Expression and modelling of stratigraphic sequence distortion

Sedimentary Geology 178 (2005) 159 – 186

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Expression and modelling of stratigraphic sequence distortion C. Robina,*, D. Roubya,1, D. Granjeonb,2, F. Guillocheaua,1, P. Allemandc,3, S. Raillardd,4 a

Ge´osciences Rennes, UMR 6118 CNRS, Universite´ de Rennes 1, Campus de Beaulieu, 35042 Rennes Ce´dex, France b Institut Franc¸ais du Pe´trole, 1-4 Avenue du Bois-Pre´au, 92852 Rueil-Malmaison, France c Universite´ de Lyon 1, UMR4570, 43 bvd du 11 Novembre 1918, 69622 Villeurbanne, Ce´dex, France d TotalFinaElf, Avenue Larribau, 64018 Pau Ce´dex, France Received 30 March 2004; received in revised form 8 March 2005; accepted 13 April 2005

Abstract This study describes the lateral variability of stratigraphic sequences under changing conditions of subsidence and/or sediment supply. Changes in sediment supply, subsidence and absolute sea-level result in at least three modes of distortion of a depositional sequence. Our study of a case history from offshore West Africa, together with numerical analysis, provides insights as to how changes in subsidence and sedimentation may be extracted from the stratigraphic record. We analysed a listric fault/raft system, located on the Congo–Cabonda margin, resulting from gravity-induced extension of a mixed siliciclastic/carbonate platform sequence. Within this system, quantitative analysis was carried out on high-resolution stratigraphic sequences of five wells, as well as on their geometry, on the relative duration of prograding compared to retrograding half-cycles, and on the timing of the inversion of these half-cycle trends. Parameters were defined to quantify the distortion of depositional sequences that results from either (i)spatial variations in subsidence and sedimentation rates (bspatial distortionQ D), or (ii) superposition of two frequencies of stratigraphic cyclicity (bcycle superposition distortionQ DV). Using a numerical model, we investigate the distortion of depositional sequences showing two superposed scales of cyclicity, in response to spatial and temporal variations of subsidence and sedimentation rate. We show that: (i) Spatial variations in subsidence rate can result in modification of the timing of trend inversion: the onset of progradation may be delayed and the onset of retrogradation may occur earlier in the more rapidly subsiding areas. * Corresponding author. Tel.: +33 2 23 23 57 27; fax: +33 2 23 23 61 00. E-mail addresses: [email protected] (C. Robin), [email protected] (D. Rouby), [email protected] (D. Granjeon), [email protected] (F. Guillocheau), [email protected] (P. Allemand), [email protected] (S. Raillard). 1 Fax: +33 2 23 23 61 00. 2 Fax: +33 1 47 52 70 67. 3 Fax: +33 4 72 72 86 77. 4 Fax: +33 5 59 83 40 00. 0037-0738/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sedgeo.2005.04.003

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(ii) The sedimentation rate can modulate the amount of distortion related to subsidence by amplifying, limiting, compensating or even inverting the temporal offset. (iii) Spatial variations in sedimentation rate alone may also induce changes in the timing of trend inversion: the onset of progradation may take place sooner and the onset of retrogradation may be delayed in the area showing the fastest sedimentation rate. (iv) The amount of distortion depends not only on the sedimentation and subsidence rates, but also on the maximum rate of sea-level change causing the depositional sequence, that is, it depends on the period and amplitude of the sequence. The faster the sea-level change (i.e. the higher the frequency and the larger the amplitude), the weaker is the distortion produced. (v) In terms of resulting time lag, numerical analysis shows that the distortion of high frequency depositional sequences (genetic units) is negligible. These sequences can therefore be safely used to correlate time lines across the studied area, whereas use of lower frequency sequences with significant spatial distortion could lead to significant errors in correlation. We use these results to interpret the distortion observed in the case study, in terms of temporal and spatial variations of subsidence and sedimentation rates. In this case, complex temporal and spatial variations in subsidence and sedimentation rates lead to variations of the distortion of stacks of genetic unit. This distortion produced difference in timing of the onset and duration of the inversion of trend within prograding and retrograding half-cycles. D 2005 Elsevier B.V. All rights reserved. Keywords: Sequence stratigraphy; Stratigraphic sequence; Accommodation; Subsidence rate; Sedimentation rate; Sea-level change

1. Introduction Since sedimentary basins are located at the interface between the lithosphere and the atmosphere/ hydrosphere, they document the complex interactions between the processes of deformation at various scales and external processes such as sea-level and climate fluctuations (acting at different time scales). These interactions are recorded as stratigraphic sequences of multiple orders, corresponding to cycles of shoreline progradation (seaward stepping) and retrogradation (landward stepping). Such cycles are controlled by variations of sediment supply, eustasy and subsidence of the basin basement. The way in which these factors combine to drive shoreline movements has been formalized in terms of variations in the ratio between two independent parameters: accommodation [A] (i.e. the space available for sedimentation located between the basement and the base level; Jervey, 1988) and sediment supply [S] (e.g. Cross, 1988; Homewood et al., 1992). Accommodation is controlled by variations of absolute sea-level, as well as by vertical displacements of the basement in response to local and regional deformation. By definition, eustatic changes of sea-level only vary with time and are

homogeneous in space (i.e. over the whole basin), whereas deformation (and sediment supply) varies both in time and space (i.e. within the basin). Therefore, synchronous spatial variations of accommodation [A] are driven by deformation. On the other hand, the large number of parameters controlling sedimentation (erosion, transport, in situ production) lead to complex spatial and temporal variations of sediment supply [S]. The aim of this paper is to examine the effects of spatial variations of the [A] / [S] ratio on stratigraphic sequences (Jervey, 1988; Wehr, 1993; Martinsen and Helland-Hansen, 1995; Catuneanu et al., 1998), and, as a consequence, how differential subsidence and sedimentation can be extracted from the stratigraphic record. The links between stratigraphic sequences and these variations may be examined by a quantitative analysis of their geometry, the relative duration of their prograding and retrograding phases and the timing of trend inversion (i.e. the onset of progradation or retrogradation). The study of a listric fault/raft system produced by gravity-induced extension in a mixed siliciclastic/carbonate platform setting on the West African margin, using an integrated approach combining structural geology, sequence stratigraphy and numerical model-

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ling, allows us to compare stratigraphic architecture across growth faults. (1) We characterize the variations in the geometry of depositional sequences across the growth faults and determine the types of distortion that can be observed; (2) We use these observations to calibrate a numerical simulation of the influence of variations in subsidence and sedimentation rate on the geometry of depositional sequences at various scales, and then determine the conditions of sediment supply, subsidence rate and periodicity of absolute sea-level change for which the observed distortions can be reproduced. This study builds on previous work (Rouby et al., 2002, 2003) in which: (1) based on a 3-D seismic data set and well logs, we established the detailed sequence stratigraphic framework and the evolution through time of the 3-D geometry of the fault system; (2) from 3-D restoration and accommodation variation measurements, we quantified the displacement rates related to deformation as well as the sedimentation rates.

