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Stratigraphic modelling of turbidite prospects to reduce exploration risk
Dave Waltham
1
, Noah Jaffey
2,3
, Stuart MacLean
2
and Valentina Zampetti
2,3
1
Department of Geology, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK
(e-mail: d.waltham@gl.rhul.ac.uk)
2
Midland Valley Exploration Ltd, 144 West George Street, Glasgow G2 2HG, UK
3
Present address: Shell International Exploration and Production B.V., Kessler Park 1, 2288 GS Rijswijk,
The Netherlands
ABSTRACT: This article presents an integrated workflow to model the evolution of
ancient turbidity currents on a 3D structurally reconstructed palaeo-seafloor,
allowing ancient turbidite sediment distributions to be estimated. Effective use of
such approaches requires efficient model-inversion procedures so that model
parameters (e.g. flow dimensions, densities etc.) can be estimated from any available
data. It is shown that a directed Monte Carlo approach (i.e. a simple genetic search
algorithm) is very effective. A case study of a Mesozoic prospect in the UK North
Sea shows the power of these methods to discriminate between potentially attractive
sediment-source locations. The main power of this approach lies in its ability to
exclude many, otherwise attractive, sedimentation scenarios.
KEYWORDS: turbidity currents, forward modelling, inverse modelling, 3D reconstruction
INTRODUCTION
The increasing importance of turbidite reservoirs over the last
30 years (Weimer et al. 2000) has resulted in the development of
seismic-based techniques for the evaluation of sand distribution
in turbidity current deposits. Seismic attribute maps have
proved to be particularly effective, as have isopach maps
formed between closely spaced horizons (see Fonessu (2003)
for excellent examples of both these techniques). This paper
proposes an alternative approach which does not require
the exceptionally high data quality necessary for attribute/
isopach-based techniques. In outline, the workflow involves:
+3D seismic interpretation of major horizons and faults;
+structural reconstruction of the palaeo-seafloor at the time
of turbidite deposition;
+modelling of turbidity current flow across the palaeo-
seafloor to determine areas of sand deposition;
+re-imposition of the structural deformation to move the
predicted turbidite deposits back to their present-day
locations.
This paper will describe this methodology in more detail and
demonstrate its application to a specific North Sea prospect.
Three-dimensional structural reconstruction of palaeo-
surfaces is now a well-established technique (Barr 1985; Buddin
et al. 1997; Williams et al. 1997; Rouby et al. 2000; Yin &
Groshong 2006), which has been developed from 2D ‘section
balancing’ (see Buchanan (1996) for a review). The objective is
to reconstruct the palaeogeometry of a volume by successively
stripping off each top layer and removing the effects of faulting,
folding and compaction on all remaining horizons.
Forward modelling of particulate gravity currents is equally
well established in the geological literature, where it has been
used for modelling pyroclastic flows and debris flows as well as
for turbidity currents (Parker et al. 1986; Dade & Huppert 1995;
Mulder et al. 1997; Huppert 1998; Iverson & Denlinger 2001;
Pratson et al. 2001; Waltham & Davison 2001; Bursik et al. 2005;
Sheridan et al. 2005). However, stratigraphic modelling in
general has not yet established itself as a widely used technique
in hydrocarbon exploration. It should be noted that in forward
modelling the distribution of deposits is predicted from sup-
plied parameters, such as flow volume and density. In inverse
modelling, on the other hand, parameters are determined from
knowledge of the sediment distribution. In practice, both
forward and inverse modelling are needed since modelling
parameters are first estimated from any limited data that may be
available and then forward modelling allows the consequences
of these parameters to be determined for areas lacking data.
The aim of this paper is to present a general methodology
for using stratigraphic models to reduce risk. This paper
concentrates upon an inverse modelling strategy which is quite
general and could be adapted for use with many other forward
models. The details of this particular stratigraphic model are
therefore not central to the argument but, for completeness, are
outlined in the Appendices. The next section describes a
specific Moray Firth (UK North Sea) exploration problem and
outlines the structural reconstruction used as input for subse-
quent stratigraphic modelling. The paper then discusses the
inverse modelling approach and shows how it resolves Moray
Firth exploration issues.
