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Similarity measure of sedimentary successions and its application in inverse stratigraphic modeling

Similarity measure of sedimentary successions and its application in inverse stratigraphic modeling Pet. Sci. (2017) 14:484–492 DOI 10.1007/s12182-017-0174-1 ORIGINAL PAPER Similarity measure of sedimentary successions and its application in inverse stratigraphic modeling Taizhong Duan Received: 28 October 2016 / Published online: 24 July 2017 The Author(s) 2017. This article is an open access publication Abstract This paper presents a unique and formal method Keywords Similarity quantification  Sedimentary of quantifying the similarity or distance between sedi- succession  Inverse stratigraphic modeling  Global mentary facies successions from measured sections in optimilization  Syntactic approach outcrop or drilled wells and demonstrates its first applica- tion in inverse stratigraphic modeling. A sedimentary facies succession is represented with a string of symbols, or 1 Introduction facies codes in its natural vertical order, in which each symbol brings with it one attribute such as thickness for the Quantitative study of sedimentary successions unavoidably facies. These strings are called attributed strings. A simi- involves the formal description of discrete or symbolic larity measure is defined between the attributed strings properties such as facies, rock texture, or structure, and based on a syntactic pattern-recognition technique. A until recent our ability has been still very limited on how to dynamic programming algorithm is used to calculate the quantify such type properties, for instance, the difference similarity. Inverse stratigraphic modeling aims to generate or similarity between sedimentary facies successions from quantitative 3D facies models based on forward strati- measured sections in outcrop, or drilled sections. More graphic modeling that honors observed datasets. One of the traditional ways to do such comparison are almost exclu- key techniques in inverse stratigraphic modeling is how to sively either qualitative or graphic such as a simple quantify the similarity or distance between simulated and description ‘‘the two facies successions look very similar,’’ observed sedimentary facies successions at data locations or a plot representation usually used as shown in Fig. 1.On in order for the forward model to condition the simulation the other hand, economically efficient recovery of natural results to the observed dataset such as measured sections or resources such as oil and gas demands better and formal drilled wells. This quantification technique comparing quantification of geological models such as geocellular sedimentary successions is demonstrated in the form of a modeling in reservoir characterization. cost function based on the defined distance in our inverse In hydrocarbon reservoir modeling, geostatistical stratigraphic modeling implemented with forward model- methods currently dominate, to large extent due to their ing optimization. data conditioning capacity, that is, the model can honor the observed dataset easily. In contrast, forward stratigraphic modeling has yet to be accepted as the major modeling technique in reservoir facies modeling as it seems it should have. Forward stratigraphic modeling is geological process & Taizhong Duan based and is more relevant to petroleum reservoirs duantz.syky@sinopec.com (Bosence and Waltham 1990; Granjeon and Joseph 1999; Petroleum Exploration and Production Research Institute, Griffiths et al. 2001), and this method has been developed Sinopec, 31 Xueyuan Road, Haidian District, Beijing 100083, since the 1960s (Harbaugh and Bonham-Carter 1970), China compared to the geostatistics also developed since the 1960s (Matheron 1962, 1989). One of the main reasons Edited by Jie Hao 123 Pet. Sci. (2017) 14:484–492 485 (a) (b) (m) Forward stratigraphic model 10  Subsidence, sea level 3D SS Carbonate production simulation G  Erosion, transport result 9 1 Deposition Inversion engine Adjust a. model parameters b. model itself if needed NO Observed Closely Simulation dataset: extracted == matched Drill wells 4 dataset ??? YES B Converged result: analysis/utilization Fig. 2 ISM core techniques and workflow: 1 forward modeling generates 3D simulations; 2 comparing technique quantifies the matching between the simulated and observed datasets; 3 inversion engine updates new simulations for a better match SS SS a proper distance measure can speed up inversion and make it more robust and can lead to better practical application of Fig. 1 Typical graphic representation of sedimentary facies succes- geological process-based modeling in general. This paper sion. Patterns and gray scale represent different facies: a two ‘‘very similar’’ measured facies sections (modified from Kerans et al. 1994); presents a unique and formal method of quantifying the b a plot channel model (modified from Cant and Walker 1976), with distance between sedimentary facies successions based on ordered facies codes to form a string presentation of the channel a discrete, or symbolic computing technique, combined facies: SSABCBEDFG (right) or SSACBCBDFGSS (left) with other numerical techniques. The formal definition of the distance measure between sedimentary facies succes- why this technique has been delayed to dominate in pet- roleum reservoir modeling is its inability to implement data sions will be presented first, and then, its application as the cost function in ISM will be demonstrated. conditioning. Since late 1990s, similar techniques but under different names were proposed to overcome the inability and initiated a new research front in computa- 2 Definition of sedimentary facies successions tional stratigraphy and sedimentology, such as inverse distance stratigraphic modeling (ISM) (Griffiths et al. 1996; Les- senger and Cross 1996; Cross and Lessenger 1999; Duan 2.1 Formal representation of sedimentary facies et al. 2001a; Imhof and Sharma 2006; Charvin et al. 2009; successions Griffiths 2009; Charvin et al. 2011), adaptive modeling (Duan et al. 1998), modeling optimization (Bornholdt and In order to quantify the distance or similarity between Westphal 1998; Wijns et al. 2003, 2004), or model cali- bration (Falivene et al. 2014). However, the progress of sedimentary facies successions, it is essential to define what a sedimentary facies succession is and what is the these techniques, all of which will be called ISM afterward for simplicity, has been limited, and one of the major distance between the sedimentary facies successions or the sedimentary facies successions distance (SFSD) formally hurdles is still the data conditioning. and quantitatively. A gene-typing technique was proposed Therefore, in current version of ISM as shown in Fig. 2, for correlation of petrophysically derived numerical among others, a critical technique needed to enhance the lithologies between boreholes (Bakke and Griffiths 1989; procedure is how to quantify the similarity or distance Griffiths and Bakke 1990). A syntactic methodology between simulated results and the observed dataset. That is, 123 486 Pet. Sci. (2017) 14:484–492 developed in pattern recognition (Fu 1982) was first pro- DNA or RNA in biology. Therefore, we believe this posed to formally describe the language of sedimentary technique has great potential in improving stratigraphic rocks (Griffiths 1990), due to its ability in characterizing the inverse modeling. naturally discrete or symbolic feature of sedimentary facies. The more detailed syntactic approach to the analysis of 2.2 Definition of distance between sedimentary sedimentary