2. Influence of deformation on sedimentation processes Since deformation processes control accommodation, the stratigraphic infill of sedimentary basins provides a high-resolution record of kinematics of their deformation as successively deposited sedimentary layers are involved in different stages of their development. The geometry of syn-deformational strata is commonly used to determine the kinematics and growth processes of deformation structures (for extensional systems: e.g. Steckler and Watts, 1978; McCulloh, 1988; Cartwright et al., 1998; Rouby et al., 1993, 2000; for compressional systems: e.g. Poblet and Hardy, 1995; Ford et al., 1997; Bourgeois et al., 1997) generally assuming that sedimentary layers can be considered as passive markers recording only the deformation (fill-to-the-top model, which assumes

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that all the available space is systematically filled up by sedimentation). However, numerous studies have demonstrated the influence of deformation on sedimentary processes not only on the thickness distribution of sedimentary bodies in time and space (e.g. Watts et al., 1982; Sinclair and Allen, 1992; Horton and DeCelles, 1997; Morley, 1999; McLeod et al., 2000; Young et al., 2001; among many others) but also on (1) the depositional profile and sedimentary environments, and, (2) the stratigraphic architecture of the basin infill (Fig. 1). 2.1. Influence of deformation on the depositional profile If sedimentation [S] is insufficient to fill the space available [A] created by subsidence (i.e. due to folding or faulting), then deformation structures may modify the surface topography. That is to say, deformation can affect the depositional profile of contemporaneous sedimentary systems and, in so doing, the sedimentary facies (e.g. Edwards, 1995; Doglioni and Prosser, 1997; Doglioni et al., 1998; Fig. 1b). Many studies in the literature have discussed the influence of fold and faults on depositional profiles and drainage systems in continental domains (mainly in continental rifts: e.g. Leeder and Gawthorpe, 1987; Benedicto et al., 1999; Morewood and Roberts, 1999; Gawthorpe and Leeder, 2000; Peakall et al., 2000; Trudgill, 2002; or in foreland basins: e.g. Burbank et al., 1996; Williams et al., 1998; Gupta, 1997; Horton and DeCelles, 2001; Lo`pezBlanco, 2002); or affecting deltaic or turbiditic deposits (e.g. Allen et al., 1991; Butler et al., 1997; Haughton, 2000; Kneller and Buckee, 2000; Sinclair, 2000). 2.2. Influence of deformation on the stratigraphic architecture As pointed out above, deformation processes cause spatial and temporal variations of subsidence rates that lead to spatial and temporal variations of the [A] / [S] ratio, i.e., they result in the modification of the expression of stratigraphic sequences. 2.2.1. Temporal variations of accommodation Tectonically driven stratigraphic sequences (i.e. related to temporal variations of subsidence rates) have been proposed for single sequences (in foreland

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a) SEDIMENTARY THICKNESS

b) SEDIMENTARY FACIES

c) STRATIGRAPHIC SEQUENCES t4 t3 t2 t1 RETROGRADATION

proximal facies

PROGRADATION

distal facies

t0

Fig. 1. Influence of deformation on sedimentary processes. Topographic variations associated with deformation structures (faults or folds) result in differential subsidence that might modify (a) the distribution of sedimentary bodies in time and space (thickness), (b) the depositional profile and (c) the stratigraphic architecture (depositional sequences).

basins: e.g. Gupta and Allen, 1999; Takano, 2002; in rifts; e.g. Prosser, 1993; Gawthorpe et al., 1997; Gupta et al., 1999) as well as for multiple sequences (in foreland basins: e.g. Blair and Bilodeau, 1988; Burbank et al., 1988; Heller et al., 1988; Jordan and Flemings, 1991; Verge`s et al., 1998; Ito et al., 1999; ` lvaro et al., 2000; Catuneanu and Elango, 2001; in A rifts; e.g. Razin et al., 1996; Quiquerez et al., 1997). 2.2.2. Spatial variations of accommodation Spatial variations of subsidence rate associated with deformation will also induce spatial variations of accommodation [A] that can laterally modify the expression of stratigraphic sequences driven by other mechanisms (such as eustasy or basin-scale deformation). Several types of spatial modifications of the expression of stratigraphic sequences have been described in the literature. (1) As pointed out by Edwards (1995), the most obvious effect of deformation is an overall

change in the thickness of stratigraphic sequences (Fig. 2a). (2) Spatial modifications of accommodation variation curves may be produced by the subsidence related to deformation structures (e.g. Posamentier and Allen, 1993b; Gawthorpe et al., 1994). Such an effect has been proposed to explain lateral changes in the sedimentological expression of stratigraphic surfaces (maximum flooding surfaces, flooding surface, unconformities, sequence boundaries, etc. . ..) and, as a consequence, the geometry of stratigraphic sequences (e.g. Crumeyrolle et al., 1991; Edwards, 1995; Lickorish and Butler, 1996; Dromart et al., 1998; Hardy and Gawthorpe, 1998; Dreyer et al., 1999; Gawthorpe et al., 2000; Newell, 2000). This effect has also been described for basin-scale deformation, in foreland basins, where variations in subsidence/uplift rates observed away from the main deformation thrust (within a foredeep, forebulge, backbulge, etc.) strongly modify the expression of stratigraphic

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a) SEQUENCE THICKNESS

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Fig. 2. Modes of distortion of the stratigraphic architecture by deformation. Differential subsidence can modify (a) the thickness of the whole depositional sequence, (b) the thickness ratio of progradational and retrogradational half-sequences and (c) the temporal occurrence of the onset of progradation (MFS) or retrogradation (FS). For each case, the distortion is illustrated in actual geometry and thickness (space vs. space top diagram) and in a Wheeler diagram (time vs. space bottom diagram). Note that, in the third case, the onset of retrogradation (FS) does not occur at the same time at both locations.

sequences and the stacking pattern between proximal or distal locations (e.g. Goodman and Brett, 1994; Joy et al., 2000; Willis, 2000; Castle, 2001), even leading to inversion of the prograding/retrograding phases (e.g. Heller et al., 1988; Posamentier and Allen, 1993a; Catuneanu et al., 1997). Indeed, spatial variations of subsidence can enhance or reduce the effects of absolute sea-level variations and, in doing so, alter the thickness ratio of prograding and retrograding phases (the P / R ratio; Fig. 2b). (3) Furthermore, as implied in the cases described above and as explicitly pointed out by Wehr (1993), Catuneanu et al. (1998), Catuneanu

(2002) and Castelltort et al. (2003), spatial variations of subsidence might also result in lateral variation of the timing of trend inversions (Fig. 2c). These authors demonstrated that faster subsidence rates associated with a fold or a fault could locally delay the onset of progradation and lead to an earlier onset of retrogradation. For a given variation of absolute sea-level, the duration of relative sea-level fall will be shorter in areas of faster subsidence, that is to say, the onset of retrogradation will occur earlier (compare locations A and B on Fig. 3b). In addition, Wehr (1993), Martinsen and Helland-Hansen (1995) and

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Fig. 3. Sketch of embedding and spatial distortions of depositional sequences. (a) Embedding distortion: by temporal variation of accommodation [A] as defined by Cross (1988) and Guillocheau (1991, 1995), resulting from the superimposition of two frequencies of sea-level change. Top diagrams show the high and low frequency components of absolute sea-level variations and their superimposition. Bottom diagram shows the geometry of resulting depositional sequences for constant subsidence and sedimentation rates. (b) Spatial distortion: distortion by spatial variation of accommodation rate. Top diagrams show the absolute sea-level variations, the rates of subsidence and the resulting accommodation variations at two different locations. Bottom diagram shows the geometry of resulting depositional sequences for a constant sedimentation rate. See details in text.

Catuneanu et al. (1998) pointed out that spatial variations in sedimentation rate related to a delta might locally shift the onset of progradation to an earlier time and delay the onset of retrogradation.

section the different modes of distortion proposed in the literature.

