HIERARCHICAL SIMULATION OF MULTIPLE-FACIES RESERVOIRS USING MULTIPLE-POINT GEOSTATISTICS

Transcription

1 HIERARCHICAL SIMULATION OF MULTIPLE-FACIES RESERVOIRS USING MULTIPLE-POINT GEOSTATISTICS AREPORT SUBMITTED TO THE DEPARTMENT OF PETROLEUM ENGINEERING OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE By Amisha Maharaja June 2004

2 I certify that I have read this report and that in my opinion it is fully adequate, in scope and in quality, as partial fulfillment of the degree of Master of Science in Petroleum Engineering. André Journel (Principal advisor) ii

3 Abstract Joint simulation with presently available multiple-point geostatistical simulation algorithms leads to poor shape reproduction as the number of facies increases. This is because the training image (Ti) cannot depict with enough replicates all alternative patterns that can be found. Moreover, the size of the Ti itself and that of the template required to capture large-scale structures in the Ti is limited due to memory restriction, especially in 3D. Hierarchical simulation of facies with distinct shapes and spatial continuity is proposed to overcome the disadvantages of joint simulation. The idea is to identify a hierarchy of deposition within the reservoir using geologic rules of deposition and simulate the facies accordingly. This amounts to first simulating the large-scale structures and then simulating smaller structures conditional to the pre-simulated large-scale structures. The hierarchical approach is demonstrated using synthetic 2D and 3D meandering fluvial reservoirs and the results are compared with that of joint simulation. Finally, the hierarchical simulation techinique is applied to a real life dense data set from the Rhine-Meuse delta. The advantage of hierarchical approach is better reproduction of large-scale structures by reducing the size of the Ti and the number of facies being simulated simultaneously. Most of the memory demand during mp-simulation comes from the size of the search-tree, which in turn depends on the size of the Ti, the size of the data template, and the number of facies in the Ti. Large-scale structures require a bigger template to capture them, however that increases the size of the search tree. In hierarchical simulation, the increase in the size of the search tree due to a larger template is compensated by the reduction in the number of facies being simulated simultaneously. Once the large-scale structures are simulated the smaller structures iii

4 are simulated conditioned to the former. The Ti for simulating the small-scale features need not be very large and rich, thereby further reducing the size of search tree. The hierarchical approach is geologically sound because the sequence of simulations follows the natural sequence of deposition. iv

5 Acknowledgements v

6 Contents Abstract Acknowledgements Table of Contents List of Tables List of Figures iii v vi viii ix 1 Introduction 1 2 Joint Simulation Comments Hierarchical Simulation Hierarchical simulation of three facies Comments Hierarchical simulation of four facies Two-step cookie-cut simulation method Three-step hierarchical simulation method Two-step hierarchical simulation method D Example Implementation of hierarchical simulation 27 vi

7 5 Rhine-Meuse Delta Case Study Introduction to the Data Set Results and Discussion Conclusions and Future Work Conclusions Future work References 32 vii

8 List of Tables 2.1 Input parameters and simulation statistics for the joint simulation of 2, 3 and 4 facies, see Figures 2.2, 2.4, 2.6, and 2.8. The servo-system parameter is in [0,1], with 1 corresponding to maximum control. The parameters Nodes in data template and Radii of search ellipsoid determine the size and geometry of the data template. Average nodes retained in template indicates the average number of nodes that were retained in the data template Input parameters and simulation statistics for the Two-step cookie-cut approach Input parameters and simulation statistics for the Three-step hierarchical approach Input parameters and simulation statistics for the Two-step hierarchical approach viii

9 List of Figures 1.1 The morphological elements of a meandering-river system Binary training image of channels (100 x 300). Green: channel 22%; grey: floodbasin 78% Unconditional realization using two facies training image (100 x 100). Green: channel 29.9%; grey: floodbasin 70.1% Three facies training image (100 x 300). Green: channel 22%; blue: levee 7%; grey: floodbasin 71% Joint simulation of three facies using a 3-facies training image (100 x 100). Green: channel 28.9%; blue: levee 7.6%; grey: floodbasin 63.5% Three facies training image (100 x 300). Green: channel 20%; red: crevasse 7%; grey: floodbasin 73% Joint simulation of three facies using a 3-facies training image (100 x 100). Green: channel 29.8%; red: crevasse 7.1%; grey: floodbasin 63.1% Four facies training image (100 x 300). Green: channel 21%; blue: levee 6%; red: crevasse 5%; grey: floodbasin 68% Joint simulation of four facies using a 4-facies training image (100 x 100). Green: channel 25%; blue: levee 8%; red: crevasse 5%; grey: floodbasin 61.8% Hierarchical simulation of only levee and floodbasin using a 3-facies Ti (100 x 100). Channels were previously simulated (Figure 2.2) and set as hard data. Green: channel 30%; red: levee 7%; grey: floodbasin 63% 17 ix