STRUCTURAL RECONSTRUCTION AND THE
EXPLORATION PROBLEM
Client confidentiality prevents specification of the precise
location and age of the dataset, which covers a Mesozoic
prospect lying near the Moray Firth region of the North Sea off
northeast Scotland. The 3D seismic data cover an area of
c.2025 km, within which ten key reflectors were picked and
depth-converted by the client company.
Structural reconstruction was undertaken by removing the
top interval and decompacting the remaining beds (Sclater &
Petroleum Geoscience, Vol. 14 2008, pp. 273–280 1354-0793/08/$15.00 2008 EAGE/Geological Society of London
Christie 1980). The effect of thermal subsidence and isostasy
was removed by bulk rotation of the entire model. The
deformation at every new upper surface was restored using
vertical shear and fault block translation and rotation. This
sequence of operations was repeated for each successive upper
surface until the formation of interest reached the seafloor.
Figure 1 shows the resulting accommodation space estimate.
Note that this represents the maximum bathymetric variation
that could have existed at this point in time assuming that space
had been created rapidly and slowly filled in. Figure 1 also
shows the locations of five wells for which there are net-to-
gross (NTG) estimates (defined as fractional thickness of clean
sand) in the overlying formation, based upon interpretation of
wireline logs. The NTG values themselves are given in Table 1.
Note that wells C and D have good sand content but that the
remaining wells contain no clean sand at all. The exploration
issue was to determine a sediment transport path to the high
NTG wells (C, D) that did not place sand in the remaining wells
(A, B, E). The ultimate objective was to improve understanding
of the exploration risk associated with further drilling in the
area.
The key features of the resulting surface are a general
deepening towards the north and the existence of a possible
north–south channel supplying sand to wells C and D. A flow
starting at the south-entry point in Figure 1 and moving along
this channel towards wells C and D could result in high-quality
sand deposits within this channel and so it is important to
investigate whether such a scenario is plausible. Figure 2 shows
a forward model of such a flow. However, note that there are
no significant barriers to this flow reaching wells A, B and E.
Hence, the question becomes: does the absence of sand in wells
A, B and E preclude the possibility that sediment was supplied
to wells C and D by a flow along the channel? If this sediment
path is not viable then an alternate sediment supply for
locations C and D is needed. The stratigraphic modelling
therefore needs to investigate two alternatives: (1) sediment
supply is from the south and deposits good quality sand in the
channel; (2) sediment supply is from close to the sand-rich wells
and never enters the channel. Flow-entry points for both
scenarios are indicated in Figure 1.
Fig. 1. Reconstructed palaeo-seafloor used for modelling turbidity currents. Large arrows indicate the flow-entry locations for the two flows
modelled in this paper. Note the narrow channel used by the south-entry flow.
Table 1. Net-to-gross at the well locations shown in Figure 1, together with the calculated
coarse fractions for the best-fit numerical model
Location Well Best model
NTG coarse fraction
A00
B00
C 0.64 0.642
D 0.39 0.379
E00
rms.error 0.005
NTG, net-to-gross
D. Waltham et al.274
INVERSE MODELLING OF TURBIDITY
CURRENTS
The forward model is controlled by the seafloor bathymetry
together with the 14 parameters described in Table 2. If these
parameters are given, then the forward model can be used to
predict the coarse-fraction distribution across the modelled area
for comparison with the known NTG in the wells. Note that
this comparison is valid only if coarse-deposits are predomi-
nantly clean (i.e. not contaminated by simultaneous fine depo-
sition) and if the multiple, successive flows responsible for a
particular turbidite unit are fairly similar to one another (i.e. the
properties of a single deposit are reasonably representative of
the bulk properties of stacked multiple deposits). The first
assumption could be avoided by using a more sophisticated
measure of forward-model NTG but the second assumption is
fundamental since modelling of (possibly) hundreds of very
different flows would require the specification of thousands of
parameters (i.e. 14 for each flow).
For the specific case of the Moray Firth problem investigated
in this paper, it might be thought that finding 14 parameters
given five constraints (i.e. the NTG values at the well locations)
is a severely under-constrained problem. However, all 14
parameters are constrained to a greater or lesser extent by
physical reasonableness. For example, a flow with an initial
thickness of 10 km would not be plausible. Given loose
constraints, it is perfectly possible for a nonlinear problem to
have no solutions at all (e.g. the parabola y=x
2
has no solution
for real xgiven the loose constraint that ymust be negative).