successions for reservoir characterization can facies successions be found in Duan, Griffiths, and Johnsen (Duan et al. 1999, 2001a), among others. In the syntactic approach, a To define a distance or similarity measure between sedi- sedimentary facies succession (one-dimensional vertical mentary facies successions, it is important to understand what stratigraphic section) is represented in a string of facies characters are essential in distinguishing different sedimen- symbols, each of which contains one or more attributes, i.e., tary facies successions and what would be fundamental a string with attributes in symbolic computation language requirements for a distance definition mathematically. (Duan et al. 2001a). For instance, the bottom to top facies From the point of view of sedimentology and stratigra- type of a typical channel sedimentary facies succession can phy, comparison of sedimentary facies successions should be coded by symbols as shown in Fig. 1b, whereas the code account for following aspects: (1) facies types and their of each facies can be associated with a number or an attri- division in sections, including special facies such as ero- bute representing the thickness of each facies in the sional surfaces; (2) the thickness of each identified and parentheses as in the following attributed string of codes: divided facies (maybe repeated) in sections; and (3) the vertical order or sequence of the coded facies in each SS(0.5) A(3.0) C(0.5) B(2.5) C(1.5) B(1.3) E(0.5) D(0.4) sedimentary facies successions. When it is said that two F(1.5) G(0.3) SS(0.1) sedimentary facies successions are equivalent, it means all Besides the normal facies, the erosional surfaces are three of the above-mentioned characters should be the treated as a special facies. SS(0.1) and SS(0.5) mean two same. For instance, following are two code strings X and Y erosional surfaces with 0.1 and 0.5 million years time gap, from neighboring sedimentary successions: respectively, whereas A(3.0) and C(0.5) mean 3.0-m X: SS(0.5) A(3.0) C(0.5) B(2.5) C(1.5) B(1.3) E(0.5) trough-cross-bedded sandstone facies and 0.5-m tabular- D(0.4) SS(0.15) cross-bedded sandstone facies, respectively. Of course, Y: SS(0.5) A(3.0) C(0.5) B(2.5) C(1.5) B(1.3) E(0.5) each code may have more than one attribute, either D(0.4) F(1.5) G(0.3) SS(0.1) numerical or symbolic, such as grain size, color, fossils, and mineral composition. It should be noticed that facies They are not equal, because firstly the string-X lacks coding and attributing can be compensated with each other. ‘‘F(1.5) G(0.3),’’ implying that part of the channel top For instance, if you already account for grain size in facies deposits may be eroded away. Secondly, its top SS attribute coding such as conglomerate, coarse sandstone, fine 0.15 is different from the string-Y’s 0.1, implying that the sandstone, siltstone, and mudstone, you may not need to time gap represented by the SS of the string-X is 0.05 repeat the same information by adding a grain size attribute million years longer than the string-Y’s. for the above facies code. For the following two strings: The similar approach was first proposed for stratigraphic U: B(2.0) D(2.0) F(2.0) and correlation between drilling wells or outcrop sections, V: F(2.0) D(2.0) B(2.0) which suffered from being incapable of handling the facies change problem between sections, and has been almost They are not equal, because the vertical order of the forgotten in the geological community. The so-called facies code is different, though three facies types and their facies change dilemma in automatic strata correlation is thickness are the same. Actually, string-U may represent that two sections of strata with the same facies can be upper channel deposits, while string-V may represent cre- defined equivalent, and two sections of strata with totally vasse splay deposits. different facies also need to be defined equivalent if facies A distance measure of sedimentary facies successions change occurs between, which is not mathematically was first proposed in a syntactic approach (Duan et al. 2001a) which accounts for all three aspects of the above- sound. However, in stratigraphic inversion, similarity quantification between simulated and observed successions mentioned characters in sedimentary succession compari- naturally avoids the facies change problem, two succes- son. It is based on a series of syntactic distance measures sions (simulated and observed) to be compared are fun- such as the Levenshtein distance (Levenshtein 1966), damentally the same. As we understand, the application of generalized Levenshtein distance (Fu 1986), and distance this technique to the stratigraphic inverse modeling is like between attributed strings (Fu 1986). For details of the the similar technique’s application to the comparison of definition, refer to Duan et al. (2001b) from Definition 1 123 Pet. Sci. (2017) 14:484–492 487 through to Definition 5 and related concepts. However, for y: a(2)b(7)c(3) continuation and readability, the main definition is repro- z: b(7)c(3) duced as follows. a ¼ 1; b ¼ 1 Definition 1 Let x and y be two attributed strings, xði; jÞ¼ 0 i ¼ j x(i, j) = 1 i = j x = a a ...a 1 2 n x(k,0) = x(0, k) = 1; i, j, k = {a, b, c} y = b b ...b 1 2 m 0 00 0 00 k ¼ k ðc ; c Þ¼ 1 c ¼ c continuation j C j j j j 0 00 0 00 Corresponding attributes of x and y are denoted as: k ¼ k ðc ; c Þ 1 c 6¼ c insertion j I j j j j 0 00 0 00 0 1 2 k 1 2 k 1 2 k k ¼ k ðc ; c Þ 1 c 6¼ c substitution j S x ¼ a a  a a a  a  a a  a j j j j 1 1 1 2 2 2 n n n 0 1 2 k 1 2 k 1 2 k Then, the minimum-length alignment is y ¼ b b  b b b  b  b b  b 1 1 1 2 2 2 n n n x [abc] x [9 7 3] It is assumed that each terminal symbol has k attributes. y [abc] y [2 7 3] The distance between x and y is defined as: z [abc] z [0 7 3] AS GL A A0 d ðÞ x; y ¼ ad ðÞ x; y þ bd ðÞ x; y or bd ðÞ x; y and distances are GL where a and b are two positive weights; d ðÞ x; y is the AS GL A d (x, y) = ad (x, y) ? bd (x, y) GL A generalized Levenshtein distance between x and y; and d (x, y) = 0; d (x, y) = |2 - 9| ? |7 - 7| ? |3 - 3| = 7; AS d ðÞ x; y is the attribute distance between x and y after d (x, y) = a 9 0 ? b 9 7 = 7; GL A optimal alignment with only insertion (when substitution is d (x, z) = 1; d (x, z) = |0 - 9| ? |7 - 7| ? |3 - 3| = 9; AS GL A accepted, a variant d ðÞ x; y is obtained) is carried out to d (x, z) = ad (x, z) ? bd (x, z) = 10; GL A make one string equal another syntactically. Let the d (y, z) = 1; d (y, z) = |0 - 2| ? |7 - 7| ? |3 - 3| = 2; AS GL A transferred string be: d (y, z) = ad (y, z) ? bd (y, z) = 3; AS AS AS 0 0 0 d (x, y) ? d (y, z) = 7 ? 3 C d (x, z) = 10 x ¼ c c  c ðy t t 1 2 k 00 00 00 ¼ c c  c Þfmaxfjj x ; jyjg  k ðjj x þ jyjÞg where x, y, and z are strings; a, b, c are terminal symbols or 1 2 k AS facies codes; x(i, j) weights for d (x, y), and k (i, j), k (i, C I Then A j), k (i, j) are weights for d (x, y) in symbol continuation, A 0 00 insertion, and substitution operations. d ðx; yÞ¼ k dðAðc Þ; Aðc ÞÞ j ¼ 1; 2; ...; k j j A dynamic programming algorithm (Fu 1986; Duan 0 00 0 00 where A(c ) and A(c ) are attribute vectors of c and c , et al. 2001a) is modified to calculate the SFSD as defined in j j j j respectively; dAðÞ 1; A2 can be any p-norm distance (p = 1 this section. used in our case); k are weighting coefficients defined as: 0 00 0 00 1. k ¼ k ðc ; c Þ¼ 1c ¼ c (continuation) j C j j j j 0 00 0 00 2. k ¼ k ðc ; c Þ 1 c 6¼ c (insertion) j I j j j j 3 Application of sedimentary facies successions 0 00 0 00 3. k ¼ k ðc ; c Þ 1 c 6¼ c (substitution) j S j j j j distance in inverse stratigraphic modeling Note that when insertion occurs, the inserted symbol’s Distance measures between sedimentary facies successions attributes are assigned zero. If substitution is allowed and can be of significant application in ISM (Duan et al. occurred, attributes remain the same in related symbols. 