10 3.2 Hierarchical simulation of only crevasse and floodbasin using a 3-facies Ti (100 x 100). Channels are previously simulated (Figure 2.2). Green: channel 30%; red: crevasse 8%; grey: floodbasin 62% Combining Figures 3.1 and 3.2 using cookie-cut. Crevasse erodes levees and floodbasin. Green: channel 30%; blue: levee 7%; red: crevasse 8%; grey: floodbasin 55% Flowchart for two-step cookie-cut simulation method Step 3 of three-step method: Simulation of only crevasse and floodbasin using a 4-facies Ti. The channels were simulated in step 1 (Figure 2.2) and levees were simulated in step 2 (Figure 3.1) and set as hard data. Simulated facies proportions:- Green: channel 30%; blue: levee 7%; red: crevasse 6%; grey: floodbasin 57% Flowchart for three-step hierarchical simulation method Step 2 of two-step hierarchical simulation method. Simulation of crevasse and floodbasin using 4-facies Ti. The channel and levee have been previously simulated in step 1 and set as hard data (Figure 2.4). Simulated facies proportions: Green: channel 28.9%; blue: levee 7.6%; red: crevasse 6.4%; white: floodbasin 57% Flowchart for two-step hierarchical simulation method Four facies training image (100 x 100 x 100). Green: channel 25%; blue: pointbar 4%; red: levee 2%; grey: floodbasin 69% Hierarchical simulation of channel, pointbar, levee and floodbasin (100 x 100 x 100). Green: channel 28%; blue: pointbar 5%; red: levee 4%; grey: floodbasin 63% x

11 Chapter 1 Introduction Stochastic simulation is widely used for generating heterogeneous reservoir models. Traditional variogram-based simulation methods utilize the correlation between only two points at a time and therefore cannot simulate curvilinear structures. Examples are SGS and SIS methods. Multiple-point geostatistical simulation techniques consider the relation between three or more points taken together and are able to reproduce curvilinear structures and account for complex patterns between the variables being simulated. The pros and cons of various methods that utilize multiple-point geostatistics is discussed in Strebelle (2000). Boolean object-based methods can reproduce the curvilinear geometries but are difficult to condition to dense data. Pixel-based methods are easier to condition to dense data from different sources. Simulated annealing, MCMC simulation and similar algorithms are pixel-based but iterative techniques and their rate of convergence is not known a priori. The first non-iterative multiple-point algorithm was suggested by Guardiano and Srivastava (1993) in which mp-statistics were obtained directly by scanning a training image (Ti). However, the algorithm was extremely CPU demanding because the entire Ti had to be scanned completely at each unsampled node to obtain conditional probability distribution for that node. Strebelle (2000, 2002) proposed an 1

12 CHAPTER 1. INTRODUCTION 2 algorithm snesim which was based on the idea of Guardiano and Srivastava of deriving probabilities directly from a Ti, however, it required scanning the Ti only once and cataloguing the conditional probabilities using a dynamic data structure called search-tree. The snesim algorithm is pixel-based, non-iterative, and general so that any random geometry can be accomodated. The size of the search-tree depends on three factors Size of the training image Size of the data template used to scan the training image Total number of facies in the training image Consider a categorical variable Z(u) valuedin1,...,k. Let N TI be the total number of locations in the Ti. Since for a given data event size j, therecannotbe more than N TI different data events in the Ti, a (crude) upper-bound of the memory demand of the search tree is: Memory Demand J min(k j,n TI ) j=1 Hydrocarbon reservoirs often consist of multiple facies with different shapes and sizes. For example, a meandering fluvial system typically contains channels, pointbars, levees, crevasse splays and floodbasin (Figure 1.1). Note that the geologic definition of facies is Accumulation of deposits that exhibits specific characteristics and grades laterally into other sedimentary accumulations that were formed at the same time but exhibit different characteristics (Leet, 1982). In this paper, the term facies refers to the morphological elements of a depositional system that have a distinct shape. The shapes and proportions of these different facies vary greatly from one fluvial system to another. Moreover, the facies are related to each other by some definite geologic rules, for example, crevasse splays must be attached to channel belt. In mpsimulation, the shapes, relative proportions and the spatial dependence between the