Fig. 2. Example flow 800 seconds after entry into the modelled area. Note relatively thick flow along channel but that a thin overbank flow is
also spreading out from the entry point.
Table 2. Parameters used in the forward model
Parameter Comments
Inflow location Easting (m)
Inflow location Northing (m)
Inflow direction (EofN)
Inflow width (m)
Inflow thickness (m) Also controls inflow velocity
(assumes Froude Number1.0)
Deposit volume (km
3
)
Flow concentration (%) Flow volume=Deposit
volume/concentration
Grain density (kg m
3
) Fixed at 2500 kg m
3
Sea-water density (kg m
3
) Fixed at 1030 kg m
3
Inflow grain distribution modal
diameter (µm)
Inflow assumed to have a
log-normal distribution
Inflow grain distribution sorting
index (ϕ)
Defined as standard deviation of ϕ
Seafloor roughness (m) Controls relation between basal
shear stress and flow speed
Sand diameter (µm) Defines coarse-fine transition. Fixed
at 250 µm
Pelagic thickness (m) Thickness of fine sediment
deposited between flows
Inverse modelling attempts to determine these parameters by finding values
that minimize the root mean square error between the NTG in the wells and
the coarse-fractions produced by the model.
Stratigraphic modelling of turbidite prospects 275
On the other hand, if a solution does exist it is likely to be
highly non-unique (e.g. consider the foregoing parabola
example but with the constraint that ybe positive). Hence, the
objective with modelling is not to determine the precise set of
parameters that produced a given deposit but, rather, to
establish whether any set of parameters could produce the
observed deposit given a specified sediment-input location.
Inverse modelling starts by defining an objective function,
i.e. a measure of misfit between model and data. In the Moray
Firth example the constraints were in the form of NTG
estimates and so an objective function given by the root mean
square (rms.) error (rmse) was defined with
rmse2=兺i=1
N共wi
ⳮmi兲2(1)
where w
i
is the NTG at the ith well and m
i
is the modelled
coarse-fraction at the same location. This, in turn, was calcu-
lated using
mi=Ci
Ci+Fi+P(2)
where C
i
is the thickness of coarse sediment modelled at the
location of the ith well, F
i
is the corresponding fine thickness
and Pis the thickness of pelagic background sediment
deposited between flows. Note that pelagic sedimentation is
therefore assumed to have a spatially uniform thickness and to
be entirely fine grained. The grain diameter defining the
threshold between coarse and fine sediments is user defined
(see Table 2) and is assumed to be known.
Hence, given a particular model output (controlled by all the
parameters in Table 2 except P), the value for Pwhich
minimizes the rms. error can be found and this is the rms. error
for that model run. In practice, equations (1) and (2) can be
evaluated very rapidly for a large number of different values for
Pand so a simple exhaustive search approach is viable. This has
the advantages of being very robust, being unaffected by local
minima and being easy to constrain within geologically plausible
limits (e.g. 0–10 m thick pelagic deposits).
This leaves the problem of minimizing rms. error with
respect to the remaining 13 parameters in Table 2 (i.e. excluding
pelagic thickness). Of these, three are assumed to be known
(the sand cut-off, the grain density and the ambient water
density) and so this leaves ten parameters to be found.
Exhaustive search is not feasible since, even searching over just
five values for each parameter, would require 5
10
10 million
simulations. A typical forward-model simulation takes the order
of one minute (on a mid-range PC) and so searching would take
approximately 20 years even at this very crude parameter-
resolution. Gradient-descent techniques (Press et al. 2002) are
likely to become trapped in shallow, local minima given the
highly nonlinear nature of the problem and so more robust
approaches are needed. Here, a Monte Carlo approach was
modified.
Starting with a user-defined range for all parameters, the
forward model was run repeatedly using sets of parameters
obtained by randomly choosing values within the allowed
range. The power of this approach was increased greatly by
making a modification that changes the Monte Carlo approach
into a very simple genetic search algorithm. The search range
within which parameters were chosen was decreased, as the
simulations proceeded, with the reduced range centred upon
the best set of parameters found so far. Hence, an initially large
parameter search-volume was gradually collapsed onto a good
solution.