1998, 2001a), among others (Duan et al. 2001a). The ISM As Duan et al. (2001b) pointed out, the distance measure can be especially beneficial to reservoir evaluation in the defined must satisfy following mathematical requirements early stage of field development when the data quality or (Kaufman and Rousseeuw 1990): amount is not enough for geological reservoir modeling to AS D(1): d ðx; yÞ 0 take advantage of geostatistics-based geomodeling AS D(2): d ðx; xÞ¼ 0 techniques. AS AS D(3): d ðx; yÞ¼ d ðy; xÞ Of great importance in ISM is quantifying mismatch AS AS AS between simulated results and the observations as shown in D(4): d ðx; zÞ d ðx; yÞþ d ðy; zÞ; Fig. 2. Our SFSD defined in previous section provides a For instance, the following calculation demonstrates the unique formal way to measure the mismatch properly: (1) it Rule D(4) validation in our SFSD, whereas it is more considers both syntactic and attribute distances between obvious to validate the rule D(1) to D(3). Let: x: a(9)b(7)c(3) 123 488 Pet. Sci. (2017) 14:484–492 facies successions, i.e., facies type and their thickness; (2) based cost function in 3D ISM. Firstly, ‘‘an observed it also naturally considers the vertical order of facies types dataset’’ of 5 pseudowell facies sections was generated by in the succession, compares the succession as a whole, and extracting from a 3D facies model (reference model) which does not need further within-succession time calibration was simulated with known model parameters by the for- between the simulated and observed successions; and (3) it ward model. Then, a series of forward model runs was set can easily be adapted to accounting for time gaps associ- up with all known parameters from the reference model ated with erosional surfaces as coded into the attribute. fixed, but one selected parameter each time that did vary To quantify the mismatch between simulated and systematically across its known value in the reference observed successions, an absolute time framework needs to model. Obviously, the change of the selected parameter to be established within the succession, so that facies/thick- the known value used in the reference model will cause the ness formed in the same time interval can be compared simulation to be different from the reference model. each other. For instance, the strata formed between time-1 Thirdly, the cost function values can be calculated against and time-2 in the simulated succession should be compared the varying selected parameter by quantifying the differ- with the strata of the same time interval, time-1 and time-2 ence between the simulations and the reference model with in the time-calibrated observed succession. The succession the extracted 5 pseudowell sections based on the SFSD simulated by forward modeling very commonly contains defined in previous section. Finally, the cross-plots time resolution, say, 5000 year (modeling time step), between cost function values and the selected parameter whereas the observed succession contains dated time res- were created as shown in Fig. 3. olution usually in million years, or at higher resolution In total, there were 29 model parameters sensitivity hundreds of thousand years. The method published so far to analyzed, with representatives shown in Fig. 3. Twenty- calculate the mismatch of simulated and observed succes- five out of 29 parameters are sensitive to the model sions is to only compare the thickness of the smallest dated inversion, with most showing the typical V-shaped curve stratigraphic units such as a strata cycle (Cross and Les- (Fig. 3a, P17 curve as an example), and a few U-shaped senger 1999; Charvin et al. 2009), or the unit thickness (Fig. 3a, P18 curve), or L-shaped (Fig. 3b). The V- or maps (Falivene et al. 2014). Of course, in these methods, U-shaped curves behave very similarly, both with a major the higher resolution of time calibration the observed minimum at the true parameter value of the reference succession has, the more accurate the comparison of the model, and if multiminimums exist, the major one is simulated and observed successions is. However, it is much more significant than other smaller ones (also almost impossible to time calibrate the observed succession shown in Fig. 3c), which makes the convergence of with an order of modeling time step resolution. Therefore, model inversion much easier, and the inverted parameter values more accurate. The L-shaped curves, only a few of the advantage of our method is very obvious, indicated by the properties (1) to (3) mentioned in the previous sec- them, usually correspond to those model parameters, the tion. Moreover, the property (2) implies that the method value change of which beyond a specific limit no longer does not need a higher-resolution chronostratigraphic makes a contribution to the simulation results, and the framework within the observed succession to quantify the true value of which is close to the limit, behaving just like mismatch better. a half-U curve. The 3D carbonate forward stratigraphic model used in The other 4 parameters seemingly insensitive to model our ISM is energy and sediment flux based (Duan et al. inversion can be called as flat-shaped (Fig. 3d, P5 and P6 2000; Shafie and Madon 2008) and can simulate progra- curves). But in fact in most cases, they are pseudoinsen- dation, aggradation, and retrogradation of a carbonate sitive, mainly caused by too large a sampling interval of platform simplified to account for the main factors con- parameter values in sensitivity analysis calculations. If trolling platform evolution such as basin subsidence, high-resolution sensitivity is carried out, they would basement flexure, sea-level change, carbonate productivity, become sensitive to inversion. As shown in Fig. 3d, P5 and sediment transport, erosion, and deposition. The 2D model P6 curves will become more like P4 if their high-resolution used is simplified from the 3D model as a tester simulating curves are calculated. mainly the subsidence, sea-level change, carbonate pro- The landscape of this cost function can be described as ductivity, and sedimentation. multimodel, stepped, probably noisy, and one-minimum dominated. This type of cost function is complex enough, 3.1 Characterization of the cost function based but can be handled well in inversion with direct search on sedimentary facies successions distance algorithms of global optimization (Ingber and Rosen 1992; Storn and Price 1997). These features of the cost function The sensitivity analysis of model parameters to the inver- based on SFSD have made our 3D model inversion pos- sion process was run studying the landscape of the SFSD- sible with reasonably stable results. 