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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access December 29, 2017

A location-based multiple point statistics method: modelling the reservoir with non-stationary characteristics

  • Yanshu Yin EMAIL logo and Wenjie Feng
From the journal Open Geosciences

Abstract

In this paper, a location-based multiple point statistics method is developed to model a non-stationary reservoir. The proposed method characterizes the relationship between the sedimentary pattern and the deposit location using the relative central position distance function, which alleviates the requirement that the training image and the simulated grids have the same dimension. The weights in every direction of the distance function can be changed to characterize the reservoir heterogeneity in various directions. The local integral replacements of data events, structured random path, distance tolerance and multi-grid strategy are applied to reproduce the sedimentary patterns and obtain a more realistic result. This method is compared with the traditional Snesim method using a synthesized 3-D training image of Poyang Lake and a reservoir model of Shengli Oilfield in China. The results indicate that the new method can reproduce the non-stationary characteristics better than the traditional method and is more suitable for simulation of delta-front deposits. These results show that the new method is a powerful tool for modelling a reservoir with non-stationary characteristics.

1 Introduction

Two-point geostatistics and the emergence of multiple point statistics (MPS) are well developed based on the basic assumption that the reservoir statistical properties do not change with location, referred to as the statistical properties of ‘stationarity’. For one of the representative stationary reservoirs, most stochastic modelling methods (including MPS) were tested in the fluvial reservoir and showed widely approved results [1,2,3,4,5,6,7]. However, in the delta reservoir, the sedimentary characteristics change along the source direction, and the model is not accepted for a fluvial deposit with the use of MPS based on the stationary hypothesis [7].

The abovementioned issue, known as the statistical property of ‘non-stationarity’, is a difficult problem in MPS. There are some improved methods, such as the geometric transformation-based method and the partition-based simulation methods [8,9,10]. Some non-stationary characteristics in a fan delta reservoir have been obtained. However, geometric transformation methods demand several parameters, such as rotation angles and width variation. These parameters can be easily obtained from an aerial map but are difficult to extract from a threedimensional map. A substitution method uses trend maps to form a three-dimensional map and constrain the modelling as auxiliary data. However, the applicability of this method is quite limited [8]. Partition simulation becomes trapped by the problem of how to reasonably divide the non-stationary domain into stationary sub-domains [9]. Furthermore, the details of how to integrate the simulation results of the sub-domains and conform with the actual sedimentary characteristics are challenging. Design and development of new methods is necessary. Among these methods, direct sampling simulation is an approach used to process the non-stationary reservoir simulation [10,11,12]. Instead of scanning the entire training image, the direct sampling method only searches a reasonable data event within a certain area. However, the choice of the data event can lead to uncertainty due to the less constraining conditional data. It is difficult to reflect the changes in the sediment patterns with the change of location of the depositions. The MPS method, based on Gibbs sampling, divides the data model into smaller portions with fewer points and rebuilds the kriging equations with the partitions [13]. This approach is essentially based on the statistical stationarity assumption and is not suitable for a non-stationary reservoir.

The Simpat method guides the modelling technology into the domain of image restoration, which models the non-stationary reservoir by pattern combination [7]. The modelling result does not achieve the expectation, but the concept of pattern-based modelling is widely propagated and a pattern-based MPS is developed for non-stationary reservoir modelling, which is an extension of the Simpat method [7, 14]. The core idea of this approach is to compare the similarity between all data events and the distance between the central position of the data events and the conditional data. A weight parameter ω is assigned to the distance function to characterize the non-stationary. From the simulation results, the non-stationary characteristics of delta reservoir are reproduced. However, the author noted that the training image size must be consistent with that of the simulation region for calculation of the distance between the similarity and the central position of the data event, which limits the application of the proposed method [16]. Furthermore, the value of weight parameter ω is subjective, and an objective assignment is needed.

In this paper, the pattern-based MPS methods are further studied and revised to form a location-based MPS method. A relative central position concept is adopted to reveal a certain sedimentary pattern in place, which is also considered to be found in the training image for the similarity of the sediment process. The dimensional difference is also calculated as a relative concept to guarantee the relative consistency between the simulated grids and the training image. A local search strategy is introduced to ensure the calculation efficiency. A synthesized model and a real reservoir of Shengli Oilfield of China are analysed to demonstrate the robustness of the revised algorithm in reproducing the phenomenon with non-stationary characteristics.