The upper curve in Figure 3 illustrates this process for the
case of the sorting index, which controls the width of the
grain-size distribution for the suspended sediments in the flow
as they enter the model. This shows the progress of inversion
for an inflow location in the northwest of the model (see
Fig. 1). In the early part of the search, the sorting index
fluctuates over the entire user-supplied range. However, as the
simulations proceed, the fluctuations become smaller and
slowly converge. The lower curve in Figure 3 shows how the
rms. error changes during the same set of simulations. The rms.
error is initially relatively large but, as improved solutions are
found, all model parameters are adjusted to concentrate search-
ing in regions of good solutions and there is overall a very
gentle decrease in the lowermost rms. error values. After
simulation of 200 flows (taking typically1–10 hours on a laptop
PC, depending upon the exact parameters ranges chosen) an
excellent match between model and wells is found, as can be
seen from the very small rms. error (0.005) and also by direct
comparison of the modelled sand fractions to the well values
(Table 1). Note that Figure 3 shows the rms. error for each of
the 200 simulations but that the final model chosen is the single
simulation that happens to have the smallest rms. error. This
will not generally be the final iteration (i.e. iteration 200 here)
but, because of the gradually narrowing search-range, will
tend be one of the later simulations (in fact, the minimum
does occur for the very last simulation here but that is a
coincidence).
The excellent match of model to data is emphasized by
Figure 4, which shows the coarse-fraction distribution across
the entire modelled area. The superimposed wells are colour-
coded using the same scheme as the underlying map and so any
data/model discrepancies would show up very clearly. Figure 5
shows a similar map but for the case where parameter-
searching was carried out for an inflow assumed to be from the
south and along the channel. Note that the match between
model and wells is very much poorer, a point emphasized by
the much higher final rms. error (0.29). Observation of the 200
individual flows that were tested during the production of
Figure 5 shows that the key problem is in getting sand into well
C without also supplying sand to well E. Flows large enough to
reach C invariably also reach E and, in fact, many of them also
flood wells A and B. Moreover, even when sand does get to
well C, well D always has the higher coarse-fraction and this
contrasts sharply with the true situation in the wells. However,
Fig. 3. The upper curve shows an example inversion run in which
the sorting index was initially allowed to vary between 0.5 and 0.9 but
with this range gradually decreasing. The lower curve shows the
evolution of the root mean square error (rmse; as a result of
progressively modifying all parameters, not just the sorting index).
Note that there are occasional good solutions (below dotted line at
0.05) throughout the inversion but that the algorithm converges onto
excellent solutions (rmsem0.01) towards the end.
D. Waltham et al.276
when the inflow comes from the north (i.e. Fig. 4) it is very easy
to supply sand to wells C and D without the flows reaching
wells A, B or E and, provided the inflow location is chosen
appropriately, it is also quite easy to obtain a higher coarse-
fraction in well C than in well D. Indeed, Figure 3 shows that
there are several different northerly-sourced solutions that
produce very low rms. errors (e.g. iteration 182 has
rmse=0.0056).
DISCUSSION
The foregoing modelling confirms the expectation that, for
some parameter constraints (e.g. a flow from the south), there
are no parameter sets which produce a good match of model to
data whilst, for others (e.g. a flow from the north), there are
multiple possible solutions. This suggests that a good way to
apply such models is to use them to exclude possibilities. In
addition, even although no single good solution can be
extracted, the modelling can be used to draw conclusions about
the likely properties of the true solution by looking at features
that all good solutions have in common. The unknown, correct
solution is likely to share those common properties. Finding
properties common to all solutions is a powerful method that
has been applied to the modelling of shallow-marine clastic
sequence stratigraphy (Waltham et al. 2003) and to the very
different problem of unravelling the causes of oceanic stron-
tium isotope fluctuations (Waltham & Grocke 2006). Applying
this methodology to turbidite deposition is a much more
complex problem, however, and remains the topic of ongoing
research.
The specific problem addressed in this paper was unusually
poorly constrained. However, the software has been used in a
similar fashion for investigating datasets containing more than
thirty wells and still produced useful outputs that matched most
wells extremely closely. It should be noted that adjacent wells
tend to have similar NTG values and so it is not correct to state
that 30 wells give 30 independent constraints. As a result, the
foregoing analysis holds and parameter searches still separate
into those having multiple solutions and those having none.