123 Pet. Sci. (2017) 14:484–492 489 (a) (b) Error. Data Error. Data p17 p18 p19 0 0 0 5 10 15 20 0 5 10 15 20 (c) (d) 1200 2000 Error. Data Error. Data p28 600 1000 p4 p5 p6 0 0 020 40 60 80 100 0 5 10 15 20 Fig. 3 Plots of cost function (Y-axis) to parameter value (X-axis) in sensitivity analysis. Function value was evaluated with systematic change of each parameter within assigned value range, while all other parameters fixed at their correct values of the reference model 3.2 Convergence behavior of stratigraphic inversion relatively simple. The inversion process had correctly based on sedimentary facies successions distance explored focusing around the true parameter value. Fig- ure 4c represents the most complex convergence process. ISM is said converged if the cost function values become The inversion had worked on a local minimum for over close to zero in acceptable ranges. A series of 2D and 3D half of its time and then focused on the other values a while ISM were run using a synthetic dataset, examining the before reaching the true value. Figure 4b represents an convergence behavior of the inverse process by tackling example between the two cases. The inversion had entered the evaluations of the SFSD-based cost function during a local minimum briefly and then jumped out and started to inversion. Both a genetic algorithm and simulated anneal- focus on the true parameter value, though finally reached ing were used as our inversion engine. the true value after working on 5 other groups of values. Our numerical experiments show that the stratigraphic Figure 4d represents a case in which even though the true inversion based on the SFSD cost function behaves very value was outside of the assigned parameter range for robustly for both 2D and 3D modeling as shown in Fig. 4. searching, the inversion still can find the best value within The behavior of the parameter inverting process can be the range. grouped into four types. Figure 4a represents a most typi- The detailed analysis of convergence behavior, together cal, straightforward parameter inversion in which the true with sensitivity analysis for each parameter, has helped value was inverted smoothly and the convergence was guide the setup of model and parameters for inversion 123 490 Pet. Sci. (2017) 14:484–492 (a) (c) (a) Dataset to extract 6 well data (b) (b) (d) Inverted case-1: Correct solution (c) Inverted case-2: Incorrect solution but 6 well data matched Fig. 4 Cost function value or error (Y-axis) to parameter value (X- (d) axis) obtained in a converged inversion. Each dot represents one function evaluation during the inversion. Horizontal lines indicate the error threshold accepted for uncertainty analysis. Axis scale is 6 sections normalized between 0 and 1 extracted Inverted sea level during our inverse modeling. For instance, those insensitive parameters in a specific time–space scale model will be Fig. 5 Comparison of simulated results and observation (synthetic excluded in inversion; more attention should be put on data). a Original carbonate dataset; b, c are two inversion results with parameters with complex behavior in determining their inverted production rate shown in Fig. 6a, d, respectively; d extracted searching ranges; and parameter ranges can be increased or facies successions of 6 pseudowells as data decreased according to converging tendency from several recover model parameter values correctly or closely with short scoping modeling runs, so that the range is wide cost function errors close to zero (errors 3.0 to 8.1; zero is enough to include possible solutions, but narrow enough to the minimum). There is almost no difference between the speed up convergence. inverted strata profile and the original (Fig. 5a, b), or a difference difficult to identify graphically in many cases. 3.3 Non-uniqueness of inverted results based For the specific parameter values, the inverted and original on sedimentary facies successions distance ones are also very close such as carbonate productivity (Fig. 6a–c). For 3D model experiments, very similar results In a physical system such as a depositional system, non- are achieved. uniqueness implies that more than one reason may cause However, non-unique solutions of the inversion are the same consequence; for instance, an observed uncom- found as indicated in Fig. 5c, where the inverted profile is formity may be caused either by a sea-level fall or by a significantly different from the original though the cost tectonic lift. For the given observed sedimentary facies function error is close to zero (3.0). The unexpected solu- successions, can there be more than one set of model tion is caused by an unexpected inverted parameter pro- parameters found meeting the forward stratigraphic model? ductivity shown in Fig. 6d, which corresponds to a much A series of runs of 2D and 3D ISM were also used to study higher productive rate than the original at specific water the convergence behavior regarding the non-uniqueness depths, causing an unusual sedimentary profile although at issue of inversion, and multiple converged results were 6 observed locations, simulated and original are very close robustly achieved. In the 2D example, six vertical sections (converged in inversion). In practical study, this type of the of the synthetic strata profile (Fig. 5a) were taken as our non-uniqueness can be recognized easily as an incorrect observed dataset (Fig. 5d), and parameter values related to solution by using additional geological information, such basin subsidence, sea-level change, and carbonate pro- as the gradient of carbonate platform slope from seismic ductivity were inverted. It can be shown that inversions can 123 Pet. Sci. (2017) 14:484–492 491 (a) (c) Rate Rate Inverted Original Original Inverted Error=3.0 Error=8.1 Water depth Water depth (b) (d) Inverted Rate Rate Original Original Inverted Error=3.7 Error=3.0 Water depth Water depth Fig. 6 Inverted carbonate production rate (Y-axis) to water depth (X-axis). a–d Curves corresponding to different error levels of inverted rates. Note d is associated with the inverted result shown in Fig. 5c. Axis scale is normalized between 0 and 1 data in our example, and it will not confuse our meaningful modeling; and (3) therefore, the distance measure or other geological interpretation of the inversion results. However, similar ones potentially would be very useful in facies or this may imply that in simpler systems, the non-uniqueness discrete feature-based inverse geological modeling. may be a more common issue in inversion. Acknowledgements This research financially was supported by The non-uniqueness of a model solution means the same Colorado School of Mines, and are supported by the Science and solution can be achieved by different sets of parameter Technology Ministry of China (2016ZX05033003), China Academy values in modeling. Replacing of one set to another set of of Sciences (XDA14010204) and Sinopec (G5800-15-ZS-KJB016). parameter values to generate the same modeling result is The Petroleum Science editors and four anonymous reviewers are thanked for their constructive comments and suggestions. WB Zhang called equivalent-effect parameter substitution. No such and PQ Lian are thanked for redrawing figures. similar, obvious non-uniqueness case as seen in our 2D inversions was found in over 10 inverted example real- Open Access This article is distributed under the terms of the izations in our 3D ISM starting with different initial points Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, of parameter space. The explanation may be that, in a distribution, and reproduction in any medium, provided you give system complex enough as in the 3D stratigraphic model- appropriate credit to the original author(s) and the source, provide a ing, non-uniqueness caused by the equivalent-effect link to the Creative Commons license, and indicate if changes were parameter substitution is less probable, compared to a made. simpler system as in the 2D model case. That is, the probability that equivalent-effect parameter substitution References can generate non-uniqueness in a modeling system may be inversely proportional to the complexity of the system. Bakke S, Griffiths CM. Interactive stratigraphic matching of petro- physically derived numerical lithologies based on gene-typing techniques. In: Collinson J, editor. Correlation in hydrocarbon exploration. London: Norwegian Petroleum Society, Graham & 4 Conclusion Trotman; 1989. p. 61–76. doi:10.1007/978-94-009-1149-9_7. Bornholdt S, Westphal H. 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Similarity measure of sedimentary successions and its application in inverse stratigraphic modeling