2 Methodology

Before describing the revised algorithm, we first review the important elements of the direct pattern-based simulation method [14]. Unlike the former MPS methods that only focus on the data event, the pattern-based method uses a location database locdbT to depict the variation of the sedimentary pattern in different locations, and hence it reveals the non-stationary characteristics of the reservoir. The similarity of patterns in the conditional data and the training images are driven by the following equation:

dnonstat=(1ω)dpat+ωdloc(1)

where ω is the weight (ω ∈ [0,1]) of the pattern location, ω = 0 denotes non-stationary simulation, and ω = 1 returns the training image itself. A precondition hypothesis of the method is that the dimension of the training image must be the same as that of the simulated grid. This requirement is the main limitation, which means that even if simulated areas of different size have the same statistical and sedimentary characteristics, the training image must not be the same and must be prepared accordingly. Indeed, only an assumption of stationarity allows for such a difference in grid sizes with the same training image, and thus the former MPS algorithm is often tested and used in fluvial deposits, which are typically statistically stationary, and the training image can be obtained easily by object-based modelling methods [14]. However, acquiring the training image with a non-stationary property is highly difficult and often requires tedious and timeconsuming manual work. The transportability of the training image with a non-stationary property is necessary, regardless of whether its size is larger or smaller than the simulated grids.

Figure 1 illustrates the theory of the direct pattern-based method and its difference from the traditional method. This method focuses on describing the internal relationship among the position of the deposition, the conditioning data and the sedimentary mode in reservoir modelling. In Figure 1, the training image is shown on the left, and the right side above shows the grids estimated with three conditional data points. If only the conditioning data are considered, at least three data events (a, b, c) match the condition in the training image. However, these data events are located in different portions of the delta front, representing different depositional modes. In traditional MPS methods such as the Snesim and Simpat methods, a random selection is drawn and pasted to the grids [7,14]. However, choosing different data events leads to different simulation results, which influences the undergoing simulation, and the simulated result might be located far away from the reservoir architecture revealed in the training image, which is considered to be a real condition by the geological analysis and information. If data event (a) is selected, it indicates that a sediment type of the main channel and bar system exists and that the flow direction is north-south and north-west, which is not the real condition in the training image and the study area. The simulated value is regarded as conditional data to conduct the subsequent modelling and finally gives an unreasonable realization. The direct pattern-based method gives a minimum distance of the data event (c) because it considers the distance between the central position of the patterns in the simulated area and the training image, which are assigned the same coordinate value in data event (c) for the constraint of the same size. However, if the data event (c) does not match the conditional data, the selection of the direct pattern-based method is limited by the distance of the central location because the conditional data must be exactly matched. A minor deviation of the central location gives a different result. Taking data events (d) and (e) as an example, data event (d) moves left 2 grids, and data event (e) moves down 1 grid of data event (c). The minimum distance is obviously data event (e), regardless of the weight ω. However, in the geological view, the data event (d) might be the first choice.

Figure 1 An illustration of the simulation of the unsimulated grid in traditional MPS methods. In the Snesim method and Simpat method, the selection of pattern is decided by the conditional data and the Monte Carlo sampling, hence a random selection of a, b, c is the modeling result. In the direct pattern-based MPS method, the distance of the position between modeling area and training image is also considered, hence the data event (c) is selected. The data event (d, e) is selected to reveal the problems faced by the direct pattern-based MPS method.
Figure 1

An illustration of the simulation of the unsimulated grid in traditional MPS methods. In the Snesim method and Simpat method, the selection of pattern is decided by the conditional data and the Monte Carlo sampling, hence a random selection of a, b, c is the modeling result. In the direct pattern-based MPS method, the distance of the position between modeling area and training image is also considered, hence the data event (c) is selected. The data event (d, e) is selected to reveal the problems faced by the direct pattern-based MPS method.

When the sizes of the modelling area and the training image are different, the direct pattern-based method collapses. An example is shown in Figure 2, which shows a node in the modelling area of the right image and a training image in the left portion. The size of the modelling area is 120×80, which is different from the 150×100 size of the training image. The coordinate of the simulated node is denoted as (32, 24), and when seeking the data event in the training image, the match point is (40, 30) because the training image is simply a zoomed image of the modelling area, and the sedimentary mode is the same. However, the direct pattern-based method gives a notably high value in this location because it uses the absolute value of the coordinate. This result interferes with the choice of the data events in the training image and results in an unsuitable realization.