We refer to this later effect as the spatial distortion of stratigraphic sequence. The term distortion has already been used referring to other processes, so for sake of clarity, we briefly review in the following

The first type of distortion, referred to here as the bcycle superposition distortionQ of stratigraphic sequences (Fig. 3a), was defined as related to temporal variations of accommodation [A] by Cross (1988)

2.3. The concept of distortion of stratigraphic sequences

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and Guillocheau (1991, 1995) and then quantified by Granjeon (1997). These authors pointed out the effect of superposed higher and lower frequencies sea-level variations. According to the timing of the high frequency sequences within the low frequency one, the relative duration of high frequency rises and falls of sea-level varies (Fig. 3a). They are identical for high frequency sequences occurring at the maxima (and minima) of the low frequency variation whereas the rise is longer during the low frequency rise and shorter during the low frequency fall. The second type of distortion, known as the bvolumetric distortionQ (i.e. half-cycle spatial thickness variation), describes the capacity of the system to preserve differentially the sediment supply (input at one end of the profile) along the depositional profile in relation to hydrodynamic processes (principle of volumetric partitioning; Homewood et al., 1992). During retrogradation ([A] / [S] N 1), sediments are trapped in the continental part of the system, while the shoreface is recorded as an erosion surface and the open marine part as a condensed surface. During progradation ([A] / [S] b 1), sediments are eroded on the continent and trapped in marine environments. For a given stratigraphic sequence (i.e. a given variation of [A] / [S] ratio), this results in different thickness ratios between progradation and retrogradation phases along the depositional profile. In this case, the trend inversions (onsets of progradation or retrogradation) occur at the same time along the whole depositional profile. The third type of distortion, referred to here as bspatial distortionQ (i.e. half-cycle phase lag), is the one observed by Wehr (1993), Martinsen and Helland-Hansen (1995), Catuneanu et al. (1998), Catuneanu (2002), and Castelltort et al. (2003) and is related to spatial variations in accommodation [A] or sedimentation rate [S] (see above). However, for this mode of distortion, no clear rule has yet been proposed for predicting the nature and the amount of distortion of stratigraphic sequences. This underlines the remaining need to examine, in a quantitative and precise way, how local variations of subsidence and sedimentation rate can modify the sequence stratigraphic architecture of contemporary deposits (changes in overall thickness of sequences in P / R ratio and in the timing of trend inversion) and, consequently, how spatial variations of subsidence

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and sedimentation can be extracted from the stratigraphic record.

3. Case example 3.1. Regional, 3D structural and stratigraphic setting The studied area is located along the West African margin, in the Congo–Cabinda Basin (Fig. 4). It corresponds to a growth fault/raft system, about 20
20 km, located within the extensional domain of a gravity-driven deformation system resulting from the gravitational spreading of post-rift sediments above a de´collement layer made up of evaporites (e.g. Duval et al., 1992; Lundin, 1992; Liro and Coen, 1995; Spathopoulos, 1996) (Fig. 4). In this study, we focus on the middle Cretaceous history of the structure (Middle Albian: 107 to 101 Ma; Figs. 4 and 5). In a previous work (Rouby et al., 2002), from a timeconverted 3-D seismic cube (20 m vertical resolution) and data from five well logs, we showed that during this period, the 3D geometry of the system corresponded to a N–S-trending depocentre located between two less subsiding areas (rafts on Fig. 5). At that time, the studied wells were located both within and outside a subsiding area (Fig. 5). We use the sequence stratigraphic architecture of the studied area for the Albian–Cenomanian defined by Rouby et al. (2003) based on the stacking pattern method (Homewood et al., 1992). Sedimentary environments and lithology are defined from well-log data (the mixed carbonate–siliciclastic platform is made up of marly clays interbedded with more or less bioclastic limestones deposited below the fair weather wavebase in offshore). Elementary stratigraphic units (bgenetic units,Q from a few metres to tens of metres thick) were defined as progradational/retrogradational cycles (i.e. shallowing/deepening upward cycles in an open marine environment) bounded by the two deepest facies (maximum flooding surfaces MFS sensu Homewood et al., 1992). The trend inversion between progradation and retrogradation is marked by the shallowest facies (flooding surface FS, sensu Homewood et al., 1992). The vertical stacking of genetic units defines hectometric thick progradational/retrogradational cycles called genetic unit sets (sensu Homewood et al., 1992).

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Fig. 4. (a) Location of the sedimentary basins off the coasts of Gabon and Congo. Dashed area shows the extent of salt basins. (b) Synthetic cross-section across the Congo–Cabinda Basin. Approximate location shown in (a). The outlined box indicates the study area. (c) Detailed cross-section across the studied area. The studied time interval is outlined in white.

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a)reconstructed isopach map of the studied interval

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Fig. 5. Geometry of the fault system during the studied time interval (Middle Albian) based on the 3D restoration of Rouby et al. (2002). (a) The reconstructed isopach map shows the graben (subsiding area materialized by the NS depocentre) between the upper and lower rafts and the location of the wells at that time. (b) Schematic EW cross-section across the studied area during the studied interval.

We assumed that genetic units can be correlated across the studied area, that is to say, they have the same cause active on a larger scale than the studied area (few km wide). This implies that a given number of genetic units in two different wells cover the same time span. By counting up genetic units between biostratigraphic markers and validating them from one well to another, we defined seven time lines for the studied interval (t0 to t7; thin black lines on Fig. 6). Time lines were thus defined using correlations based on the stacking pattern of genetic units and not by correlation of trend inversion of genetic unit sets. Note that like the genetic units, genetic unit sets can be traced out throughout the studied area. In this case, they cannot therefore be driven solely by accommodation variations related to displacement on the growth faults bounding the subsiding area (Figs. 4 and 5), that is, they must have been at least partly driven by a process operating on a larger scale than the studied system such as eustatic variations for example. Within the chronostratigraphic framework of the Albo–Cenomanian of this region, we calibrated the time lines on absolute ages using available micropaleontological data (palynology calibrated with ammo-

nites Jardine´ et al., 1974; Morgan, 1978; Doyle et al., 1982; Aurisano, 1992) and assuming that genetic unit sets are mostly driven by eustatic variations (Haq et al., 1987; Gradstein et al., 1994). A full discussion of the dating model, associated assumptions and error analysis can be found in Rouby et al. (2003). The correlated time lines (t0 to t7) define steps of about 0.5 to 2 Myr over a period of about 6.5 Myr (Fig. 6). 3.2. Subsidence and sedimentation rates In a previous work (Rouby et al., 2003), we also quantified at well locations (1) vertical displacements related to subsidence by measuring accommodation variations and (2) sedimentation rates, by measuring the preserved (decompacted) thickness of sediments for each time step. Accommodation is the vertical space available for sedimentation located between the basement and the base level taken as equivalent to the sea-level in marine domains (Jervey, 1988). It can be determined for a given time step as the sum of the (decompacted) thickness of sediments deposited during the time step and the variation of bathymetry between the base and the top of this sediment layer.

168 C. Robin et al. / Sedimentary Geology 178 (2005) 159–186 Fig. 6. Correlation of time lines on the five wells (see location on Fig. 5). Well logs (gamma ray and sonic) and interpreted lithologies are shown for each well. Depositional environments and paleodepths interpreted from elementary well log signatures (lower and upper offshore, shoreface) are shown for Well 2. In addition, depositional sequences (genetic unit sets 2 to 5 from Rouby et al., 2003) are shown at the scale of the genetic unit sets and genetic units (correlations of some genetic units are indicated as thin black lines), along with the chronostratigraphic model used to attribute absolute ages to time lines (thick black lines).

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about 60 m Myr1 on average, ranging between 20 and 140 m Myr1.

To carry out this calculation, we required: (i) the time lines determined above; (ii) the lithology and porosity of the sediments (the measured thicknesses of sediments were corrected for the dip of the strata and the compaction according to the laws established by Sclater and Christie, 1980) and (iii) estimations of paleobathymetry at each time line defined from sedimentary facies (Fig. 6). Calculated accommodation variation and sedimentation rates, assumed to be constant during each considered time step (between time lines defined above), are shown in Fig. 7. We then estimated the subsidence at well locations by correcting accommodation variations from long-term variations in the eustatic signal of Haq et al. (1987) (indeed variations in accommodation result from both absolute sea-level changes and vertical displacements related to subsidence). Averaged subsidence rates for the studied period were about 70 m Myr1 within the rafts and about 110 m Myr1 within the graben. Sedimentation rates were 101

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3.3. Distortion of the stratigraphic architecture To evaluate the influence of the subsidence on the stratigraphic architecture, we analyse the geometry of depositional sequences across the subsiding area (materialized by the depocentre on Fig. 5). To achieve this, we represent the genetic unit sets and the surfaces marking their trend inversions (FS and MFS; Fig. 8) within the stratigraphic framework defined above (time lines defined by stacking of genetic units). We then compare the wells located within the graben and the lower raft to the well located in the upper raft, which we have chosen as a reference (Figs. 5 and 8). Three observations can be made. (1) Genetic unit sets show an overall thickening in the wells located within the subsiding area 105

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Fig. 8. Transect of lithological logs of the five studied wells (see location on Fig. 5) showing variability in the geometry of genetic units sets. Time lines (thick black lines) are established by correlation of depositional sequences at the scale of genetic units (thin black lines, not all shown). For the reference well, the trend inversions of genetic unit sets are contemporaneous with time lines. For the other wells, trend inversions of genetic units sets (dashed lines) are defined independently according to the bathymetry of encountered depositional environments. For these wells, black arrows indicate the time lag for the occurrence of the trend inversion with respect to the reference well. Upward pointing arrows indicate that the trend inversion is delayed with respect to the reference well. Downward pointing arrows indicate that the trend inversion is earlier with respect to the reference well.