Duan, Taizhong
Petroleum Science , Volume 14 (3) – Jul 24, 2017
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Springer Journals
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Copyright © 2017 by The Author(s)
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Earth Sciences; Mineral Resources; Industrial Chemistry/Chemical Engineering; Industrial and Production Engineering; Energy Economics
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1672-5107
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10.1007/s12182-017-0174-1
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Pet. Sci. (2017) 14:484–492 DOI 10.1007/s12182-017-0174-1 ORIGINAL PAPER Similarity measure of sedimentary successions and its application in inverse stratigraphic modeling Taizhong Duan Received: 28 October 2016 / Published online: 24 July 2017 The Author(s) 2017. This article is an open access publication Abstract This paper presents a unique and formal method Keywords Similarity quantification  Sedimentary of quantifying the similarity or distance between sedi- succession  Inverse stratigraphic modeling  Global mentary facies successions from measured sections in optimilization  Syntactic approach outcrop or drilled wells and demonstrates its first applica- tion in inverse stratigraphic modeling. A sedimentary facies succession is represented with a string of symbols, or 1 Introduction facies codes in its natural vertical order, in which each symbol brings with it one attribute such as thickness for the Quantitative study of sedimentary successions unavoidably facies. These strings are called attributed strings. A simi- involves the formal description of discrete or symbolic larity measure is defined between the attributed strings properties such as facies, rock texture, or structure, and based on a syntactic pattern-recognition technique. A until recent our ability has been still very limited on how to dynamic programming algorithm is used to calculate the quantify such type properties, for instance, the difference similarity. Inverse stratigraphic modeling aims to generate or similarity between sedimentary facies successions from quantitative 3D facies models based on forward strati- measured sections in outcrop, or drilled sections. More graphic modeling that honors observed datasets. One of the traditional ways to do such comparison are almost exclu- key techniques in inverse stratigraphic modeling is how to sively either qualitative or graphic such as a simple quantify the similarity or distance between simulated and description ‘‘the two facies successions look very similar,’’ observed sedimentary facies successions at data locations or a plot representation usually used as shown in Fig. 1.On in order for the forward model to condition the simulation the other hand, economically efficient recovery of natural results to the observed dataset such as measured sections or resources such as oil and gas demands better and formal drilled wells. This quantification technique comparing quantification of geological models such as geocellular sedimentary successions is demonstrated in the form of a modeling in reservoir characterization. cost function based on the defined distance in our inverse In hydrocarbon reservoir modeling, geostatistical stratigraphic modeling implemented with forward model- methods currently dominate, to large extent due to their ing optimization. data conditioning capacity, that is, the model can honor the observed dataset easily. In contrast, forward stratigraphic modeling has yet to be accepted as the major modeling technique in reservoir facies modeling as it seems it should have. Forward stratigraphic modeling is geological process & Taizhong Duan based and is more relevant to petroleum reservoirs duantz.syky@sinopec.com (Bosence and Waltham 1990; Granjeon and Joseph 1999; Petroleum Exploration and Production Research Institute, Griffiths et al. 2001), and this method has been developed Sinopec, 31 Xueyuan Road, Haidian District, Beijing 100083, since the 1960s (Harbaugh and Bonham-Carter 1970), China compared to the geostatistics also developed since the 1960s (Matheron 1962, 1989). One of the main reasons Edited by Jie Hao 123 Pet. Sci. (2017) 14:484–492 485 (a) (b) (m) Forward stratigraphic model 10  Subsidence, sea level 3D SS Carbonate production simulation G  Erosion, transport result 9 1 Deposition Inversion engine Adjust a. model parameters b. model itself if needed NO Observed Closely Simulation dataset: extracted == matched Drill wells 4 dataset ??? YES B Converged result: analysis/utilization Fig. 2 ISM core techniques and workflow: 1 forward modeling generates 3D simulations; 2 comparing technique quantifies the matching between the simulated and observed datasets; 3 inversion engine updates new simulations for a better match SS SS a proper distance measure can speed up inversion and make it more robust and can lead to better practical application of Fig. 1 Typical graphic representation of sedimentary facies succes- geological process-based modeling in general. This paper sion. Patterns and gray scale represent different facies: a two ‘‘very similar’’ measured facies sections (modified from Kerans et al. 1994); presents a unique and formal method of quantifying the b a plot channel model (modified from Cant and Walker 1976), with distance between sedimentary facies successions based on ordered facies codes to form a string presentation of the channel a discrete, or symbolic computing technique, combined facies: SSABCBEDFG (right) or SSACBCBDFGSS (left) with other numerical techniques. The formal definition of the distance measure between sedimentary facies succes- why this technique has been delayed to dominate in pet- roleum reservoir modeling is its inability to implement data sions will be presented first, and then, its application as the cost function in ISM will be demonstrated. conditioning. Since late 1990s, similar techniques but under different names were proposed to overcome the inability and initiated a new research front in computa- 2 Definition of sedimentary facies successions tional stratigraphy and sedimentology, such as inverse distance stratigraphic modeling (ISM) (Griffiths et al. 1996; Les- senger and Cross 1996; Cross and Lessenger 1999; Duan 2.1 Formal representation of sedimentary facies et al. 2001a; Imhof and Sharma 2006; Charvin et al. 2009; successions Griffiths 2009; Charvin et al. 2011), adaptive modeling (Duan et al. 1998), modeling optimization (Bornholdt and In order to quantify the distance or similarity between Westphal 1998; Wijns et al. 2003, 2004), or model cali- bration (Falivene et al. 2014). However, the progress of sedimentary facies successions, it is essential to define what a sedimentary facies succession is and what is the these techniques, all of which will be called ISM afterward for simplicity, has been limited, and one of the major distance between the sedimentary facies successions or the sedimentary facies successions distance (SFSD) formally hurdles is still the data conditioning. and quantitatively. A gene-typing technique was proposed Therefore, in current version of ISM as shown in Fig. 2, for correlation of petrophysically derived numerical among others, a critical technique needed to enhance the lithologies between boreholes (Bakke and Griffiths 