13 CHAPTER 1. INTRODUCTION 3 different facies is conveyed through a Ti. Fluvial systems tend to be very complex, hence a large and rich Ti is required if all the facies are simulated jointly, which can be very memory demanding. Instead of simulating all the facies jointly, they can be simulated sequentially. Strebelle (2000) proposed a hierarchical method for simulating four fluvial facies namely, channels, levees, crevasse splays, and floodbasin. However, the specific hierarchy used to simulate the facies and the manner in which the hierarchy was implemented is different from the implementation presented in this report, see Chapter 4 for details. The hierarchy used here is based on geologic rules of deposition of the facies, see Chapter 3 Once the shapes of the facies are simulated with the correct proportion and spatial arrangement, sand and mud with varying net-to-gross can be simulated within each facies using traditional variogram based algorithms such as SGS. This is similar to first simulating the different containers and then their specific contents. The net-togross ratio, defined as the percentage of sand, is generally highest in the channels, followed by the crevasse splays and levees. Floodbasin contains the largest volume of fine sediments in the fluvial system. Because this is a pixel-based stochastic simulation, conditioning to dense data of various support sizes is easier. Moreover, a measure of uncertainty can be attained by simulating multiple realizations using the same Ti, or by using multiple realizations of several Tis depicting alternative plausible geological scenarios. Chapter 2 shows results of joint simulation of two, three and four facies reservoirs. Hierarchical simulation of three and four facies reservoirs is introduced in Chapter 3. Three alternative approaches are discussed for the 2D reservoir and the results are compared with that of joint simulation. Finally, a 3D example is presented. The implementation of hierarchical simulation in the snesim algorithm is discussed in Chapter 4. The Rhine-Meuse delta case-study is presented in Chapter 5. Conclusions and recommendations for future work are presented in Chapter 6.

14 CHAPTER 1. INTRODUCTION 4 Figure 1.1: The morphological elements of a meandering-river system.

15 Chapter 2 Joint Simulation To perform joint simulation of multiple facies, all the facies must be provided in the Ti exhibiting the proper relationship between these facies. The proportion of the facies in the Ti need not be equal to the desired simulated proportions as the latter proportions can be controlled by a servo-system. Fluvial systems can be quite complicated because facies of different shapes and sizes are involved, hence it is desirable to use a Ti that is larger than the size of the simulation grid to provide enough replicates of any particular pattern or structure. This comes at a cost of larger RAM memory demand. A binary training image consisting of channel and floodbasin (Figure 2.1) is used to simulate the realization shown in Figure 2.2. All the training images in this paper have been generated using the fluvsim algorithm developed by Deutsch and Tran (2002). It is important that the Ti provides the correct channel attributes such as, width, thickness, and sinuosity, to get the desired results. These attributes can be obtained directly from seismic amplitude maps, outcrop analogues or inferred from well data (Bridge and Tye, 2000). Moreover, in order to simulate long, thin, continuous channels, it is important to provide a large Ti. In this case, the Ti is 100 x 300 pixels, while the simulation grid is 100 x 100 pixels. The 3 facies channel, levee, and floodbasin are jointly simulated (Figure 2.4) using a 3-facies channel-levee-floodbasin Ti (Figure 2.3). Similarly, channel, crevasse, and floodbasin are jointly simulated (Figure 2.6) using the corresponding 3-facies 5

16 CHAPTER 2. JOINT SIMULATION 6 channel-crevasse-floodbasin Ti (Figure 2.5). Finally, all four facies are jointly simulated (Figure 2.8) using a 4-facies Ti (Figure 2.7). 2.1 Comments Table 2.1 summarizes the input parameters and simulation statistics for the joint simulation approach. The poor quality of results from joint simulation of more than two facies is evident from the examples in Figures 2.4, 2.6, and 2.8. The complexity of the Ti in Figure 2.5 is greater than that in Figure 2.3 because the crevasse splays have a distinctly different shape than the channels. The crevasse splays of Figure 2.5 are discontinuous, fan-like bodies, while the levees of Figure 2.3 are fairly continuous facies that border the channels with the same elongated rectangular shape. Hence, the joint simulation of a channel-levee-floodbasin system (Figure 2.4) gives better results than that of a channel-crevasse-floodbasin system (Figure 2.6). Since the Ti is three times as long as the simulation grid, the simulated channels and levees of Figure 2.4 are reasonably continuous. Moreover, the levees and crevasse splays are simulated close to the channels as specified by the training images. The servo-system has been enforced, hence the target facies proportions are correctly reproduced, see Table 2.1. The shape of the crevasse splays is adequately reproduced in both Figures 2.6 and 2.8. When the four-facies are simulated jointly (Figure 2.8), the results are much poorer as the complexity of the Ti has increased considerably. The quality of results can be improved by using a much larger Ti as the variety of patterns found in the Ti as well as their number of replicates would increase, however for the same reason the size of the search tree and hence the RAM demand will increase. Moreover, to ensure a good reproduction of the large scale features a large data template is required, which in case of 3 or more facies quickly leads to a large RAM demand, especially in 3D. Multiple-grid simulation approach (Tran, 1994) has been proposed as a work-around for simulating large-scale structure, however, a larger template size is still desirable.