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(Wells 4, 5 and 6) with respect to the wells outside (Wells 2 and 7; Figs. 5 and 8). (2) This is associated with a modification of the lithology of the sequences. Lithologies are very similar in the wells located outside the graben, i.e., Wells 2 and 7 (except for the t0– t2 interval, where a significant percentage of sand is preserved in Well 7). As a difference, within the subsiding area, the carbonates / clay ratio increases and sandy layers are preserved (Fig. 8). (3) The position of the trend inversions of genetic unit sets (thick black lines in Fig. 8) with respect to time lines (thin black lines, determined at the scale of genetic units) shows that the onset of progradation (MFS) or retrogradation (FS) does not occur at the same time across the subsiding area. For example, the onset of retrogradation (FS) for set 4 is coeval with time line t5 in Well 2, whereas it is located below (i.e. earlier) in the wells of the subsiding area (Wells 4, 5 and 6). Also, the onset of progradation (MFS) for set 5 is contemporaneous with time line t6 in Well 2, but is located above (i.e. later) in the wells of the subsiding area. This suggests that subsidence in the graben alters the timing of trend inversion. There is however no clear rule predicting whether a given type of stratigraphic surface (FS or MFS) is delayed or advanced in the subsiding area. For example, the onset of progradation (MFS) for set 4 is coeval with time line t4 in Well 2, being located below (i.e. earlier) in the wells of the subsiding area (whereas the onset of progradation for set 5 is delayed). The timing of some stratigraphic surfaces is also modified between wells located outside the subsiding area (Wells 2 and 7). For example, the onset of retrogradation (FS) of set 3 is earlier in Well 7 with respect to Well 2 (where it is coeval with time line t3). This suggests that differences in subsidence between Wells 2 and 7, even though less pronounced than with the wells of the subsiding area, may also alter the timing of trend inversions. However, as pointed out by Wehr (1993) and Martinsen and Helland-Hansen (1995), in areas of similar subsidence rate, differences of sediment supply could also cause this distortion.

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Our data allow us to identify the occurrence of trend inversion offsets and whether these shifts are forwards or backwards in time. However, we cannot measure accurately the duration of the time lag (the observed vertical offset actually depends on the sedimentation rate), since, the number of genetic units involved in the offset interval gives only a qualitative indication of the duration of the time lag. Using this criterion, we can, nevertheless, observe that for a given trend inversion within the graben, both the vertical and temporal offsets are generally different from one well to another (for example, the onset of progradation in set 4 shows an offset involving different numbers of genetic units in Wells 4, 5 and 6; Fig. 8).

4. Stratigraphic modelling The observation of this spatial distortion on the studied example raises several questions. (1) For a given sediment supply, under what conditions of subsidence is a change in the timing of trend inversion observed (i.e. the sequence is distorted)? (2) How does the rate of sedimentation modify this distortion? (3) Under which conditions, can sediment supply alone drive this distortion? (4) Is it possible to predict the time shift and the amount of distortion? Above all, (5) are the genetic units (used to establish time lines) also distorted and by how much? Further analysis of the studied area does not allow us to answer these questions because we cannot quantify exactly the observed time shifts. Moreover, we do not know the frequency and the amplitude of the regional sea-level changes controlling the depositional sequences (genetic units and genetic unit sets). Therefore, we used a numerical experimental approach to address these questions. 4.1. Model configuration and parameters In order to keep the modelling straightforward, we assumed very simple boundary conditions. (i) We used a numerical model (in 1-D) to study the influence of the variations of subsidence and sediment supply on the geometry of depositional sequences at various scales. The modelling is not

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(iv) At a given location (i.e. in 1-D), trend inversion surfaces of a sequence record the bathymetric maxima (MFS) and minima (FS). We therefore calculated the evolution of bathymetry using a schematic configuration with various rates of sediment supply and subsidence calibrated using values determined for the studied example. (v) Accommodation variations result from the superimposition of two sinusoidal signals (eustasy) and a linear variation (subsidence). We simplified the model down to two domains showing independent subsidence and sedimentation rates (Fig. 9). Both subsidence and sedimentation rates are assumed to be constant through time.

intended to reproduce the observed stratigraphic architecture, but to test quantitatively the effects of varying the duration of depositional sequences, sedimentation and subsidence rate, so we can reproduce the observed distortions. (ii) The trend inversion offsets of the genetic unit sets observed across the studied area cannot be related to a volumetric distortion because they occur over a short distance (a few km, Figs. 5 and 6) and within similar depositional environments, as indicated by the homogeneity of well logging signatures (i.e. sedimentary facies). Thus, in the following numerical approach, we do not take sediment transport processes (i.e. volumetric partitioning) into account: sedimentation is assumed to correspond to a vertical filling. (iii) The trend inversion offsets of genetic unit sets observed across the studied area cannot be related solely to a cycle superposition effect. This is because the latter may only produce temporal distortion and not the observed spatial variation in the expression of stratigraphic sequences. Nonetheless, we test the influence of the superposition of genetic units within genetic unit sets.

The variation of accommodation with time is therefore given by: 

t t aðt Þ ¼ RS t þ e1 sin 2p þ e2 sin 2p þc T1 T2 ð1Þ where a is the accommodation, t is the time (in Myr), R S is the rate of subsidence (in m Myr1 ), e 1 and e 2

a) MODEL 1 :

b) MODEL 2 :

various subsidence rates across the fault

various sedimentation rates across the fault

t4 t3 t2 t1 t0

reference

sedimentation = 60 m/Ma case 1 subsidence = 90 m/Ma case 2 subsidence = 110 m/Ma ? case 3 subsidence =130 m/Ma

t4

reference

subsidence = 110 m/Ma

t3

case 4 sedimentation = 30 m/Ma

t2 t1

case 2 sedimentation = 60 m/Ma

t0

?

case 5 sedimentation = 80 m/Ma case 6 sedimentation = 100 m/Ma case 7 sedimentation = 120 m/Ma

common: eustasy 1 (T=0.1 Ma, A= 10 m), eustasy 2 (T=2 Ma, A= 40 m), initial bathymetry = 100 m reference : subsidence = 70 m/Ma, sedimentation = 60 m/Ma

Fig. 9. Sketch of the two models of calculations of bathymetric variations (see also parameters detailed in Table 1). Two sinusoidal sea-level variations are superimposed: a high frequency at the scale of the genetic units (0.1 Myr, amplitude of 10 m) and a low frequency at the scale of genetic unit sets (2 Myr, amplitude of 40 m). The reference well is modelled with a subsidence rate of 70 m Myr1 and a sedimentation rate of 60 m Myr1. (a) Model 1: sedimentation rate is identical in both wells, and the subsidence rates range from 90 to 130 m Myr1. (b) Model 2: subsidence rate in the other well is 110 m Myr1 and sedimentation rates range from 30 to 120 m Myr1.