Figure 2 An illustration of the size effect in simulation between the direct pattern-based method and the proposed one. The relative coordinate of the simulated position is the same with the training image, but the absolute coordinate is different and may give a large distance value in calculation for the minimum distance in data event selection.
Figure 2

An illustration of the size effect in simulation between the direct pattern-based method and the proposed one. The relative coordinate of the simulated position is the same with the training image, but the absolute coordinate is different and may give a large distance value in calculation for the minimum distance in data event selection.

Figure 3 The random selection of data event in the defined distance tolerance (Rd) for no conditional data constrain.
Figure 3

The random selection of data event in the defined distance tolerance (Rd) for no conditional data constrain.

We propose a method known as the location-based method that introduces the relative distance to overcome the size limitation and reproduce the sedimentary characteristics. The location-based method, which is inspired by the relative place and distance method, can produce the correct result even with different sizes of the two images. The formula for relative distance is given as follows:

dloc=(xu/nxxT/nxtr)2+(yu/nyyT/nytr)2+(zu/nzzT/nztr)2(2)

where (xu, yu, zu) and (xT, yT, zT) are the positions of the unsimulated grid and the training image, respectively, and (nx, ny, nz) and (nxtr, nytr, nztr) are the sizes of the modelling area and the training image, respectively. The relative position is calculated by the ratio in each coordinate direction. Therefore, the distance function (2) can describe the relative distance between the unsimulated grid and the data event in the training image. In Figure 2, the distance of the match point and the unsimulated node is exactly zero using formula (2).

dloc=(24/8030/100)2+(32/12040/150)2=0

In pattern-based method, a parameter ω is assigned to evaluate the stationary with position. Additionally, the new method introduces this concept with a small difference. Because the heterogeneity in each direction as revealed by the variogram calculation might be different, the contribution to dloc in each direction is also different. A calibration weight is introduced to characterize the differences in each direction, which reflects the heterogeneity in 3-D space. The formula (2) can be rewritten as follows:

dloc=ω1(xu/nxxT/nxtr)2+ω2(yu/nyyT/nytr)2+(zu/nzzT/nztr)2(3)

Whereω1, ω2, ω3 is the weight reflecting the heterogeneity in the different direction and can be derived from the variogram calculation or from the geological analysis.

In practical modelling, data conditioning is also considered, and the method for conditioning the data is described in detail in the Snesim and Simpat method [2, 7]. For example, the Simpat method calculates the distance between the data event in a training image and the simulated data event in the study area for data conditioning [7].

dpat=[dev(xuyuzu)Pat(xTyTzT)]2(4)

Where dev (xuyuzu) is the data event surrounding the unsimulated grid at position (xu, yu, zu), and Pat (xTyTzT) is the data event centred at position (xT, yT, zT) in the training image.

The combination of formula (3) and (4) derives a total distance function (5) for the stochastic modelling. The data event in the training image with least distance value is selected and pasted onto the unsimulated area. After all of the unsimulated grid is visited, a realization is finished.

dsim=dloc+dpat(5)

When there are no data surrounding the un-simulated grid, the relative distance functions are the only selection criteria, and the simulation is certain. However, uncertainty always exists in the area of less conditional data. To describe the uncertainty, a local distance tolerance radius (Rd) is given when there are no conditional data. Any data event in Rd is a reasonable selection to paste on the un-simulated grid. At the same time, the data event in a certain position is connected with the simulated grid due to the constraint of the relative position, i.e., we consider a local strategy to select data events instead of scanning the entire training image, which speeds up the simulation programme and increases efficiency.

Finally, the multiple grid method is adopted to reflect the different level and scale of the reservoir heterogeneity characteristics [16]. The semi-random structured path method [17] is used in the conditional data constraint in modelling. The simulation begins at the area with the most conditional data and gradually expands to the area with less conditional data.

The steps are described as follows:

  1. Data pre-processing and training image construction.

  2. Setting of modelling parameters, including the size of the data template, weight value ω and the Rd.

  3. Definition of a semi-random structured path in the simulated grid.

  4. At each un-simulated grid, the surrounding conditional data events are extracted and compared with the data event in the training image. The well-matched data events are saved, including the central position coordinates.

  5. The distance function is to calculate the relative distance between the unsimulated grid and the data event saved in step (4), and the data event with the least distance is selected as the final simulation result.

  6. Local or whole substitution of the un-simulated grid is conducted with the selected data event.

  7. The process continues to the next un-simulated grid, and steps (4) to (6) are repeated until all undefined grids are simulated and the realization is finished.