17 CHAPTER 2. JOINT SIMULATION 7 Figure 2.1: Binary training image of channels (100 x 300). Green: channel 22%; grey: floodbasin 78% Figure 2.2: Unconditional realization using two facies training image (100 x 100). Green: channel 29.9%; grey: floodbasin 70.1%

18 CHAPTER 2. JOINT SIMULATION 8 Parameters 2facies 3facies 3facies 4facies Figure 2.2 Figure 2.4 Figure 2.6 Figure 2.8 Training Image 2facies 3facies 3facies 4facies Figure 2.1 Figure 2.3 Figure 2.5 Figure 2.7 Ti facies proportion(%) Channel:22 Channel:22 Channel:20 Channel:21 Levee:7 Crevasse:7 Levee:6 Crevasse:5 Cond. data proportion(%) None None None None Target proportion (%) Channel:30 Channel:30 Channel:30 Channel:23 Levee:7 Crevasse:7 Levee:7 Crevasse:7 Simulated proportion(%) Channel:29.9 Channel:28.9 Channel:29.8 Channel:24.7 Levee:7.6 Crevasse:7.1 Levee:8.4 Crevasse:5.1 Servosystem parameter # of multiple grids Nodes in data template Radii of search ellipsoid 50,50,1 50,50,1 50,50,1 50,50,1 Average nodes retained in template simulation time (sec) on 3 GB RAM machine max. RAM used (GB) Table 2.1: Input parameters and simulation statistics for the joint simulation of 2, 3 and 4 facies, see Figures 2.2, 2.4, 2.6, and 2.8. The servo-system parameter is in [0,1], with 1 corresponding to maximum control. The parameters Nodes in data template and Radii of search ellipsoid determine the size and geometry of the data template. Average nodes retained in template indicates the average number of nodes that were retained in the data template.

19 CHAPTER 2. JOINT SIMULATION 9 Figure 2.3: Three facies training image (100 x 300). Green: channel 22%; blue: levee 7%; grey: floodbasin 71% Figure 2.4: Joint simulation of three facies using a 3-facies training image (100 x 100). Green: channel 28.9%; blue: levee 7.6%; grey: floodbasin 63.5%

20 CHAPTER 2. JOINT SIMULATION 10 Figure 2.5: Three facies training image (100 x 300). Green: channel 20%; red: crevasse 7%; grey: floodbasin 73% Figure 2.6: Joint simulation of three facies using a 3-facies training image (100 x 100). Green: channel 29.8%; red: crevasse 7.1%; grey: floodbasin 63.1%

21 CHAPTER 2. JOINT SIMULATION 11 Figure 2.7: Four facies training image (100 x 300). Green: channel 21%; blue: levee 6%; red: crevasse 5%; grey: floodbasin 68% Figure 2.8: Joint simulation of four facies using a 4-facies training image (100 x 100). Green: channel 25%; blue: levee 8%; red: crevasse 5%; grey: floodbasin 61.8%

22 Chapter 3 Hierarchical Simulation 3.1 Hierarchical simulation of three facies The hierarchical simulation of 3 facies is done in two steps. The first step is joint simulation of channel and floodbasin facies using the binary Ti shown in Figure 2.1. The simulated channel pixels (Figure 2.2) are frozen as hard data. In the second step, the channel-levee-floodbasin Ti (Figure 2.3) and channel-crevasse-floodbasin Ti (Figure 2.6) are used to generate Figures 3.1 and 3.2 respectively. During the second step, the Ti-derived probability of simulating a channel pixel is set to zero so that no more channel is simulated. The servo-system correction is applied at both steps to ensure reproduction of the target proportions Comments Figure 3.1 indicates a poor reproduction of the elongated shape of the levees, while Figure 3.2 shows a good reproduction of the crevasse shape. The channels are continuous as they were simulated with a large binary Ti and a large template independently of the crevasse and levees. Both crevasse and levees are attached to the channels in spite of having been simulated independently of the channels, however, some isolated levee pixels are found in Figure 3.1. It is recommended that sequential simulation proceeds over a single grid for the hierarchical simulation of levee so that isolated levee pixels are minimized. Indeed, in the multiple-grid approach, the hard data are 12