C. Robin et al. / Sedimentary Geology 178 (2005) 159–186

are the amplitudes (in m) of the eustatic signals (with e N 0) and T 1 and T 2 are the periods (in Myr) of the eustatic signals and c, a constant corresponding to the initial bathymetry of the system. By reference with the studied area, we defined a high frequency sea-level variation (T 1 = 0.1 Ma) corresponding to the suspected scale of genetic units, and a low frequency variation (T 2 = 2 Ma) at the scale of the genetic unit sets. Amplitudes (e 1 = 10 m and e 2 = 40 m) are evaluated from similar eustatic variations according to Haq et al. (1987). The calculation increment is 4 ka and the initial bathymetry is 100 m. Using the values determined for the studied area (see Section 3.2), the reference well located in the footwall of the normal fault (or on the hinge of an anticline) is modelled with a subsidence rate of 70 m Myr1 and a sediment supply rate of 60 m Myr1. We performed two series of calculations. In Model 1, the sediment supply in the hanging wall compartment (or in the synclines rimming an anticline) is identical to the footwall and the subsidence rate ranges from 90 to 130 m Myr1 (Fig. 9a). In Model 2, the subsidence rate of the hanging wall is fixed at 110 m Myr1 and the sedimentation rates range from 30 to 120 m Myr1 (Fig. 9b). The parameters used for modelling are summarized in Table 1.

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4.2. Results The results of the modelling are presented as bathymetric curves determined at given locations (i.e. for various sedimentation and subsidence rates). Trend inversions of both scales of depositional sequences (high and low frequency) can be identified along this curve (by bathymetric maxima (MFS) and minima (FS)) (Fig. 10). Fig. 11 shows the calculated variations of bathymetry in the footwall (reference) and in the hanging wall for the various cases of sediment supply and subsidence rate described above (Models 1 and 2). The influence of subsidence variations (with identical sedimentation rate) can be observed at the scale of the genetic unit sets (compare reference and case 2 on Fig. 11a): the progradation onsets (MFS) are delayed and the retrogradation onset (FS) occurs earlier in the subsiding area (dashed vertical lines on Fig. 11a). Moreover, the larger the subsidence differences across the fault, the longer the offset of the trend inversions (compare cases 1, 2 and 3 on Fig. 11a). The sediment supply can modify this behaviour (see Fig. 11b representing accommodation variation for similar subsidence difference with various sediment supplies, Model 2). If the sediment supply rate is

Table 1 Eustasy 1 Amplitude AE1 (m) Period T 1 (Myr) Maximum rate E 1 (m Myr1) Eustasy 2 Amplitude AE2 (m) Period T 2 (Myr) Maximum rate E 2 (m Myr1)

10 0.10 628.32 40 2.00 125.66 Set 1 Reference

1

Subsidence rate R S (m Myr ) 70 Sedimentation rate S (m Myr1) 60 D (no cycle superposition effect) D eustasy 2 1.11 Time lag Dti (Myr) 0 D eustasy 1 1.02 Time lag Dti (Myr) 0 DV (with cycle superposition effect) DV eustasy 1 1.32 Time lag Dti (Myr) 0

Set 2

1 case

2 case

3 case

4 case

5 case

6 case

7 case

90 60

110 60

130 60

110 30

110 80

110 100

110 120

1.36 0.10 1.06 0.001

1.70 0.21 1.11 0.002

2.21 0.33 1.15 0.003

2.57 0.39 1.18 0.0036

1.36 0.10 1.06 0.001

1.11 0 1.02 0

0.90 0.10 0.98 0.001

1.38 0.001

1.44 0.0021

1.51 0.0032

1.54 0.0037

1.38 0.001

1.32 0

1.27 0.0010

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high

a) ABSOLUTE SEA LEVEL: short wavelength component (genetic units)

time

+

high

b) ABSOLUTE SEA LEVEL: long wavelength component (genetic unit sets)

time

high

c) SUBSIDENCE

+

time

high

d) SEDIMENTATION

+ time

= e) BATHYMETRY

genetic units MFS

depth

genetic unit sets MFS FS

time

genetic unit sets FS

genetic units MFS genetic unit sets MFS

C. Robin et al. / Sedimentary Geology 178 (2005) 159–186

reduced in the subsiding area with respect to case 2, the time shift of trend inversions is increased (case 4; Fig. 11b). On the contrary, if the sediment supply rate is progressively increased with respect to case 2, the time shift is either decreased (case 3), cancelled (case 4) or even inverted (case 5; dashed vertical lines on Fig. 11b). In addition, for areas with similar subsidence rates, variations of sediment supply produce a time shift in the trend inversion (compare cases 2 and 3 to case 7; Fig. 11b): the lower the rate of sediment supply, the longer the delay in onset progradation (MFS) and the earlier the onset of retrogradation (FS). At the scale of the genetic units, on the contrary, whatever the rate of sedimentation and subsidence, no significant modification is observed in the timing of trend inversion (thin dashed black vertical lines on Fig. 11). 4.3. Quantification of the distortion 4.3.1. Distortion of genetic unit sets D In order to analyse these results, we define a parameter quantifying the distortion of depositional sequences. In a first step, we simplify the system to a single sinusoidal variation of sea-level. The equation describing accommodation variations becomes:  t  aðt Þ ¼ RS t þ esin 2p þc ð2Þ T where a is the accommodation, t is the time (in Myr), R S is the subsidence rate, e is the amplitude (in m) and T (in Myr) is the period of the eustatic signal (with e N 0) and c is a constant corresponding to the initial bathymetry (in m) of the system. The rate of accommodation variation becomes:  t  e A ¼ RS þ Ecos 2p with E ¼ 2p ð3Þ T T where A is the rate of accommodation variation (in m Myr1), R S is the subsidence rate (in m Myr1) and E is the maximum rate of eustatic sea-level change (in m Myr1).

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The distortion D of a given depositional sequence can be defined as the ratio between the duration of the progradation Dtp (defined between a MFS and a FS) and the duration of the retrogradation Dtr (defined between a FS and a MFS), that is to say (the equation is derived in the appendix): Dtr W ¼ Dtp pW e and E ¼ 2p : T

D¼


S  RS with W ¼ arc cos E ð13Þ

This amount of distortion of a depositional sequence shows three domains of values. (i) When D = 1, the durations of progradation and retrogradation are identical, i.e. with constant sediment supply, the cycle is symmetric in terms of thickness. (ii) When D N 1, retrogradation is longer than progradation, i.e. with constant sediment supply, the cycle is asymmetric in terms of thickness, being dominated by retrogradation. (iii) When D b 1, retrogradation is shorter than progradation, i.e. with constant sediment supply, the cycle is asymmetric in terms of thickness, being dominated by progradation. Table 1 shows the values of D for the genetic unit sets (Eustasy 2) modelled in the different cases. 4.3.2. Distortion of genetic units DV In the above analysis, we did not take account of the superimposition of two sinusoidal signals due to eustatic sea-level changes. In the modelled case, high frequency sequences (genetic units) are distorted as they are superposed to low frequency sequences (genetic unit sets): this is the cycle superposition distortion defined by Cross (1988) and Guillocheau (1991, 1995; see Section 2.3). The definition of the following distortion parameter DV for genetic units

Fig. 10. Definition of the trend inversions at a given location (in 1-D) in term of bathymetry. The superimposition of (a) a high frequency sinusoidal absolute sea-level variation (genetic units) and (b) a low frequency component (genetic unit sets) onto constant (c) subsidence and (d) sedimentation rates results in (e) bathymetric variations. Onset of retrogradation of the genetic unit sets (FS; dashed vertical line) occurs at the minimum bathymetry and the onset of progradation (MFS; dotted vertical line) occurs at the maximum bathymetry. The onset of progradation of genetic units occurs at the temporary bathymetric maxima (MFS; thin dotted lines), corresponding to the high frequency sea-level variations.

