# Characterizing a Wedged Chalk Prospect in the Danish Central Graben Using Direct Probabilistic Inversion

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## Abstract

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## 1. Introduction

## 2. Method: Direct Probabilistic Inversion

**m**represents the subsurface model parameter (e.g., facies, porosity, saturation, etc.). Here, the information about

**m**is described by a probability density function (pdf). In the initial state of the inference, before taking seismic data into consideration, the information is described by the prior pdf, ρ. The prior pdf is updated with the information provided by seismic AVO data via the likelihood function, L, which measures, in terms of probability, the misfit between forward modeled g(

**m**) and measured seismic AVO data,

**d**. The c is a normalization constant, and the Bayesian (or posterior) pdf, σ, represents the updated state of inference of the subsurface model parameters,

_{obs}**m**, assimilating the prior, the AVO data and the forward modeling, g. The problem of non-uniqueness disappears when solving for a pdf and resolves many of the associated problems when interpreting standard inversion attributes, including correctly propagating uncertainty and spatial dependencies. In general, the posterior pdf cannot be solved analytically, and the inverse problem must therefore be approached by brute-force sampling or some approximation, with DPI taking the latter approach.

## 3. Study Area, Data Coverage and Reservoir Model

## 4. Inversion Setup

- Shaly overburden (ShOvb);
- Ekofisk-Tor (Zone of interest: Zi);
- Upper Hod (UHod);
- Lower Hod (LHod);
- Shaly underburden (ShUb).

- Thickness distributions are estimated to be exponential;
- Facies vertical ordering follows a first-order Markov process (Figure 7);
- The ordering statistics vary between all intervals;
- Fluid gravitational ordering is assumed;
- Older sequences are always located below younger sequences;
- The elastic properties (acoustic impedance, Vp/Vs and density) within each facies can be modeled with Gaussian distributions;
- The correlations of elastic properties within a given facies are modeled with an exponential correlation model.

## 5. Inversion Results

## 6. Implementing DPI Results into Reservoir Modeling

## 7. Conclusions

_{2}or hydrogen storage sites or geothermal resources.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of seismic inversion by the conventional two-step method (

**upper**) and a one-step (direct) probabilistic inversion method (

**lower**).

**Figure 2.**Map of the Danish Central Graben with the location of the different case studies. (

**a**) Map with structure elements during the Cretaceous, adapted with permission from Reference [9] under Creative Commons Attribution 4.0 License. (

**b**) Location of the study area within NW Europe.

**Figure 3.**Schematic workflow of the DPI inversion (adapted from Goodway et al. [17]).

**Figure 4.**Upper: prospect cartoon section showing the main elements of the hydrocarbon chalk prospect. Tor and Ekofisk reservoir pinch out against the Top Upper Hod and are charged with hydrocarbons via faults. Lower: 2D seismic crossline section through the prospect area with the corresponding Top Chalk surface in MapView showing nearby wells. TWT: two-way time.

**Figure 5.**N–S extracted section through the 3D static model and the wedged chalk prospect, with modeled porosity values. Note that the reservoir units have a finer layering and the highest porosities, and they pinch out to the south. Note also the decrease in porosity in the deeper northern part of the model as a result of increased burial compaction.

**Figure 6.**Facies class legend. The presented facies classes focus on the three main zones within the target interval: Ekofisk-Tor (Zone of interest: Zi), Upper Hod (UHod) and Lower Hod (LHod).

**Figure 7.**(

**a**) Stratigraphic subdivision from Surlyk, et al. [23]. The target interval is within Late Cretaceous. (

**b**) Histogram of observed facies thickness distribution. (

**c**) Logarithm to the probability of thicknesses larger than a given thickness. The linear trend indicates that thicknesses are exponentially distributed in line with a vertical Markov spatial model [10].

**Figure 8.**Markov model transition probability matrix. Probabilities are given in volume fractions (v/v); shOvb and shUb are shale overburden and underburden, respectively. The information in the matrix is used locally when generating facies windows for generating likelihood models.

**Figure 9.**Prior model setup for the Siah-NE-1X (

**left**), Nora-1 (

**middle**) and Elin-1 (

**right**) wells. For each well from left-to-right, shale volume (vclay), porosity (por) and water saturation logs; observed (or reference) facies profile; and the prior probabilities as a function of time/depth. The four seismic surfaces are plotted: Top Chalk, Top Upper Hod, Top Lower Hod and Base Chalk.