23 CHAPTER 3. HIERARCHICAL SIMULATION 13 relocated to the nearest grid node, which combined with the dropping of nodes in the data template can cause levees to be simulated in the floodbasin detached from the channels. 3.2 Hierarchical simulation of four facies In the case of 4 facies, the hierarchical simulation can be done in three ways. In all three hierarchical approaches the natural sequence of development of the fluvial facies is adhered to, which is as follows: The main channel belt forms prior to levees and crevasse splays and all sediments are deposited inside the channel belt, hence in the first two hierarchical approaches the channels are simulated prior to levees and crevasse. When excessive sediments are supplied, they cannot be contained within the channel belt and they spill over to form levees and floodbasin deposits. Consequently, in the first two approaches levee is simulated after channel, while in the last approach it is simulated jointly with channel. If the channel has a high sinuosity, the levees can be breached and crevasse splays are formed adjacent to the channel belt, hence in all three approaches, crevasse is simulated last Two-step cookie-cut simulation method In this method, a 4-facies training image is not needed. The results from hierarchical simulation of channel-levee-floodbasin (Figure 3.1) and channel-crevasse-floodbasin (Figure 3.2) are cookie-cut one onto the other such that the crevasse erodes the levee: this results in a simulated 4-facies image (Figure 3.3). This approach is geologically correct because the crevasse splays form by breaching the natural levees. After cookiecut the proportion of the levee will be lower because some of the levee pixels are replaced by crevasse pixels. The flowchart in Figure 3.4 summarizes the simulation procedure and Table 3.1 summarizes the input parameters and simulation statistics for this method.

24 CHAPTER 3. HIERARCHICAL SIMULATION 14 Parameters step 1 step2 step2 step3 Figure 2.2 Figure 3.1 Figure 3.2 Figure 3.3 Training Image 2facies 3facies 3facies 4facies Figure 2.1 Figure 2.3 Figure 2.5 N/A Ti facies proportion(%) Channel:22 Channel:22 Channel:20 N/A Levee:7 Crevasse:7 Cond. data prop.(%) None Channel:100 Channel:100 N/A Levee:0 Crevasse:0 Target proportions (%) Channel:30 Channel:30 Channel:30 N/A Levee:7 Crevasse:7 Simulated proportions (%) Channel:29.9 Channel: 29.9 Channel:29.9 Channel:29.9 Levee:7.1 Crevasse:8.2 Levee:7 Crevasse:8.2 Servosystem parameter N/A # of multiple grids N/A Nodes in data template N/A Radii of search ellipsoid 50,50,1 50,50,1 50,50,1 N/A Average nodes retained in template N/A simulation time (sec) N/A on 3 GB RAM machine max. RAM used (GB) N/A Table 3.1: Input parameters and simulation statistics for the Two-step cookie-cut approach Comments In Figure 3.3 the channels are continuous and both levee and crevasse are simulated close to the channels. Since a 4-facies Ti is not used, the RAM required is smaller than that for joint simulation of the four facies. This allows the use of a larger template to capture the large-scale features. The shape of the crevasse is well reproduced. Since crevasse is simulated independently of levee using a 3-facies channel-crevassefloodbasin Ti, it is not affected by the isolated levee pixels. Simulation of channelfloodbasin took 7.2 seconds while that of levee-floodbasin took 2.1 seconds and that of crevasse-floodbasin took 3.9 seconds on a 2.0 GHz desktop with 3 GB RAM.

25 CHAPTER 3. HIERARCHICAL SIMULATION Three-step hierarchical simulation method The 4 facies can also be simulated in three steps as follows: First only the channel and floodbasin are simulated using a binary Ti and the simulated channels as set as hard data (Figure 2.2). In the second step, only levees and floodbasin are simulated (Figure 3.1) using a 3-facies channel-levee-floodbasin Ti (Figure 2.3). The simulated levees are set as hard data together with the simulated channels from step 1. Finally, crevasse and floodbasin are simulated (Figure 3.5) using a 4-facies channel-crevasselevee-floodbasin Ti (Figure 2.7). In the second step, the Ti-derived probability of simulating a channel pixel is set to zero so that no more channel is simulated. Similarly, in step three, the Tiderived probabilities of simulating both channel and levee pixels is set to zero so that no more channel and levees are simulated. Servo-system correction is applied at all three steps to ensure reproduction of target proportions. The procedure for this method is summarized in the flowchart in Figure 3.6. Table 3.2 summarizes the input parameters and simulation statistics for this method. Comments In Figure 3.5 the channels are continuous as they were simulated using a large binary Ti and a large template. Since the crevasse simulation is conditioned to previously simulated channel as well as levee values, some crevasse pixels are simulated next to the isolated levee pixels, which is permitted by the Ti. The target proportions of the four facies are well reproduced because the servo-system is applied. The total simulation took 14.4 seconds Two-step hierarchical simulation method Comparing the results of joint versus hierarchical simulation of channel-levee-floodbasin system, it is observed that the levees are better simulated jointly with channels since they have the same elongated shape as the channels. Hence, in this last approach, the levees are simulated jointly with channel and floodbasin in the first step using a 3-facies channel-levee-floodbasin Ti. The simulated channel and levee pixels are then