1989; procedure is how to quantify the similarity or distance Griffiths and Bakke 1990). A syntactic methodology between simulated results and the observed dataset. That is, 123 486 Pet. Sci. (2017) 14:484–492 developed in pattern recognition (Fu 1982) was first pro- DNA or RNA in biology. Therefore, we believe this posed to formally describe the language of sedimentary technique has great potential in improving stratigraphic rocks (Griffiths 1990), due to its ability in characterizing the inverse modeling. naturally discrete or symbolic feature of sedimentary facies. The more detailed syntactic approach to the analysis of 2.2 Definition of distance between sedimentary sedimentary successions for reservoir characterization can facies successions be found in Duan, Griffiths, and Johnsen (Duan et al. 1999, 2001a), among others. In the syntactic approach, a To define a distance or similarity measure between sedi- sedimentary facies succession (one-dimensional vertical mentary facies successions, it is important to understand what stratigraphic section) is represented in a string of facies characters are essential in distinguishing different sedimen- symbols, each of which contains one or more attributes, i.e., tary facies successions and what would be fundamental a string with attributes in symbolic computation language requirements for a distance definition mathematically. (Duan et al. 2001a). For instance, the bottom to top facies From the point of view of sedimentology and stratigra- type of a typical channel sedimentary facies succession can phy, comparison of sedimentary facies successions should be coded by symbols as shown in Fig. 1b, whereas the code account for following aspects: (1) facies types and their of each facies can be associated with a number or an attri- division in sections, including special facies such as ero- bute representing the thickness of each facies in the sional surfaces; (2) the thickness of each identified and parentheses as in the following attributed string of codes: divided facies (maybe repeated) in sections; and (3) the vertical order or sequence of the coded facies in each SS(0.5) A(3.0) C(0.5) B(2.5) C(1.5) B(1.3) E(0.5) D(0.4) sedimentary facies successions. When it is said that two F(1.5) G(0.3) SS(0.1) sedimentary facies successions are equivalent, it means all Besides the normal facies, the erosional surfaces are three of the above-mentioned characters should be the treated as a special facies. SS(0.1) and SS(0.5) mean two same. For instance, following are two code strings X and Y erosional surfaces with 0.1 and 0.5 million years time gap, from neighboring sedimentary successions: respectively, whereas A(3.0) and C(0.5) mean 3.0-m X: SS(0.5) A(3.0) C(0.5) B(2.5) C(1.5) B(1.3) E(0.5) trough-cross-bedded sandstone facies and 0.5-m tabular- D(0.4) SS(0.15) cross-bedded sandstone facies, respectively. Of course, Y: SS(0.5) A(3.0) C(0.5) B(2.5) C(1.5) B(1.3) E(0.5) each code may have more than one attribute, either D(0.4) F(1.5) G(0.3) SS(0.1) numerical or symbolic, such as grain size, color, fossils, and mineral composition. It should be noticed that facies They are not equal, because firstly the string-X lacks coding and attributing can be compensated with each other. ‘‘F(1.5) G(0.3),’’ implying that part of the channel top For instance, if you already account for grain size in facies deposits may be eroded away. Secondly, its top SS attribute coding such as conglomerate, coarse sandstone, fine 0.15 is different from the string-Y’s 0.1, implying that the sandstone, siltstone, and mudstone, you may not need to time gap represented by the SS of the string-X is 0.05 repeat the same information by adding a grain size attribute million years longer than the string-Y’s. for the above facies code. For the following two strings: The similar approach was first proposed for stratigraphic U: B(2.0) D(2.0) F(2.0) and correlation between drilling wells or outcrop sections, V: F(2.0) D(2.0) B(2.0) which suffered from being incapable of handling the facies change problem between sections, and has been almost They are not equal, because the vertical order of the forgotten in the geological community. The so-called facies code is different, though three facies types and their facies change dilemma in automatic strata correlation is thickness are the same. Actually, string-U may represent that two sections of strata with the same facies can be upper channel deposits, while string-V may represent cre- defined equivalent, and two sections of strata with totally vasse splay deposits. different facies also need to be defined equivalent if facies A distance measure of sedimentary facies successions change occurs between, which is not mathematically was first proposed in a syntactic approach (Duan et al. 2001a) which accounts for all three aspects of the above- sound. However, in stratigraphic inversion, similarity quantification between simulated and observed successions mentioned characters in sedimentary succession compari- naturally avoids the facies change problem, two succes- son. It is based on a series of syntactic distance measures sions (simulated and observed) to be compared are fun- such as the Levenshtein distance (Levenshtein 1966), damentally the same. As we understand, the application of generalized Levenshtein distance (Fu 1986), and distance this technique to the stratigraphic inverse modeling is like between attributed strings (Fu 1986). For details of the the similar technique’s application to the comparison of definition, refer to Duan et al. (2001b) from Definition 1 123 Pet. Sci. (2017) 14:484–492 487 through to Definition 5 and related concepts. However, for y: a(2)b(7)c(3) continuation and readability, the main definition is repro- z: b(7)c(3) duced as follows. a ¼ 1; b ¼ 1 Definition 1 Let x and y be two attributed strings, xði; jÞ¼ 0 i ¼ j x(i, j) = 1 i = j x = a a ...a 1 2 n x(k,0) = x(0, k) = 1; i, j, k = {a, b, c} y = b b ...b 1 2 m 0 00 0 00 k ¼ k ðc ; c Þ¼ 1 c ¼ c continuation j C j j j j 0 00 0 00 Corresponding attributes of x and y are denoted as: k ¼ k ðc ; c Þ 1 c 6¼ c insertion j I j j j j 0 00 0 00 0 1 2 k 1 2 k 1 2 k k ¼ k ðc ; c Þ 1 c 6¼ c substitution j S x ¼ a a  a a a  a  a a  a j j j j 1 1 1 2 2 2 n n n 0 1 2 k 1 2 k 1 2 k Then, the minimum-length alignment is y ¼ b b  b b b  b  b b  b 1 1 1 2 2 2 n n n x [abc] x [9 7 3] It is assumed that each terminal symbol has k attributes. y [abc] y [2 7 3] The distance between x and y is defined as: z [abc] z [0 7 3] AS GL A A0 d ðÞ x; y ¼ ad ðÞ x; y þ bd ðÞ x; y or bd ðÞ x; y and distances are GL where a and b are two positive weights; d ðÞ x; y is the AS GL A d (x, y) = ad (x, y) ? bd (x, y) GL A generalized Levenshtein distance between x and y; and d (x, y) = 0; d (x, y) = |2 - 9| ? |7 - 7| ? |3 - 3| = 7; AS d ðÞ x; y is the attribute distance between x and y after d (x, y) = a 9 0 ? b 9 7 = 7; GL A optimal alignment with only insertion (when substitution is d (x, z) = 1; d (x, z) = |0 - 9| ? |7 - 7| ? |3 - 3| = 9; AS GL A accepted, a variant d ðÞ x; y is obtained) is carried out to d (x, z) = ad (x, z) ? bd (x, z) = 10; GL A make one string equal another syntactically. Let the d (y, z) = 1; d (y, z) = |0 - 2| ? |7 - 7| ? |3 - 3| = 2; AS GL A transferred string be: d (y, z) = ad (y, z) ? bd (y, z) = 3; AS AS AS 0 0 0 d (x, y) ? d (y, z) = 7 ? 