3 Case studies

3.1 A synthesis case

Because the newly designed method is used to reproduce the non-stationary sedimentary reservoir characteristics, the non-stationary modern deposits in Poyang Lake of China are selected to construct a synthetic threedimensional training image for algorithm testing.

Poyang Lake is a typical modern shallow delta with the characteristics of distributary mouth bars and channels. On the way to the lake, the channels are affected by terrain, slope, water capacity, etc. Unloaded sands form the mouth bars in the estuary, the channels are branched by the mouth bars, and additional mouth bars are formed towards the lake. Finally, the horn distributary mouth bar sedimentary characteristics are formed (Figure 4). The hydrodynamic force becomes gradually weaker, and the deposit thickness of the distributary channels becomes thinner from the estuary to the lake, resulting an increase in the ratio of width/thickness. The direction of the distributary channels spreads widely, and the apex angle is approximately 78°. The number of mouth bars increases gradually lakeward, and the direction is variable and consistent with the channel direction. The sedimentary type shows typical non-stationary geological characteristics and complex heterogeneities on sedimentary reservoir.

Figure 4 The sedimentary pattern of delta in Poyang Lake (China).
Figure 4

The sedimentary pattern of delta in Poyang Lake (China).

The partial area of Poyang Lake is digitized under geological interpretation to construct the training images using the Petrel software (Figure 5). The training image grid size is 199×199×10. This training image is a reference map used to assess the efficiency of the algorithm, i.e., an existing image to be reproduced using the proposed algorithm [15].

Figure 5 A synthesized 3-D training image of Poyang Lake (China).
Figure 5

A synthesized 3-D training image of Poyang Lake (China).

Figure 6 shows the simulation result using the traditional Snesim method in Petrel software. The data template size is 10×10×3 with a multi-grid strategy. Distributary channel bifurcation and variability of the flow direction in the model are not reproduced, and the spatial configuration of the mouth bars and distributary channels is not obvious in the established geological model. The model cannot reveal the non-stationary characteristics in the training image. The newly designed method is also used to construct a model with the same data template and multi-grid strategy (Figure 7). According to Figure 7, the sedimentary characteristics of the overall delta front are well reproduced. Under the influence of the mouth bars, the distributary channels branch and the number increase towards the lake. The space configuration of distributary channels and mouth bars is reflected accurately. In addition, the boundaries between the delta front and the front delta are also reflected clearly. The established geological model is highly similar to the sedimentary model of modern Poyang Lake, which proves that the newly designed method can reproduce the non-stationary reservoir geological characteristics and describe the reservoir sedimentation mode of the delta front.

Figure 6 Model of Poyang Lake constructed by Snesim method.
Figure 6

Model of Poyang Lake constructed by Snesim method.

Figure 7 Model of Poyang Lake constructed by the location-based MPS.
Figure 7

Model of Poyang Lake constructed by the location-based MPS.

Furthermore, 25 pseudo-wells are designed, and the corresponding data of the sedimentary facies are extracted to demonstrate validation of the new designed method under conditional data constraints (Figure 8). A realization is shown in Figure 9. Obviously, the newly designed method is able to establish the model with non-stationary characteristics under the constraint of the well data.

Figure 8 Synthesized well logs with microfacies of Poyang Lake.
Figure 8

Synthesized well logs with microfacies of Poyang Lake.

Figure 9 A 3-D model of Poyang Lake with conditional data.
Figure 9

A 3-D model of Poyang Lake with conditional data.

As a typical characteristic of the newly designed method, the size of the training image can differ the image of the study area. A training image size of 150×100×10 is constructed to test the proposed method, the Snesim method and the Simpat method. For simplicity, the previously used training image is rescaled to give comparable characteristics and sediment type (Figure 10a), but the dimensions of the channel and mouth bar are different from the previously used one with a reduction of approximately 75% in width and 50% in length, although the thickness is the same. Because the modelling area has a large spacing distance between wells, there are more distributary channels and mouth bars than in the training image, which maintains the same dimension between the modelling area and the training image. The Snesim method, Simpat method and the proposed method are executed, and the results are shown in Figure 10b. Due to its stationarity hypothesis, the Snesim method cannot reflect the sedimentary type and the characteristics of the bifurcation, and the dimensions of the modelling channel and the mouth bar are larger than in the training image. The Simpat method produces a much better result than the Snesim in which the delta shape is reflected and the bifurcation of the channels can be recognized, but not obviously (Figure 10c). Selected mudstones, which are not seen in the real-world training image, are reproduced in the location near the accommodation channel. The dimensions of the mouth bar and the channel are approximately 4 cells and 1 cell larger than the one in the training image. The main reason for this result is that the size difference between the study area and the training image is not considered, which means that the selection of the pattern is not suitable and can produce a much larger channel/mouth bar or certain unreasonable mudstones. Indeed, the rescaled well data show a much larger distance than the training image and give an indication of much more bifurcation of the channel and mouth bars in the real reservoir, which is well revealed by the proposed method. The dimension of the simulated channel and mouth bar is well matched with the one in the training image (Figure 10d). The realization shows that the proposed method is flexible and can solve the problem of forecasting of the reservoir with non-stationary statistics.

