case, the constructed model surface is relatively smooth, and the surface fluctuation is continuous. The quality of the DEM entirely depends on the number and density of measured boreholes, and there are two deficiencies: (1) When the number of measured boreholes is small or unevenly distributed, the constructed DEM is rough, and the surface fluctuation is not apparent. Especially in the area where a large number of measured boreholes are missing, there is a significant error between the constructed DEM and the actual ground. (2) If the drilling time of the measured boreholes is too long, their ground elevation may be inconsistent with the actual surface elevation, which makes the current situation of the constructed surface DEM poor, and it is difficult to realize the seamless integration of the constructed 3D model and the latest DEM.

Second, the latest high-precision DEM is used for modeling. In this case, the advantage is that it can easily realize the seamless integration of the 3D loose-layer model and the latest DEM. The disadvantage is that the ground elevation of the measured borehole is inconsistent with that of the DEM and needs further optimization.

The actual drilling time is a reference standard for choosing which DEM to use. If the time is too early, the first DEM is suitable. In contrast, the time is very close, and the second DEM is acceptable. In addition, if the 3D loose-layer model does not need to consider the actual surface fluctuation and is mainly used for excavation analysis and bearing capacity calculation, using the first DEM is more appropriate. In contrast, if the 3D model is mainly used for the above-ground and underground 3D scene visualization, the second DEM is recommended to ensure the effective integration between underground, ground, and above-ground 3D models.

4.2.1 Applicability of modeling objects

First, the modeling method of this article is designed for a large-scale modeling area with alternating distributions of bedrock and loose layers. Starting from the development law of the loose layer, the method fully considers the five stratum distribution modes of the loose layer.

Second, a pure development area of a loose layer is also applicable. When the modeling area does not contain bedrock, there is no need to provide bedrock boundary data in the modeling process.

Finally, the method of this article is also applicable for 3D modeling in bedrock areas with relatively simple geological structures. For example:

1. Both horizontal and monoclinic structures have prominent overlapping distribution characteristics, similar to the distribution mode of loose strata. Figure 16 shows a 3D model of a bedrock area that contains six kinds of rocks. The strata of this area are stacked horizontally and can be divided into 14 stratigraphic units according to the stratigraphic sequence.

2. Vertical fold, inclined fold, and other folds without stratigraphic inversion. For those geological structures, their rock mass is in a continuous distribution state, which is very suitable for using superposition technology to calculate stratigraphic thickness.

Figure 16

Example of bedrock modeling: (a) 3D exploration boreholes with 14 stratigraphic units, and (b) 3D solids.

4.2.2 Applicability of modeling area

When a modeling area is too large, the modeling efficiency will decline and generate a model file with an excessive amount of data. In this case, the modeling area can be divided into multiple modeling units using the administrative units or geomorphic units and then modeling unit by unit. In addition, if there are islands or gaps in the modeling area (Figure 17), the modeling area can also be dispersed into multiple modeling units for modeling unit by unit.

Figure 17

Applicability of modeling area: (a) contains holes but is complete and can be modeled, (b) or (c) are dispersed geometric surfaces, which should be dispersed into a single region and modeled separately.

4.2.3 Applicability of modeling data

In this article, the measured boreholes in the modeling area need to meet certain constraints to participate in the modeling method.

1. In a modeling area, the numbering rule of the strata identifiers of the measured boreholes should be consistent.

2. The direction of the measured boreholes should be vertical. The stratum thickness in the inclined borehole is inconsistent with the actual stratum thickness. The inclined boreholes need to be processed as vertical boreholes before participating in modeling.

3. When the measured boreholes are insufficient, the virtual boreholes can be generated based on the geological profile and participate in the 3D modeling together with the measured boreholes.