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a) MODEL 1 - Bathymetry variations for various subsidence rates 500

The timing of trend inversion genetic unit sets is altered

400

Depth (m)

FS

The timing of genetic units is not significantly altered case 3

MFS

case 2

genetic units

300

MFS genetic unit sets MFS FS

200

case 1

Reference

100

0 0

1

2

3

4

Age (Ma)

5

b) MODEL 2 - Bathymetry variations for various sedimentation rates 500

The timing of trend inversion genetic unit sets is altered

400

FS

The timing of genetic units is not significantly altered

case 4

MFS

Depth (m)

genetic units

case 2

MFS

300

genetic unit sets MFS FS

case 5

200

Reference & case 6

100

case 7 0 0

1

2

3

4

Age (Ma)

5

Fig. 11. Variations of bathymetry computed for (a) Model 1 and (b) Model 2 (parameters values detailed in Table 1). The reference location is shown in black and the various cases of subsidence and sedimentation rates are shown in colours. Trend inversions at the scale of genetic unit sets (flooding surfaces FS and maximum flooding surfaces MFS) are shown (coloured vertical lines), as well as maximum flooding surfaces at the scale of genetic units (thin dotted lines). The timing of trend inversions at the scale of genetic unit sets is different at the reference location (black vertical line) and in the other cases (coloured vertical lines), whereas the timing of trend inversions at the scale of genetic units is similar (thin dashed vertical lines).

C. Robin et al. / Sedimentary Geology 178 (2005) 159–186

takes this effect into account (Eq. (23) is derived in the appendix): Dtr WV ¼ Dtp p  WV 

S  RS þE2 S  RS E2 with arc cos bWVbarc cos E1 E1 DV¼

ð23Þ where S and R S are the sedimentation and subsidence rates (in m Myr1), and E 1 and E 2 are the maximum rates of eustatic sea-level change (in m Ma1) for the high and low frequency variations, respectively. For a given scale of genetic units, DV varies between a maximum and a minimum (range of variation of W), depending on the position of the genetic unit within the lower frequency sequence (genetic unit set). Table 1 shows the maximum values of DV of the genetic units for the different modelled cases. 4.3.3. Time lag associated with the distortion If the distortion values of a given depositional sequence are different between two locations (whether DV or D), the trend inversions are not synchronous and the associated temporal offset can be determined. This time lag for a trend inversion (Dti), for example for the onset of progradation (MFS), is the difference in duration of retrogradation between the two locations, which is defined as (Eq. (16) is derived in the appendix):  Dti ¼ Dtr  Dtrref ¼ T

D Dref  1 þ D 1 þ Dref


ð26Þ

with Dtr and Dtrref are the durations of retrogradation (in Myr) at the considered location and at the reference location, respectively. D and D ref are the distortion values in the considered location and the reference location, respectively, while T is the period of the considered eustatic sea-level change (in Myr). The definition of this parameter indicates that the closer the D values at the two locations, the shorter the time lag for the trend inversion (Dti). Furthermore, it should be pointed out that the shorter the period of sea-level change (T), the shorter the time lag of the trend inversion. Table 1 shows values of Dti for the genetic unit sets (Eustasy 1) and maximum values of Dti for the genetic units (Eustasy 2) in the different modelled cases with respect to the reference.

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5. Discussion This highly simplified model (bathymetric variation in 1-D, relative sea-level corresponding to two superimposed sinusoidal signals for eustasy and a linear variation for subsidence, sedimentation corresponding to a vertical filling at constant rate) provides us with guide rules for the behaviour of a stratigraphic system in a given stratigraphic state ([A] / [S] ratio) in terms of sequence distortion. 5.1. Distortion of genetic units In the theoretical model, at the scale of genetic units (high frequency sea-level variations), whatever the rates of sedimentation and subsidence, no significant modification in the timing of trend inversion is observed (thin dashed black vertical lines in Fig. 11). The distortion of genetic units is related to both the cycle superposition effect and variations in rates of subsidence and sedimentation. The distortion parameter D only quantifies this latter effect (see D values for Eustasy 1 on Table 1), showing that the distortion of genetic units is actually not zero (D p 1). However the values are systematically closer to one (i.e. they are less distorted) than in the case of genetic unit sets (compare D values for Eustasy 1 and 2 in Table 1). Indeed, distortion depends not only on sedimentation and subsidence rates (S and R S in Eqs. (13) and (23)), but also on the rate of eustatic sea-level change (E in Eqs. (13) and (23)), that is to say, on the frequency and amplitude of the sea-level variations (T and A E in Eq. (3)). Thus, the amount of distortion of a given depositional sequence depends on the duration of sealevel change that caused it: for a given sedimentation and subsidence rate, the higher the maximum rate of eustatic sea-level change E (E being always positive), the lower the distortion D. Note that the longer the wavelength (A E ) or the shorter the period (T) of a sealevel variation, the higher the maximum rate E. In other words, a given variation of subsidence across a tectonic structure (fault or anticline) may induce a (significant) distortion only of given scales of depositional sequences. This potential amount of distortion is then modulated by the rate of sedimentation. Taking into account the distortion caused by the superposition of genetic units into genetic unit sets (DV values for Eustasy 1 on Table 1), the maximum

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distortion of genetic units can be equivalent or even larger than observed for genetic unit sets (compare DV values for Eustasy 1 and D values for Eustasy 2 on Table 1). However, in terms of time lag, the maximum duration of the offset of genetic units is of the order of thousands of years, even for large values of distortion (see Dti values for Eustasy 1 on Table 1). 5.2. Distortion of genetic unit sets 5.2.1. Influence of subsidence rate At the scale of genetic unit sets (high frequency sea-level variation in numerical models), Model 1 shows that the subsidence difference across a fault (or the anticline) delays the onset of progradation and advances the onset of retrogradation in the most subsiding area (Fig. 12a). This can be illustrated theoretically by the superimposition of a sinusoidal (regional) sea-level variation, a constant sediment supply and two different subsidence rates (Wehr, 1993; Castelltort et al., 2003; Fig. 12a). The onset of progradation occurs when the sedimentation rate equals the decreasing accommodation rate (the onset of retrogradation when it equals the increasing accommodation rate). For a constant sedimentation rate, high subsidence rates limit the accommodation reduction related to sinusoidal sea-level changes and enhance the creation of accommodation. Thus, in the most subsiding area, the onset of progradation is delayed (and the onset of retrogradation advanced) with respect to the less subsiding area (Fig. 12a). This results in a higher distortion parameter value in the subsiding area (compare D values for case 1 and the reference well in Table 1). Also, the greater the difference in subsidence between the two locations, the larger the distortion parameter value in the subsiding area (compare D values for Eustasy 2 for cases 1, 2 and 3 in Table 1) and, consequently, the longer the time lag of trend inversion (dashed vertical coloured lines in Fig. 12a; compare Dti values for Eustasy 2 for cases 1, 2 and 3 in Table 1). 5.2.2. Influence of sedimentation rate Model 2 illustrates that, for a given subsidence difference across the fault (or anticline), the sedimentation rate modifies the time lag of trend inversion (Fig. 12b). The faster the sedimentation rate in the subsiding area, the lower the distortion parameter

value in the subsiding area (compare D values for Eustasy 2 of cases 4, 5 and 6 in Table 1) and the shorter the time lag of trend inversion (dashed vertical coloured lines in Fig. 12b; compare Dti values for Eustasy 2 of cases 1, 2 and 3 in Table 1). Case 7 of Model 2 indicates that, when the sedimentation rate is higher than the subsidence rate, the distortion parameter (D) falls below 1 (Table 1). Consequently, the sense of time lag of trend inversion is reversed, that is to say, the onset of progradation is advanced (the onset of retrogradation delayed) in the subsiding area (see green dashed vertical line in Fig. 12b and Dti value for Eustasy 2 in Table 1). The theoretical model superimposing a sinusoidal (regional) sea-level variation, a constant subsidence and two different sedimentation rates, demonstrates that a difference in sediment supply alone may result in a time lag of trend inversion (Wehr, 1993; Martinsen and HellandHansen, 1995; Fig. 12b). To sum up, the slower the subsidence rate (or the faster the sedimentation rate), the longer the progradational phase of the depositional sequence (lower D with D b 1), whereas the faster the subsidence (or the slower the sedimentation) the longer the retrograding phase (higher D and D N 1). 5.2.3. Observed distortion of genetic unit sets We can use our simplified theoretical modelling to discuss the observed distortion of genetic unit sets in the studied area in terms of spatial variations in subsidence and sedimentation rates. Nevertheless, we must bear in mind some restrictions. Our theoretical model is uni-dimensional and we do observe the behaviour of a natural system along vertical lines (1D) corresponding to the five studied wells (Fig. 8). Subsidence and sedimentation rates are constant in time in the theoretical model. This is evidently not true in the studied area, where subsidence and sedimentation rates vary both in time and space (Fig. 7). However, subsidence and sedimentation rates were assumed to be constant over each considered time increment (measured averaged values on Fig. 7). Thus, the initial conditions of the model are satisfied during each studied time interval. Since the rate of subsidence is on average higher in the graben than on the raft (70 m Myr1 in the rafts and 110 m Myr1 in the graben; Fig. 7), according to the theoretical model, we expect a systematic delay of