Figure 10 The modeling results with different size of the study area and training image using the Snesim method, the Simpat method and the proposed one (a. training image, a rescaled map of the image of Figure 5; b.realization of Snesim method; c. realization of Simpat method; d. realization of the location-based method)... continued on the next page.
Figure 10

The modeling results with different size of the study area and training image using the Snesim method, the Simpat method and the proposed one (a. training image, a rescaled map of the image of Figure 5; b.realization of Snesim method; c. realization of Simpat method; d. realization of the location-based method)... continued on the next page.

A statistical analysis is performed to investigate the reproduction of the vertical facies proportion, which is a typical indicator of stationarity or non-stationarity (Figure 11). The conditional data show an increase of the channel proportion and a decrease of the mouth bar. The mudstones occupy the entire area in the top layer and the bottom layer. The proportion of facies shows the same trend, which can be considered as nonstationary characteristics. The realization of Snesim shows a smoothing line from the base to the top, which is a typical stationary statistic. The Simpat shows a trend that is much closer to the conditional data. The location-based method exactly reproduces the proportion of facies and shows the same trend as in the conditional data. Therefore, the location-based method is efficient for reproduction of the real non-stationary statistics.

Figure 11 The vertical proportion of microfacies of the conditional data and realization (a. Conditional data; b. Snesim method; c. Simpat method; d. Location-based method), the location-based method reproduces the vertical trend of the conditional data very well, the Simpat generally good and the Snesim the worst.
Figure 11

The vertical proportion of microfacies of the conditional data and realization (a. Conditional data; b. Snesim method; c. Simpat method; d. Location-based method), the location-based method reproduces the vertical trend of the conditional data very well, the Simpat generally good and the Snesim the worst.

Finally, the calculation time of the new method is approximately 20 minutes, whereas that of the Snesim in Petrel software is approximately 3 minutes. The calculation efficiency is on the same level.

3.2 A real case

To further test the proposed method, an example of a delta reservoir in the Sheng-2 area of the Shengli Oilfield is presented. A lake delta was developed in the target area, and the water depth is approximately 40 metres. The single mouth bar has a lobate shape. Multiple distributary channels formed a set of mouth bars, and their combination shows a larger lobate shape and complex superimposition types and heterogeneities (Figure 12).

Figure 12 The microfacies characteristics of the areal (A) and profiles (B) in Shengli Oilfield (China).
Figure 12

The microfacies characteristics of the areal (A) and profiles (B) in Shengli Oilfield (China).

The simulation domain is located in the bottom of the entire target area (Figure 13), and there are 31 wells in this area. Nine types of architectural elements are recognized through detailed geological study, including the distributary channel, mouth bar, back bar, bar edge, inter bar, mudstone, mud interlayer, petrophysical intercalation and calcareous intercalation. For the main purpose of solving the nonstationary statistics, this time, the training image used is the same size as the area and is constructed by the real reservoir data, which can be tested easily [15]. Truncated Gaussian simulation and manual editing are used to construct a three-dimensional training image (Figure 13). The training image represents the reservoir sedimentary characteristics and is also faithful to the well data for convenience in the comparison and testing. In the training image, the distributary channels are divergent from northeast to southwest. The mouth bars exist on both sides of the channels in the horizontal direction and up and down in profile. The relationship of the channels and mouth bars shows diversity and variation along the channels, which are typically non-stationary statistics. With the constraint of well data and the training image, threedimensional modelling is performed using a 7×7×3 data template and a 2 multi-grid strategy. Figure 13 shows the models constructed by the Snesim and the newly designed method, and their differences are obvious. The model from Snesim cannot reflect the trend from the bottom to the top because it smoothed the variation due to the stationary hypothesis. In addition, the Snesim method encounters difficulty in reproducing the change of the relationship between the distributary channels and mouth bars in the horizontal direction. In contrast, the new method under the constraint of the relative position reveals the vertical and horizontal structure characteristics quite well. The simulated reservoir geological model has good comparability with the training image.