Hence, it is not believed that the algorithm discussed above is
useful only in very under-constrained problems.
The algorithm presented here for parameter searching has
therefore proved to be extremely robust. It also has the
significant advantages of being easy to program (and hence
more likely to be error-free) and of being ideal for distributed
processing. This search algorithm has been run simultaneously
on 20 different PCs and produced excellent results in a small
fraction of the time needed by a single PC. Hence, using a PC
cluster, it is quite possible to produce good fits of model to data
within a few minutes. The search algorithm also appears to be
very efficient in that it can find reasonable solutions (when they
exist) in as few as 20 iterations. However, this may be a
consequence of the high degeneracy in solutions and remains
the subject of further research.
In conclusion, a novel combination of structural reconstruc-
tion and turbidity current simulation has been presented,
Fig. 4. Modelled coarse-fraction for the best-fit run with the north-entry point. Circles show the well NTG values using the same colour code
as used for the model coarse-fraction.
Stratigraphic modelling of turbidite prospects 277
together with an effective inversion procedure. It has been
demonstrated that these tools can be used to eliminate potential
sediment-path scenarios and hence reduce the risk associated
with possible further drilling.
Noble Energy (Europe) Ltd is acknowledged for generously allowing
publication of the data used in this paper.
APPENDIX A: GENERAL GRAVITY FLOW
EQUATIONS
The starting equations are (Acheson 1990):
Conservation of mass for an incompressible fluid
兺
i=1
3)ui
)xi
= 0 (A1)
where u
i
is velocity in the x
i
-direction.
Conservation of momentum (Cauchy’s equations of motion)
)uj
)t+兺
i=1
3
ui
)uj
)xi
=Tj
⬘+gj(A2)
where jis1,2or3;tis time; g
i
is gravity and
Tj
‘=1
兺
i=1
3)Tij
)xi
(A3)
where T
ij
is stress in the i-direction on the plane perpendicular
to the j-direction.
With depth-averaged equations, the study is interested only
in j=1 or 2 assuming j=3 corresponds to the vertical direction.
This removes the gravity term from the right-hand side of
equation (A2). Integrating equations (A1) and (A2) over the
flow depth, H, and using Leibniz’s theorem (Abramowitz &
Stegun 1965) then produces
兺
i=1
2
)
)xi
共Hu
¯i兲+)H
)t= 0 (A4)
and
)
)t共Hu¯j兲+兺
i=1
2
)
)xi
共Hu¯iu¯j兲=HT j
⬘(A5)
where an overbar indicates a depth-averaged quantity. Finally,
combining equations (A4) and (A5) gives
)
)t共u
¯j兲+兺
i=1
2
u¯i
)
)xi
共u¯j兲=Tj
⬘(A6)
which is simply a 2D version of equation (A2). Equations (A4)
and (A6) are used to model the evolution of flow thickness and
flow velocity, respectively. These expressions are quite general
as they make no assumptions concerning rheological properties
Fig. 5. Modelled coarse-fraction for a run with a south-entry point. Circles show the well NTG values using the same colour code as used for
the model coarse-fraction.
D. Waltham et al.278
(e.g. non-zero viscosity or yield strength), flow style (e.g.
laminar, turbulent or granular) or about how flow density varies
in time and space. These factors enter only via the stress term
on the right-hand side of equation (A6).
APPENDIX B: STRESSES IN A TURBULENT
NEWTONIAN FLOW
A Newtonian fluid is defined by the stress relationship
(Acheson 1990)
Tij =ⳮpij +
共
)uj
)xi
+
)ui
)xj
兲
(B1)
where pis pressure and µis viscosity. For the turbulent flow
case considered in this paper, velocities should be understood
as being ensemble averages (i.e. averages over a large numbers
of identical flows) whilst µis the eddy viscosity. For the
low-concentration turbidity current case the flow density is
nearly constant (i.e. equal to water density,
w
) so that equations
(A3) and (B1) yield
Tj
⬘⬇1
w
共
)p
)xj
+ⵜ2uj+j共H兲ⳮj共0兲
兲
(B2)
where ⵜ2⬅)2
)x1
2+)2
)x2
2and j=)uj
)x3
is horizontal shear stress in
the jdirection.