26 CHAPTER 3. HIERARCHICAL SIMULATION 16 frozen as hard data (Figure 2.4). Finally, only crevasse and floodbasin are simulated (Figure 3.7) using a 4-facies channel-levee-crevasse-floodbasin Ti (Figure 2.7) by setting to zero the Ti-derived probability of simulating channel and levee pixels. The servo-system correction is applied at both steps to ensure reproduction of target proportions of the four facies. The flowchart in Figure 3.8 summarizes the procedure and Table 3.3 summarizes the input parameters and simulation statistics for this method. Comments In Figure 3.7 the continuity of the levees is improved and isolated pixels are minimized by simulating the levees with the channels. The channels are reasonably continuous in spite of using a 3-facies Ti, because the levee has the same elongated shape as the channel. The shape of the crevasse is well reproduced and they are attached to the channels. The target proportions of the four facies are well reproduced because the servo-system is applied. The first step took 4.9 seconds, the second step took 5.1 seconds D Example The hierarchical simulation approach is applied to a synthetic 3D meandering channel reservoir with four facies namely, channel, pointbar, levee, and floodbasin. The training image is 100 x 100 x 100 grid blocks (Figure 3.9). In Figure 3.9, the channels are sinuous and have a U-shaped cross-section. The pointbars are deposited in the inner bends of the channel and in cross-section they occur along one margin of the channel. The levees have an elongated shape with a triangular cross-section and are deposited adjacent to the channel. Some of the channels are amalgamated, hence the pointbar is found between two channels. Thus, the pointbar needs to be simulated jointly with the channel so that it can be simulated inside channels as depicted by the Ti. The simulation grid is 100 x 100 x 100 grid blocks. The two-step hierarchical approach is used to generate the final 4-facies realization. In step 1, only the channel, pointbar and floodbasin are simulated using the corresponding 3-facies Ti. That

27 CHAPTER 3. HIERARCHICAL SIMULATION 17 Ti was obtained from Figure 3.9 by merging the levee facies with the floodbasin. Furthermore, only the first 50 layers of the Ti were used for the simulation to reduce the RAM requirement by reducing the size of the search tree. In step 2, only levees and floodbasin are simulated conditioned to the previously simulated channel and pointbar. The first 50 layers of the original 4-facies Ti are used for this simulation. During the second step, the Ti-derived probability of simulating a channel or pointbar pixel is set to zero so that no more channel or point-bar is simulated. Comments Figure 3.10 shows the final 4-facies realization. Both channel and pointbar are wellreproduced in plan view and cross-section. Channels become discontinuous when they thin out vertically. The sinuosity of the channel is discernable and the pointbar is simulated in the inner bend of the channel as specified by the Ti. It was possible to capture this large-scale structure by using a large data template in combination with the multiple-grid appraoch. By using hierarchy, the problem was reduced to simulation of 3 facies instead of 4 facies in the first step, which enabled using a large data template, which might not have been possible if the 4 facies had been jointly simulated. It was difficult to simulate continuous levees because in the Ti they are fragmented as they thin upward. Figure 3.1: Hierarchical simulation of only levee and floodbasin using a 3-facies Ti (100 x 100). Channels were previously simulated (Figure 2.2) and set as hard data. Green: channel 30%; red: levee 7%; grey: floodbasin 63%

28 CHAPTER 3. HIERARCHICAL SIMULATION 18 Figure 3.2: Hierarchical simulation of only crevasse and floodbasin using a 3-facies Ti (100 x 100). Channels are previously simulated (Figure 2.2). Green: channel 30%; red: crevasse 8%; grey: floodbasin 62% Figure 3.3: Combining Figures 3.1 and 3.2 using cookie-cut. Crevasse erodes levees and floodbasin. Green: channel 30%; blue: levee 7%; red: crevasse 8%; grey: floodbasin 55%