Figure 13 Training images of the study area (left: study area; middle: well data; right: training image).
Figure 13

Training images of the study area (left: study area; middle: well data; right: training image).

Figure 14 Comparison of Snesim method and location-based MPS method (left: real reservoir model; middle: location-based MPS method; right: the Snesim).
Figure 14

Comparison of Snesim method and location-based MPS method (left: real reservoir model; middle: location-based MPS method; right: the Snesim).

From the view of the well section of the realization (Figure 15), the model of Snesim shows a larger difference from the actual reservoir. The reservoir on the vertical distribution is relatively uniform, and it is difficult to observe the obvious vertical trend. However, the new method can clearly exhibit the vertical distribution of the reservoir trend, in which the mudstones are present from the bottom to the top are and gradually substituted with mouth bars and distributary channels. The reservoir space distribution has good consistency with the actual reservoir distribution, and the newly designed method is more reasonable.

Figure 15 Comparison in profile of Snesim method and location-based MPS method (up: real reservoir model; middle: location-based MPS method; bottom: Snesim).
Figure 15

Comparison in profile of Snesim method and location-based MPS method (up: real reservoir model; middle: location-based MPS method; bottom: Snesim).

Figure 16 shows the vertical facies proportion of the conditional data and the realizations. The location-based method gives a better reproduction of the facies proportion and trend than the Snesim method. The proportion of the main facies (such as mouth bar) shows an average absolute error of 3.1%, whereas the Snesim error between the conditional data and the realizations is 9.1%. This result proves that the location-based method improves the reproduction of statistical characteristics and the accuracy of the realization.

Figure 16 The vertical proportion of microfacies of the conditional data and realization. The location-based method has reproduced the proportion of facies much better than the Snesim method.
Figure 16

The vertical proportion of microfacies of the conditional data and realization. The location-based method has reproduced the proportion of facies much better than the Snesim method.

Additionally, the calculation efficiency is compared between the two methods, and the time is approximately 3 hours and 40 minutes for the new method and approximately 34 minutes for the Snesim. The new method should be optimized in future work.

4 Conclusions

This study primarily investigates the traditional MPS methods and efficiency in modelling of the non-stationary reservoir, and a new location-based MPS method is pro- posed and compared with the traditional method. From this study, some conclusions can be drawn as follows:

  1. The traditional MPS methods, such as Snesim and Simpat, are suitable for modelling of fluvial deposits with stationary statistical characteristics. These methods are not useful for modelling delta deposits because of the non-stationary statistical characteristics.

  2. The direct pattern-based MPS method can solve the reproduction of the reservoir with nonstationary statistical characteristics only under the strict constraint of the same size for the modelling area and the training image. This constraint limits the flexibility and wider use of the method.

  3. A new location-based MPS method is proposed for simulating the non-stationary reservoir. This method adopts the concept of relative position to alleviate the strict demand for the same size of the modelling area and the training image. Hence, the flexibility is improved, and real reservoir use is expected.

  4. A local search strategy is also introduced to ensure the calculation efficiency, which is often an obstacle in the real world. The modelling time is compared, and the new method is slightly time-consuming but within the range of acceptance.

  5. Two examples of Poyang Lake and Shengli Oilfield are proposed to illustrate and compare the new method and the traditional ones. The new method produces a much better realization than the traditional methods. The average error of the mouth bar proportion is 3.1%, much less than the 9.1% of the Snesim method, which shows the high accuracy of the realization. The new method is much more efficient in modelling of the nonstationary reservoir.

Acknowledgement

We greatly appreciate the editors and two anonymous reviewers for their valuable and constructive comments and suggestions, which helped us to improve our manuscript. We are also thankful for the generous financial support from the National Natural Science Foundation of China (No. 41572081), the National Scientific Important Project (No. 2016ZX05031-002-001), the Teachers Foundation for the Youth in Yangtze University and the open project of Shandong Provincial Key Laboratory of Depositional Mineralization & Sedimentary Minerals (No. DMSM201404). Professor Dali Yue of China University of Petroleum is thankful for discussions on the deposits of Sheng-2 area in Shengli Oilfield.

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Received: 2017-3-28
Accepted: 2017-11-10
Published Online: 2017-12-29

© 2017 Y. Yin and W. Feng

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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