3 C d (x, z) = 10 x ¼ c c  c ðy t t 1 2 k 00 00 00 ¼ c c  c Þfmaxfjj x ; jyjg  k ðjj x þ jyjÞg where x, y, and z are strings; a, b, c are terminal symbols or 1 2 k AS facies codes; x(i, j) weights for d (x, y), and k (i, j), k (i, C I Then A j), k (i, j) are weights for d (x, y) in symbol continuation, A 0 00 insertion, and substitution operations. d ðx; yÞ¼ k dðAðc Þ; Aðc ÞÞ j ¼ 1; 2; ...; k j j A dynamic programming algorithm (Fu 1986; Duan 0 00 0 00 where A(c ) and A(c ) are attribute vectors of c and c , et al. 2001a) is modified to calculate the SFSD as defined in j j j j respectively; dAðÞ 1; A2 can be any p-norm distance (p = 1 this section. used in our case); k are weighting coefficients defined as: 0 00 0 00 1. k ¼ k ðc ; c Þ¼ 1c ¼ c (continuation) j C j j j j 0 00 0 00 2. k ¼ k ðc ; c Þ 1 c 6¼ c (insertion) j I j j j j 3 Application of sedimentary facies successions 0 00 0 00 3. k ¼ k ðc ; c Þ 1 c 6¼ c (substitution) j S j j j j distance in inverse stratigraphic modeling Note that when insertion occurs, the inserted symbol’s Distance measures between sedimentary facies successions attributes are assigned zero. If substitution is allowed and can be of significant application in ISM (Duan et al. occurred, attributes remain the same in related symbols. 1998, 2001a), among others (Duan et al. 2001a). The ISM As Duan et al. (2001b) pointed out, the distance measure can be especially beneficial to reservoir evaluation in the defined must satisfy following mathematical requirements early stage of field development when the data quality or (Kaufman and Rousseeuw 1990): amount is not enough for geological reservoir modeling to AS D(1): d ðx; yÞ 0 take advantage of geostatistics-based geomodeling AS D(2): d ðx; xÞ¼ 0 techniques. AS AS D(3): d ðx; yÞ¼ d ðy; xÞ Of great importance in ISM is quantifying mismatch AS AS AS between simulated results and the observations as shown in D(4): d ðx; zÞ d ðx; yÞþ d ðy; zÞ; Fig. 2. Our SFSD defined in previous section provides a For instance, the following calculation demonstrates the unique formal way to measure the mismatch properly: (1) it Rule D(4) validation in our SFSD, whereas it is more considers both syntactic and attribute distances between obvious to validate the rule D(1) to D(3). Let: x: a(9)b(7)c(3) 123 488 Pet. Sci. (2017) 14:484–492 facies successions, i.e., facies type and their thickness; (2) based cost function in 3D ISM. Firstly, ‘‘an observed it also naturally considers the vertical order of facies types dataset’’ of 5 pseudowell facies sections was generated by in the succession, compares the succession as a whole, and extracting from a 3D facies model (reference model) which does not need further within-succession time calibration was simulated with known model parameters by the for- between the simulated and observed successions; and (3) it ward model. Then, a series of forward model runs was set can easily be adapted to accounting for time gaps associ- up with all known parameters from the reference model ated with erosional surfaces as coded into the attribute. fixed, but one selected parameter each time that did vary To quantify the mismatch between simulated and systematically across its known value in the reference observed successions, an absolute time framework needs to model. Obviously, the change of the selected parameter to be established within the succession, so that facies/thick- the known value used in the reference model will cause the ness formed in the same time interval can be compared simulation to be different from the reference model. each other. For instance, the strata formed between time-1 Thirdly, the cost function values can be calculated against and time-2 in the simulated succession should be compared the varying selected parameter by quantifying the differ- with the strata of the same time interval, time-1 and time-2 ence between the simulations and the reference model with in the time-calibrated observed succession. The succession the extracted 5 pseudowell sections based on the SFSD simulated by forward modeling very commonly contains defined in previous section. Finally, the cross-plots time resolution, say, 5000 year (modeling time step), between cost function values and the selected parameter whereas the observed succession contains dated time res- were created as shown in Fig. 3. olution usually in million years, or at higher resolution In total, there were 29 model parameters sensitivity hundreds of thousand years. The method published so far to analyzed, with representatives shown in Fig. 3. Twenty- calculate the mismatch of simulated and observed succes- five out of 29 parameters are sensitive to the model sions is to only compare the thickness of the smallest dated inversion, with most showing the typical V-shaped curve stratigraphic units such as a strata cycle (Cross and Les- (Fig. 3a, P17 curve as an example), and a few U-shaped senger 1999; Charvin et al. 2009), or the unit thickness (Fig. 3a, P18 curve), or L-shaped (Fig. 3b). The V- or maps (Falivene et al. 2014). Of course, in these methods, U-shaped curves behave very similarly, both with a major the higher resolution of time calibration the observed minimum at the true parameter value of the reference succession has, the more accurate the comparison of the model, and if multiminimums exist, the major one is simulated and observed successions is. However, it is much more significant than other smaller ones (also almost impossible to time calibrate the observed succession shown in Fig. 3c), which makes the convergence of with an order of modeling time step resolution. Therefore, model inversion much easier, and the inverted parameter values more accurate. The L-shaped curves, only a few of the advantage of our method is very obvious, indicated by the properties (1) to (3) mentioned in the previous sec- them, usually correspond to those model parameters, the tion. Moreover, the property (2) implies that the method value change of which beyond a specific limit no longer does not need a higher-resolution chronostratigraphic makes a contribution to the simulation results, and the framework within the observed succession to quantify the true value of which is close to the limit, behaving just like mismatch better. a half-U curve. The 3D carbonate forward stratigraphic model used in The other 4 parameters seemingly insensitive to model our ISM is energy and sediment flux based (Duan et al. inversion can be called as flat-shaped (Fig. 3d, P5 and P6 2000; Shafie and Madon 2008) and can simulate progra- curves). But in fact in most cases, they are pseudoinsen- dation, aggradation, and retrogradation of a carbonate sitive, mainly caused by too large a sampling interval of platform simplified to account for the main factors con- parameter values in sensitivity analysis calculations. If trolling platform evolution such as basin subsidence, high-resolution sensitivity is carried out, they would basement flexure, sea-level change, carbonate productivity, become sensitive to inversion. As shown in Fig. 3d, P5 and sediment transport, erosion, and deposition. The 2D model P6 curves will become more like P4 if their high-resolution used is simplified from the 3D model as a tester simulating curves are calculated. mainly the subsidence, sea-level change, carbonate pro- The landscape of this cost function can be described as ductivity, and sedimentation. multimodel, stepped, probably noisy, and one-minimum dominated. This type of cost function is complex enough, 3.1 Characterization of the cost function based but can be handled well in inversion with direct search on sedimentary facies successions distance algorithms of global optimization (Ingber and Rosen 1992; Storn and Price 1997). These features of the cost function The sensitivity analysis of model parameters to the inver- based on SFSD have made our 3D model inversion pos- sion process was run studying the landscape of the SFSD- sible with reasonably stable results. 