b) MODEL 2 - distortion by variation of the sedimentation rates

a) MODEL 1 - distortion by variation of the subsidence rate HIGH

HIGH

ABSOLUTE SEA LEVEL (eustasy or regional deformation)

ABSOLUTE SEA LEVEL (eustasy or regional deformation)

time

time

LOW

HIGH

S time

SEDIMENTATION [S]

SEDIMENTATION [S]

LOW

high subsidence location

LOW

SUBSIDENCE

low subsidence location (reference well) time

SUBSIDENCE

LOCAL SUBSIDENCE UPLIFT

high subsidence location

HIGH

low subsidence location (reference well)

ACCOMMODATION [A] (RELATIVE SEA-LEVEL)

S high sedimentation location low sedimentation location S (reference well) time

LOCAL SUBSIDENCE

time UPLIFT HIGH

ACCOMMODATION [A] (RELATIVE SEA-LEVEL)

time LOW

time LOW

MFS FS RATES OF ACCOMMODATION VARIATION AND SEDIMENTATION [A]/[S] RATIO (STRATIGRAPHIC SEQUENCES)

S

time

high subsidence location delayed MFS, advanced FS low subsidence location (reference well)

MFS FS RATES OF ACCOMMODATION VARIATION AND SEDIMENTATION [A] /[S] RATIO (STRATIGRAPHIC SEQUENCES)

S S time

high sedimentation location Advanced MFS, delayed FS low sedimentation location (reference well)

C. Robin et al. / Sedimentary Geology 178 (2005) 159–186

HIGH

LOW

Fig. 12. (a) Theoretical analysis of Model 1: distortion of the timing of trend inversions of genetic unit sets involving the superimposition of a sinusoidal absolute sea-level variation (green), a constant sediment supply (orange) and different subsidence rates (blue and red). Onset of progradation (MFS) occurs when the rate of sedimentation equals the decreasing rate of accommodation creation, while the onset of retrogradation (FS) occurs when it equals the increasing rate of accommodation creation. By construction, the MFS (dotted lines) are delayed and the FS (dashed lines) advanced in the most subsiding area. (b) Theoretical analysis of Model 2: distortion of the timing of trend inversions of genetic unit sets involving the superimposition of a sinusoidal absolute sea-level variation (green), a constant subsidence rate (purple) and different sedimentation rates (orange and red). By construction, the MFS are advanced and the FS delayed in areas undergoing more rapid sedimentation. See explanation in text. Modified after Castelltort et al. (2003).

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the onsets of progradation (MFS) and earlier onsets of retrogradation (FS). This is generally true (one major exception is discussed below) and applies, for example, to the onset of retrogradation (FS) for sets 3 and 4, and the onset of progradation (MFS) for set 5 (Fig. 8). We can therefore interpret the observed diachronisms as resulting from spatial variations of subsidence rates across the studied area. As already pointed out, a major exception is the onset of progradation (MFS) of set 4 that occurs earlier in the subsiding area (Fig. 8). This could be related either to a decrease of accommodation creation rate in the graben (see the increase of subsidence rate in modelled case 1 with respect to the reference case on Fig. 11a) or an increase of sediment supply preserved in the graben (see the increase of sedimentation rate in modelled case 6 with respect to the reference case on Fig. 11b). In the studied area, the considered time interval does not correspond to a decrease in the rate of accommodation creation but rather an increase of sedimentation rate preserved in the graben (see subsidence and sedimentation rates measured for time interval t3-t4 on Fig. 7). We can therefore interpret this observed diachronism as the result of an increase of sedimentation rate in the subsiding area resulting in the obliteration and even the inversion of the subsidence effect. For a given trend inversion, the temporal offset can be different from one well of the graben to another (see, for example, the onset of retrogradation (FS) for set 4 that shows an offset involving different numbers of genetic units in Wells 4, 5 and 6; Fig. 8). This can be interpreted as resulting from spatial variations of accommodation and sedimentation that produce, for a given trend inversion, spatial variations in the duration of the offset with respect to the reference well. To summarize, complex temporal and spatial variations in subsidence and sedimentation rates in the studied area lead to variations of the distortion of genetic unit sets. This produces a variety of trend inversion behaviours in terms of whether they occur earlier or later and in duration. 5.3. Correlation of time lines The numerical analysis provides evidence that both genetic unit sets and genetic units are distorted by lateral changes in subsidence and sedimentation rates.

In addition, genetic units are distorted by being superposed to genetic unit sets. In the cases modelled here (calibrated on the studied example), the duration of the time lag from one location to another falls in the range of hundreds to thousands of years for the genetic unit sets and thousands of years for the genetic units. For an average sedimentation rate estimated in the studied area at around 100 m Myr1 , these time lags will result in a vertical offset in the range of tens of metres for genetic unit sets and tens of centimetres for genetic units. The resolution of well-log data allows us to observe the distortion of genetic unit sets, while the vertical expression of the time lag of the genetic units falls within the resolution range of the tools. In the studied area, we can therefore consider the expression of the distortion of genetic units related to variations of subsidence and sedimentation rates across the graben as negligible. Thus, the distortion of genetic units has no effect on correlations based on the stacking pattern of genetic units, so the definition of time lines will remain accurate. Even if this type of analysis was carried out on field data, we should point out that it would be possible to improve the vertical resolution in the reading of the sedimentary record, however, a sedimentary system will only rarely record a variation at the scale of a few thousand years. On the other hand, correlations based on lower frequency depositional sequences (over 1 Myr in duration; for example, depositional sequences driven by deformation processes) could result in significant errors if the considered area is associated with spatial variations of subsidence and/or sedimentation rates. The present study confirms that the stacking pattern approach should be carried out using high frequency depositional sequences (less than 1 Myr in duration, e.g. Posamentier et al., 1988; Van Wagoner et al., 1988, 1990; Homewood et al., 1992; Wehr, 1993), i.e., depositional sequences driven by climatic variations.