29 CHAPTER 3. HIERARCHICAL SIMULATION 19 Parameters step 1 step2 step3 Figure 2.2 Figure 3.1 Figure 3.5 Training Image 2facies 3facies 4facies Figure 2.1 Figure 2.3 Figure 2.7 Ti facies proportion(%) Channel:22 Channel:22 Channel:21 Levee:7 Levee:6 Crevasse:5 Cond. data prop.(%) None Channel:100 Channel: 76.6 Levee:0 Levee:22.9 Crevasse:0 Target proportion(%) Channel:30 Channel:30 Channel:30 Levee:7 Levee:7 Crevasse:7 Simulated proportion(%) Channel:29.9 Channel:29.9 Channel:29.9 Levee: 7.1 Levee:7.1 Crevasse:6.2 Servosystem parameter # of multiple grids Nodes in data template Radii of search ellipsoid 50,50,1 50,50,1 50,50,1 Average nodes retained in template simulation time (sec) on 3 GB RAM machine max. RAM used (GB) Table 3.2: Input parameters and simulation statistics for the Three-step hierarchical approach

30 CHAPTER 3. HIERARCHICAL SIMULATION 20 Parameters 3facies 4facies Figure 2.4 Figure 3.7 Training Image 3facies 4facies Figure 2.3 Figure 2.7 Ti facies proportion(%) Channel:22 Channel:21 Levee:7 Levee:6 Crevasse:5 Cond. data proportion(%) None Channel:79.2 Levee:20.8 Crevasse: 0 Target proportion (%) Channel:30 Channel:23 Levee:7 Levee:7 Crevasse:7 Simulated proportion(%) Channel:28.9 Channel:28.9 Levee:7.6 Levee:7.9 Crevasse:6.4 Servosystem parameter # of multiple grids 3 3 Nodes in data template Radii of search ellipsoid 50,50,1 50,50,1 Average data retained in template simulation time (sec) on 3 GB RAM machine max. RAM used (GB) Table 3.3: Input parameters and simulation statistics for the Two-step hierarchical approach

31 CHAPTER 3. HIERARCHICAL SIMULATION 21 Figure 3.4: Flowchart for two-step cookie-cut simulation method

32 CHAPTER 3. HIERARCHICAL SIMULATION 22 Figure 3.5: Step 3 of three-step method: Simulation of only crevasse and floodbasin using a 4-facies Ti. The channels were simulated in step 1 (Figure 2.2) and levees were simulated in step 2 (Figure 3.1) and set as hard data. Simulated facies proportions:- Green: channel 30%; blue: levee 7%; red: crevasse 6%; grey: floodbasin 57% Figure 3.6: Flowchart for three-step hierarchical simulation method

33 CHAPTER 3. HIERARCHICAL SIMULATION 23 Figure 3.7: Step 2 of two-step hierarchical simulation method. Simulation of crevasse and floodbasin using 4-facies Ti. The channel and levee have been previously simulated in step 1 and set as hard data (Figure 2.4). Simulated facies proportions: Green: channel 28.9%; blue: levee 7.6%; red: crevasse 6.4%; white: floodbasin 57%

34 CHAPTER 3. HIERARCHICAL SIMULATION 24 Figure 3.8: Flowchart for two-step hierarchical simulation method

35 CHAPTER 3. HIERARCHICAL SIMULATION 25 Figure 3.9: Four facies training image (100 x 100 x 100). Green: channel 25%; blue: pointbar 4%; red: levee 2%; grey: floodbasin 69%

36 CHAPTER 3. HIERARCHICAL SIMULATION 26 Figure 3.10: Hierarchical simulation of channel, pointbar, levee and floodbasin (100 x 100 x 100). Green: channel 28%; blue: pointbar 5%; red: levee 4%; grey: floodbasin 63%

37 Chapter 4 Implementation of hierarchical simulation All runs in this paper were done using the snesim algorithm (Strebelle, 2000, 2002). In order to use hierarchical simulation, modifications to the existing snesim program were necessary. To illustrate these changes, consider the two-step hierarchical approach discussed in the paper. In the first step, the 3 facies, channel, levee and floodbasin are simulated using the corresponding 3-facies Ti. The simulated channel and levee pixels should be extracted from the simulation output file along with their location and saved in a separate file so that they can be supplied as hard data for the next step. This step requires no modification in the program. In the second step, a 4-facies channel-levee-crevassefloodbasin Ti is used to simulate only two facies, crevasse and floodbasin. In order to accomplish this, the following must be done at each location to be simulated: Get the Ti-derived proportions for a given data event Set the local probability of channel and levee to zero Re-standardize correspondingly the probabilities of crevasse and floodbasin Apply the servo-system correction to crevasse and floodbasin One last modification is required when obtaining the Ti-derived proportions from the search tree. When an uninformed node is encountered in the data template, the 27