For a thin flow, in which characteristic horizontal scales are
generally much greater than flow thickness, a hydrostatic
approximation is appropriate for calculation of the pressure
gradient in equation (B2). In addition, for simplicity, it is
assumed that flow concentration is constant so that
)p
)xj
=)p
)xj
=g)h
)xj
(B3)
where is the density contrast between the flow and the
ambient fluid and his the height of the flow top.
The eddy-viscosity term in equation (B2) is formally equiva-
lent to a velocity-diffusion equation and this therefore simply
smoothes the resulting velocity field. The diffusion coefficient is
given by µ/
w
which depends upon the flow speed and could
be anisotropic. However, for simplicity, this is treated as a
constant modelling parameter chosen to ensure numerical
stability.
The horizontal shear stress terms in equation (B2) can be
estimated using mixing-length theories (Duncan et al. 1960).
Full details are given in Waltham (2008) where it is shown that
the basal shear stress is given by
=ⳮw
共
ku
¯
bln共Hf/z0兲
兲
2
(B4)
where kis von Kármán’s constant (note: the clear-water value
of 0.4 is unlikely to be greatly in error for the low-
concentration flows considered in this paper); bis a constant of
order one; fis the turbulent boundary-layer fractional-thickness
and is around 0.05 (Kneller et al. 1999); and z
0
is the laminar
sub-layer thickness which can be related directly to the ob-
served roughness of the channel floor since the size of the
elements generating channel floor roughness is approximately
30z
0
(Raudkivi 1998). For the 2D (after depth-averaging) flows
used here, equation (B4) may be generalized to
j=ⳮwu¯jV
共
k
bln共Hf/z0兲
兲
2
(B5)
where Vis the depth-averaged speed (i.e. V2=u¯ 1
2+u¯ 2
2). Thus,
the basal friction is controlled by two constants; bwhich is
assumed to be unity and z
0
/f2r/3 where ris the scale of the
seafloor roughness. Note that this approach to calculating basal
shear stress is formally equivalent to using a Chezy-type friction
law except that the resulting Chezy-coefficient has a weak
dependency on flow thickness and, more importantly, it is
related directly to a quantifiable parameter (i.e. the seafloor
roughness).
APPENDIX C: PARTICULATE CURRENT
MODIFICATIONS
The preceding algorithm applies to any turbulent, thin, gravity
underflow and so, to complete the description, one needs to
add processes specifically related to sediment suspension and
deposition. The modelling is concerned with deposition from
the distal parts of the flow, thus it is assumed that the flow is
fully developed in terms of sediment and ambient-fluid entrain-
ment so that sediment deposition is the dominant process.
For sediment suspension, the turbulence-generated rms.
fluctuations in vertical velocity should exceed the fall velocity,
v
k
(where kruns over all grain diameters), (Raudkivi (1998), but
see Leeder et al. (2005) for a critique) and laboratory studies
(e.g. Bagnold 1966; Kneller et al. 1999) show that the rms.
fluctuations are of similar magnitude to the shearing velocity,
u*, given by
=wu*
2(C1)
An entirely equivalent suspension criterion is that the Rouse
number should be less than 2.5 (Rouse 1937; Allen 1997) since
this also leads to the expectation of a suspension threshold
when fall-velocity approximately equals the shear velocity. The
sedimentation rate, s
k
, is therefore zero for u*>v
k
but equal to
c
k
v
k
when the flow is stationary (where c
k
is the volumetric
concentration of grain-size k). The simplest mathematical
model consistent with these end-members is
sk=ck共vk
ⳮu*兲u*#vk(C2)
s=0 u*>vk(C3)
The still-water fall velocity can be calculated using a large
number of different formulae but, for the first-order model
described in this paper, Stokes’s Law is adequate. Each grain
diameter is independently modelled using a modified form of
equation (A4) which incorporates sediment loss:
兺
i=1
2
)
)xi
共Lku¯i兲+
)Lk
)t=ⳮsk(C4)
where L
k
is the sediment load associated with grain-size k, i.e.
Lk=ckH(C5)
Flow thickness is then recalculated using
H=c兺
k
Lk(C6)
where the overall flow concentration, c, is assumed to be fixed
for consistency with earlier model assumptions.
Stratigraphic modelling of turbidite prospects 279
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