123 Pet. Sci. (2017) 14:484–492 489 (a) (b) Error. Data Error. Data p17 p18 p19 0 0 0 5 10 15 20 0 5 10 15 20 (c) (d) 1200 2000 Error. Data Error. Data p28 600 1000 p4 p5 p6 0 0 020 40 60 80 100 0 5 10 15 20 Fig. 3 Plots of cost function (Y-axis) to parameter value (X-axis) in sensitivity analysis. Function value was evaluated with systematic change of each parameter within assigned value range, while all other parameters fixed at their correct values of the reference model 3.2 Convergence behavior of stratigraphic inversion relatively simple. The inversion process had correctly based on sedimentary facies successions distance explored focusing around the true parameter value. Fig- ure 4c represents the most complex convergence process. ISM is said converged if the cost function values become The inversion had worked on a local minimum for over close to zero in acceptable ranges. A series of 2D and 3D half of its time and then focused on the other values a while ISM were run using a synthetic dataset, examining the before reaching the true value. Figure 4b represents an convergence behavior of the inverse process by tackling example between the two cases. The inversion had entered the evaluations of the SFSD-based cost function during a local minimum briefly and then jumped out and started to inversion. Both a genetic algorithm and simulated anneal- focus on the true parameter value, though finally reached ing were used as our inversion engine. the true value after working on 5 other groups of values. Our numerical experiments show that the stratigraphic Figure 4d represents a case in which even though the true inversion based on the SFSD cost function behaves very value was outside of the assigned parameter range for robustly for both 2D and 3D modeling as shown in Fig. 4. searching, the inversion still can find the best value within The behavior of the parameter inverting process can be the range. grouped into four types. Figure 4a represents a most typi- The detailed analysis of convergence behavior, together cal, straightforward parameter inversion in which the true with sensitivity analysis for each parameter, has helped value was inverted smoothly and the convergence was guide the setup of model and parameters for inversion 123 490 Pet. Sci. (2017) 14:484–492 (a) (c) (a) Dataset to extract 6 well data (b) (b) (d) Inverted case-1: Correct solution (c) Inverted case-2: Incorrect solution but 6 well data matched Fig. 4 Cost function value or error (Y-axis) to parameter value (X- (d) axis) obtained in a converged inversion. Each dot represents one function evaluation during the inversion. Horizontal lines indicate the error threshold accepted for uncertainty analysis. Axis scale is 6 sections normalized between 0 and 1 extracted Inverted sea level during our inverse modeling. For instance, those insensitive parameters in a specific time–space scale model will be Fig. 5 Comparison of simulated results and observation (synthetic excluded in inversion; more attention should be put on data). a Original carbonate dataset; b, c are two inversion results with parameters with complex behavior in determining their inverted production rate shown in Fig. 6a, d, respectively; d extracted searching ranges; and parameter ranges can be increased or facies successions of 6 pseudowells as data decreased according to converging tendency from several recover model parameter values correctly or closely with short scoping modeling runs, so that the range is wide cost function errors close to zero (errors 3.0 to 8.1; zero is enough to include possible solutions, but narrow enough to the minimum). There is almost no difference between the speed up convergence. inverted strata profile and the original (Fig. 5a, b), or a difference difficult to identify graphically in many cases. 3.3 Non-uniqueness of inverted results based For the specific parameter values, the inverted and original on sedimentary facies successions distance ones are also very close such as carbonate productivity (Fig. 6a–c). For 3D model experiments, very similar results In a physical system such as a depositional system, non- are achieved. uniqueness implies that more than one reason may cause However, non-unique solutions of the inversion are the same consequence; for instance, an observed uncom- found as indicated in Fig. 5c, where the inverted profile is formity may be caused either by a sea-level fall or by a significantly different from the original though the cost tectonic lift. For the given observed sedimentary facies function error is close to zero (3.0). The unexpected solu- successions, can there be more than one set of model tion is caused by an unexpected inverted parameter pro- parameters found meeting the forward stratigraphic model? ductivity shown in Fig. 6d, which corresponds to a much A series of runs of 2D and 3D ISM were also used to study higher productive rate than the original at specific water the convergence behavior regarding the non-uniqueness depths, causing an unusual sedimentary profile although at issue of inversion, and multiple converged results were 6 observed locations, simulated and original are very close robustly achieved. In the 2D example, six vertical sections (converged in inversion). In practical study, this type of the of the synthetic strata profile (Fig. 5a) were taken as our non-uniqueness can be recognized easily as an incorrect observed dataset (Fig. 5d), and parameter values related to solution by using additional geological information, such basin subsidence, sea-level change, and carbonate pro- as the gradient of carbonate platform slope from seismic ductivity were inverted. It can be shown that inversions can 123 Pet. Sci. (2017) 14:484–492 491 (a) (c) Rate Rate Inverted Original Original Inverted Error=3.0 Error=8.1 Water depth Water depth (b) (d) Inverted Rate Rate Original Original Inverted Error=3.7 Error=3.0 Water depth Water depth Fig. 6 Inverted carbonate production rate (Y-axis) to water depth (X-axis). a–d Curves corresponding to different error levels of inverted rates. Note d is associated with the inverted result shown in Fig. 5c. Axis scale is normalized between 0 and 1 data in our example, and it will not confuse our meaningful modeling; and (3) therefore, the distance measure or other geological interpretation of the inversion results. However, similar ones potentially would be very useful in facies or this may imply that in simpler systems, the non-uniqueness discrete feature-based inverse geological modeling. may be a more common issue in inversion. Acknowledgements This research financially was supported by The non-uniqueness of a model solution means the same Colorado School of Mines, and are supported by the Science and solution can be achieved by different sets of parameter Technology Ministry of China (2016ZX05033003), China Academy values in modeling. Replacing of one set to another set of of Sciences (XDA14010204) and Sinopec (G5800-15-ZS-KJB016). parameter values to generate the same modeling result is The Petroleum Science editors and four anonymous reviewers are thanked for their constructive comments and suggestions. WB Zhang called equivalent-effect parameter substitution. No such and PQ Lian are thanked for redrawing figures. similar, obvious non-uniqueness case as seen in our 2D inversions was found in over 10 inverted example real- Open Access This article is distributed under the terms of the izations in our 3D ISM starting with different initial points Creative Commons Attribution 4.0 International License (http://crea tivecommons.org/licenses/by/4.0/), which permits unrestricted use, of parameter space. 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