38 CHAPTER 4. IMPLEMENTATION OF HIERARCHICAL SIMULATION 28 original snesim program considers the possibility of having any one of the four facies that exist in the Ti. However, in step 2, since only crevasse and floodbasin can be simulated, only these two facies should be considered. Indeed channel and levee have already been simulated in step 1 and set as hard data, hence the uninformed node in the data template cannot be these two facies.

39 Chapter 5 Rhine-Meuse Delta Case Study 5.1 Introduction to the Data Set 5.2 Results and Discussion 29

40 Chapter 6 Conclusions and Future Work 6.1 Conclusions Hierarchical simulation gives better results than joint simulation when more than three facies are involved. This approach is highly recommended for both algorithmic and geological considerations. Three alternative approaches for simulating four facies hierarchically were demonstrated. As the number of facies increases, the alternatives to simulate them hierarchically also increases. The actual hierarchy of simulation steps should be guided by the geological rules of deposition. In case of the levees, which have elongation similar to channels, their joint simulation with channels produces better results than a hierarchical simulation, in which some isolated pixels can be generated. Thus in similar depositional contexts, it might be better to simulate similar shaped facies together. The shape of crevasse splays is very different from that of the channels and levees. Hence, simulating crevasse separately from channels and levees gives better results. Thus, facies with different shapes and continuity should be simulated separately. A 4-facies training image can be used to simulate only two facies as was done in 30

41 CHAPTER 6. CONCLUSIONS AND FUTURE WORK 31 the two-step hierarchical simulation method, see Figure 3.7. However, the relationship between the different facies is extracted from the complex four-facies training image. This concept can be extended to other depositional environments with multiple facies. During hierarchical simulation, facies such as levee and crevasse splays are attached to the channels as specified in the training image, even though the levee and crevasse are simulated independently of the channels. In this particular case-study, the two-step hierarchical simulation method (Figure 3.7) gives the best results out of the three hierarchical methods, because channels and levees are simulated jointly, and a 4-facies Ti was used to condition the relative position of crevasse and floodbasin. 6.2 Future work Consider the synthetic 2D fluvial reservoir with four facies and assume that conditioning data is available for all of them. In the current implementation, during step 1 of the two-step hierarchical approach the crevasse data are merged with the floodbasin data, so that only channel, levee and floodbasin data exist. However, crevasse data are indirect indicator of a channel located nearby. This information is ignored when the crevasse data are merged with floodbasin data. To utilize the information carried by the crevasse data, the grid can be preprocessed to flag the simulation nodes that are within a fixed distance d of the crevasse data. The distance d depends on the size and shape of the crevasse and should be provided by the geologist. During simulation if the location to be simulated is flagged, then the Ti-derived proportion of channel P (A B) is increased by 10% by multiplying P (A B) by a factor 1.1. The 10% percent value increase is an arbitrary choice. The upper bound is set to 1 so that we do not get proportions greater than 1. P = min(1,p(a B) 1.1) (6.1) Step 2 would then proceed as described in section

42 Bibliography [1] Bridge, J.S. and Tye, R.S. Interpreting the dimensions of ancient fluvial channel bars, channels and channel belts from wireline-logs and cores. AAPG Bulletin, vol.84, no.8, pp , August [2] Deutsch, C.V. and Tran, T.T. Fluvsim: a Program for Object-based Stochastic modeling of fluvial depositional Systems. Computers and Geosciences, 2002, no. 28, pp [3] Guardino, F. and Srivastava, R.N. Multivariate geostatistics: Beyond bivariate moments. In A. Soares (ed.), Geostatistics-Troia, vol.1, Kluver Academic Publ. Dordrecht, pp , [4] Leet, L.D. Physical Geology. Englewood Cliffs, NJ: Prentice-Hall, sixth edition, [5] Strebelle, S. Conditional simulation of complex geological structures using multiple-point statistics. Math. Geology, 2002, vol. 34, no. 7, pp [6] Strebelle, S. Sequential Simulation Drawing Structures from Training Images. PhD thesis, Stanford University, Stanford, CA, [7] Tran, T. Improving variogram reproduction on dense simulation grids. Computers and Geosciences, 20(7), ,