6. Conclusions Using a numerical analysis approach combined with the observation of natural example (a system of normal growth faults), we studied the influence of subsidence and sediment supply on the expression of stratigraphic sequences. We characterized the distortion of depositional sequences in terms of the over-

C. Robin et al. / Sedimentary Geology 178 (2005) 159–186

all thickness of the sequence, the relative thickness of progradational and retrogradational phases ( P / R ratio), and the timing of trend inversions (onsets of progradation and retrogradation). We defined parameters quantifying the distortion of depositional sequences, whether resulting from spatial variations in subsidence and sedimentation rates (spatial distortion D) and from the superimposition of two scales of sea-level variation (cycle superposition distortion DV). We used a numerical model to investigate the distortion of two scales of superposed depositional sequences in response to spatial and temporal variations of subsidence and sedimentation rates. (i) We show that spatial variations in subsidence rate lead to modifications in the timing of trend inversion: the onset of progradation is delayed and the onset of retrogradation occurs earlier in the most subsiding area. (ii) The rate of sedimentation can modulate the amount of distortion related to subsidence: it can amplify, limit, compensate or even invert this temporal offset. (iii) Spatial variations in sedimentation rate alone also produce changes in the timing of trend inversion: the onset of progradation occurring earlier and the onset of retrogradation being delayed in areas showing faster sedimentation rates. (iv) The amount of distortion not only depends on the sedimentation and subsidence rates but also on the maximum rate of sea-level change causing the depositional sequence, that is, it depends on the period and amplitude of the sequence. The faster the sea-level change (i.e. the shorter the period and the larger the amplitude), the lower the amount of distortion produced. (v) In terms of resulting time lag, numerical analysis demonstrates that the distortion of the high frequency depositional sequences (genetic units) is negligible. These sequences can therefore be safely used to correlate time lines across the studied area, in contrast with the low frequency sequences (genetic unit sets) whose spatial distortion could lead to significant errors in the correlation of time lines. We use these results to interpret the distortion observed in the studied cases in terms of temporal

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and spatial variations of subsidence and sedimentation rates. In the studied area, complex temporal and spatial variations in subsidence and sedimentation rates result in variations of the distortion of genetic unit sets producing a variety of trend inversion behaviours in terms of whether they occur earlier or later and in duration.

Acknowledgements We are grateful to Total Congo for financial support of this study and for providing us with the data set, as well as Total for technical support. We thank P. Homewood, O. Catuneanu and A. Miall for positive and thorough reviews. Dr M.S.N. Carpenter post-edited the English style.

Appendix A. Derivation of Eq. (13) For a single sinusoidal variation of sea-level superimposed on a constant subsidence rate, the equation describing accommodation variations is:  t  aðt Þ ¼ RS t þ esin 2p þc ð2Þ T where a is the accommodation, t is the time (in Myr), R S is the rate of subsidence, e is the amplitude (in m) and T (in Myr) is the period of the eustatic signal (with e N 0) and c is a constant corresponding to the initial bathymetry (in m) of the system. The rate of accommodation variation is:  t  e A ¼ RS þ Ecos 2p with E ¼ 2p ð3Þ T T where A is the rate of accommodation variation (in m Myr1), R S is the subsidence rate (in m Myr1), and E is the maximum rate of eustatic sea-level change (in m Myr1). The rate of sediment supply is given by S¼

ds dt

ð4Þ

where S is the rate of sediment supply (in m Myr1) and ds is the (decompacted) thickness (in m) of sediment deposited during the time step dt (in Myr).

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The rate of accommodation variation is given by: A¼

ds db þ : dt dt

ð5Þ

Substituting for A (Eq. (3)) and ds / dt (Eq. (4)) in Eq. (5), the evolution of the bathymetry at a given location is given by:  t  db ¼ RS þ Ecos 2p  S: dt T

Appendix B. Derivation of Eq. (23) ð6Þ

The trend inversion surfaces of depositional sequences mark the bathymetric minimum (FS) and maximum (MFS), and are therefore defined when: db ¼0 dt

ð7Þ

ð8Þ

For these two surfaces, we define two critical times t max,n for MFS and t min,n for FS, since: tmax;n



W W ¼ nþ T and tmin;n ¼ n  T 2p 2p 

S  RS with W ¼ arc cos E and  1b

ð9Þ




S  RS b1: E

ð10Þ

ð11Þ

Note that, when condition Eq. (13) is not true, the rate of bathymetric variation is never zero, implying a constant rate of accommodation creation ((S  R S) / E) b 1 or destruction ((S  R S) / E) N 1, that is, the trend inversion surfaces are not defined. The durations of retrogradation Dtr (defined between a FS and a MFS) and progradation Dtp (defined between a MFS and a FS) are given by: W pW Dtr ¼ T and Dtp ¼ T: p p

To analyse genetic units stacked into genetic unit sets, we assume in Eq. (1) that accommodation can be expressed by the superimposition of two sinusoidal variations (eustasy) and a linear variation (subsidence), 

t t aðt Þ ¼ RS t þ e1 sin 2p þ e2 sin 2p þc T1 T2 ð1Þ

that is to say, following Eq. (6), when: 
t S  RS cos 2p : ¼ T E

the progradation and the retrogradation phases, that is to say: 
Dtr W S  RS ¼ with W ¼ arc cos D¼ Dtp pW E e ð13Þ and E ¼ 2p : T

ð12Þ

The distortion D of a given depositional sequence can be defined as the ratio between the durations of

where a is the accommodation, t is the time (in Myr), R S is the rate of subsidence (in m Myr1), e 1 and e 2 are the amplitudes (in m) of the eustatic signals (with e N 0) and T 1 and T 2 are the periods (in Myr) of the eustatic signals and c is a constant corresponding to the initial bathymetry of the system. Assuming that T 2 is large enough with respect to T 1 to allow a linear derivation of the second sinusoid: 

t t0 t  t0 þ 2p sin 2p ¼ sin 2p ð14Þ T2 T2 T2 


t t0 t  t0 sin 2p ¼ sin 2p cos 2p T2 T2 T2 

t0 t  t0 þ cos 2p sin 2p ð15Þ T2 T2 


t t0 t  t0 t0 cos 2p sin 2p csin 2p þ 2p : T2 T2 T2 T2 ð16Þ Substituting Eq. (16) in Eq. (1), the accommodation variation can be written as: 
 
t t0 aðt Þ ¼ RS t þ e1 sin 2p þ e2 sin 2p T1 T2 

t  t0 t0 þ 2p cos 2p þc ð17Þ T2 T2

C. Robin et al. / Sedimentary Geology 178 (2005) 159–186

Following this, the rate of accommodation variation is:  

t t0 A ¼ RS þ E1 cos 2p þ E2 cos 2p T1 T2 ei ð18Þ with Ei ¼ 2p Ti where A is the rate of accommodation variation (in m Myr1), R S is the subsidence rate (in m Myr1), E 1 and E 2 are the maximum rates of eustatic sea-level change (in m) for the high and low frequency variations, respectively. Substituting A (Eq. (18)) and ds / dt (Eq. (4)) in Eq. (5), the evolution of the bathymetry at a given location is given by: 

db t t0 ¼ RS þ E1 cos 2p þ E2 cos 2p  S: dt T1 T2 ð19Þ Substituting Eq. (19) into Eq. (7), we can define the critical times t max,n for MFS and t min,n for FS as: 

WV WV tmax;n ¼ n þ T and tmin;n ¼ n  T 2p 2p ð20Þ 

with WV ¼ arc cos

t0 S  RS  E2 cos 2p T2 E1


! : ð21Þ

In contrast with Eq. (13), in which W is constant, WV varies with time (t 0) between two extrema, for: 
t0  1bcos 2p b1: ð22Þ T2 The distortion parameter of a genetic unit superposed to a genetic unit set is: Dtr WV ¼ Dtp p  WV 

S  RS þE2 S  RS  E2 with arc cos bWVbarc cos E1 E1 DV ¼

ð23Þ where S and R S are the sedimentation and subsidence rates (in m Myr1), while E 1 and E 2 are the maximum

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rates of eustatic sea-level change (in m Myr1) for the high and low frequency variations, respectively.

Appendix C. Derivation of Eq. (26) To compare the durations of retrogradation in the two locations, we define: D¼

Dtr Dtr ¼ Dtp T  Dtr

ð24Þ

where Dr and Dp are the durations of retrogradation and progradation, respectively, and T is the period of the eustatic sea-level change. Following this, the duration of retrogradation becomes: Dr ¼

DT : 1þD

ð25Þ

Therefore, the time lag for the onset of progradation (MFS), that is, the difference of durations of retrogradation between the two locations, is defined as: 
D Dref  Dti ¼ Dtr  Dtrref ¼ T ð26Þ 1 þ D 1 þ Dref where Dtr and Dtrref are the durations of retrogradation (in Myr) in the considered location and the reference location respectively, D and D ref are the distortion values in the considered location and the reference location respectively, and T is the period of the considered eustatic sea-level change (in Myr).

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