**Figure 10.**Acoustic impedance (AI) versus Vp/Vs: (

**a**) data from the NW-Adda-1X that contain shear sonic measurements; (

**b**) the corresponding rock physics model likelihoods (RPMs) defined from the data and fluid substitution modeling. The large gray ellipse represents the over- and underburden RPMs. Refer to Figure 6 for the facies colors.

**Figure 11.**Realizations of elastic AI and Vp/Vs to approach rock physics models at seismic scale. Upper row: estimated reflectivities and well data from NW-Adda-1X. Middle and lower rows: simulated reflectivities and elastic data. Refer to Figure 6 for the facies colors.

**Figure 12.**Statistical wavelets extracted from seven partial angle stacks extracted from a fixed time window around the Upper Cretaceous.

**Figure 13.**The 10–15° angle-stack seismic data from a composite seismic line through the nearby wells and the wedged chalk prospect (dashed black line in MapView). Well trajectories intersected by the composite line are plotted. The four seismic surfaces used are plotted in different colors.

**Figure 14.**Deterministic inversion result—acoustic impedance (AI) along the composite seismic line. Well trajectories intersected by the composite line are plotted with AI log data where available. The four seismic surfaces used are plotted in black.

**Figure 15.**Deterministic inversion result—Vp/Vs ratio from a composite seismic line. Well trajectories intersected by the composite line are plotted with Vp/Vs log data where available. The four seismic surfaces used are plotted in black.

**Figure 16.**Direct probabilistic inversion result along the composite line. The individual colors represent the most likely geological facies’ inverted for (see Figure 6) and is from left to right: (1) shale, (2) high-porosity chalk with oil, (3) high-porosity chalk with brine, (4) medium-porosity chalk with oil, (5) medium-porosity chalk with brine and (6) low-porosity chalk. Well trajectories intersected by the composite line are displayed as black lines. The four seismic surfaces used are plotted in black.

**Figure 17.**Inversion results for the Siah-NE-1X (

**left**), Nora-1 (

**middle**) and Elin-1 (

**right**) wells. For each well, from left-to-right, shale volume (vclay), porosity (por) and water saturation logs; observed (or reference) facies profile; and the posterior probabilities as a function of time/depth. The four seismic surfaces are plotted: Top Chalk, Top Upper Hod, Top Lower Hod and Base Chalk.

**Figure 18.**Comparison of (

**a**) traditional geostatistical reservoir modeling method, extrapolating data from well logs; and (

**b**) prediction of high-porosity chalk facies from the DPI tool.

Name Seismic Marker | Architecture Element | Horizon Type |
---|---|---|

Top Chalk Group/Top Ekofisk Fm. | Top Reservoir | Erosional |

Top Upper Hod | Base Reservoir | Conformable |

Top Lower Hod | Base onlapping surface | Conformable |

Base Chalk Group/Base Hidra Fm. | Base lower seal | Base |

Preparation phase (input) | |

Conditioning of input seismic CDP gathers Angle stacking Preparation of well database Seismic well ties Wavelet estimation | |

Inversion phase (Output) | |

Deterministic AVO inversion Output: Acoustic impedance (AI) and Vp/Vs (elastic properties) | Direct probabilistic inversion (DPI) Output: Probabilities of facies classes: shale, high-porosity chalk with oil, high-porosity chalk with brine, medium-porosity chalk with oil, medium-porosity chalk with brine, low-porosity chalk |

Comparison phase |

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**MDPI and ACS Style**

Bredesen, K.; Herbert, I.; Smit, F.; Jakobsen, A.F.; Frykman, P.; Bruun, A. Characterizing a Wedged Chalk Prospect in the Danish Central Graben Using Direct Probabilistic Inversion. *Geosciences* **2022**, *12*, 194.
https://doi.org/10.3390/geosciences12050194

**AMA Style**

Bredesen K, Herbert I, Smit F, Jakobsen AF, Frykman P, Bruun A. Characterizing a Wedged Chalk Prospect in the Danish Central Graben Using Direct Probabilistic Inversion. *Geosciences*. 2022; 12(5):194.
https://doi.org/10.3390/geosciences12050194

**Chicago/Turabian Style**

Bredesen, Kenneth, Ian Herbert, Florian Smit, Ask Frode Jakobsen, Peter Frykman, and Anders Bruun. 2022. "Characterizing a Wedged Chalk Prospect in the Danish Central Graben Using Direct Probabilistic Inversion" *Geosciences* 12, no. 5: 194.
https://doi.org/10.3390/geosciences12050194