Porosity and Permeability Prediction through Forward Stratigraphic Simulations Using GPM and Petrel: Application in Shallow Marine Depositional Settings

The forward stratigraphic simulation approach is applied to forecast porosity and permeability trends in the Volve field subsurface model. Variograms and synthetic well logs from the forward stratigraphic model were combined with known data to guide porosity and permeability distribution. Building a reservoir model that fits data at different locations comes with high levels of uncertainty. Therefore, it is critical to generate an appropriate stratigraphic framework to guide lithofacies and associated petrophysical distribution in a subsurface model. The workflow adopted is in three parts; first, simulation of twenty scenarios of sediment transportation and deposition using the geological process modeling (GPM) software developed by Schlumberger. Secondly, an estimation of the extent and proportion of lithofacies proportions in the stratigraphic model using the property calculator tool in Petrel. Finally, porosity and permeability values were assigned to corresponding lithofacies-associations in the forward stratigraphic model to produce a forward stratigraphic-based porosity and permeability model. Results show a lithofacies distribution model, which depends on sediment diffusion rate, sea level variation, flow rate, wave processes, and tectonic events. This observation is consistent with the natural occurrence, where variation in sea level, sediment supply, and accommodation control stratigraphic sequences. Validation wells, VP1 and VP2 located in the original Volve field model and the forward stratigraphicbased models show a significant similarity, especially in the porosity models. These results suggest that forward stratigraphic simulation outputs can be used together with geostatistical modeling workflows to improve subsurface property representation in reservoir models.

Introduction 1 depositional model like the natural stratigraphic framework in a shallow marine depositional setting. 79 Therefore, obtaining a 3-dimensional stratigraphic model that shows a similar stratigraphic sequence 80 observed in the seismic data allows us to deduce variogram parameters to serve as input in actual 81 subsurface property modeling. 82 Twenty forward stratigraphic simulations were produced in the geological process modeling (GPM TM ) 83 software to illustrate depositional processes that resulted in the build-up of the reservoir interval under 84 study. By the fourth simulation, there was a development of stratigraphic patterns that shows similar 85 sequences as those observed in seismic, hence the decision to constrain the simulation to twenty scenarios.   stratigraphic pattern as observed in seismic or well data provides a 3-dimensional framework to constrain 102 subsurface property representation that conforms with the real-world property distribution trends. In clastic sedimentation, the movement of sediments relies on equations from the original SEDSIM 104 developed in Stanford University (Harbaugh, 1993). Sediment movement, erosion, and deposition is 105 governed by a simplified Navier Stokes equation. "Simplified" because the Navier-Stokes equation in its 106 original form define sediment movement in a 3-dimensions differential form, while the flow equation in 107 GPM TM is 2-dimensional with an arbitrary input of flow depth. Kieft et al. (2011) describe the influence 108 of a combination of fluvial and wave processes in the genetic structure of sediments in the Hugin 109 formation. These geological processes are rapid, depending on accommodation generated by sea-level 110 variation and or sediment composition and flow intensity. The deposition of sediments into a geological 111 basin and its response to post-depositional sedimentary or tectonic processes are significant in the ultimate 112 distribution of subsurface lithofacies and petrophysics. Therefore, several input parameters for the 113 forward simulation to attain a stratigraphic output that fits existing knowledge of paleo-sediment 114 transportation and deposition into the study area (see Table 2). The forward simulation at all stages 115 portrayed geological realism concerning stratigraphic sequence, but it also revealed some limitations, 116 such as instability in the simulator when more than three geological processes run concurrently. Given 117 this, the diffusion and tectonic processes remained constant whiles varying the steady flow, unsteady 118 flow, and sediment accumulation processes in each simulation run.

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The steady flow process in GPM simulates flows that change slowly over a period, or sediment transport 121 scenarios where flow velocity and channel depth do not vary abruptly e.g. rivers at a normal stage, deltas, 122 and sea currents. Considering the influence of fluvial activities during sedimentation in the Hugin 123 formation, it is significant to capture its impact on the resultant simulated output.

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The unsteady flow process can simulate periodic flows such as turbidites where the occurrence is not 125 regular, and the velocity of flow changes abruptly over time. The unsteady flow process applies several 126 fluid elements driven by gravity and friction against the hypothetical topographic surface. Otoo and 127 Hodgetts (2019) illustrate how the unsteady process in GPM TM attains realistic distribution of lithofacies 128 units in a turbidite fan system. Although the steady and unsteady flow governing equations distantly rely on the Navier-Stokes equations, the steady flow is quite distinct, as it uses a finite difference numerical 130 method for faster computation and to also illustrate the frequency of flow that is characteristic in channel 131 flow such as rivers. The finite difference method applies an assumption that flow velocity is constant 132 from channel bottom to surface. In contrast, the unsteady flow uses the particle method from SEDSIM3 133 to solve the sediment concentration in flow and sediment transport capacity (Tetzlaff & Harbaugh 1989).

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The simplified equation in GPM TM attempts to solve the problem of "shallow-water free-surface flow"  The equation that control steady and unsteady flow is expressed through: Where: h is flow depth, t is time, and Q the horizontal flow velocity vector.
Where: is the Lagrangian derivative of flow relative to time, g is gravity, H is the water surface 143 elevation, c2 is the fluid friction coefficient, is the water density, c1 is the water friction coefficient and 144 h is the flow depth.

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The Manning's equation is applied to relate flow, slope, flow depth and hydraulic radius channels with a 146 constant cross-section for the steady flow process. Manning's formula states: Where: V is the flow velocity, k is the unit conversion factor, n is the Manning's coefficient which 149 depends on channel rugosity, Rh is the hydraulic radius and S is the slope.

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As mentioned earlier, the unsteady flow process uses the particle method equation, which relies on the 151 assumption that erosion and deposition depend on the balance between the flow's transport capacity and the "effective sediment concentration". The equation for multiple-sediment transport in flow is given as 153 follows: Where: Aem is the effective sediment concentration of mixture, lks is the sediment concentration of each 156 type, and f1,ks is the transportability of each sediment type.

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The transport capacity of a sediment type is expressed by equations (5) and (6). Let consider 158 R = (A -Aem)f2,ks Where f2,ks is the erosion-deposition rate coefficient for sediment type ks. For every sediment type ,ks, 160 the formula for transporting sediment of different grain sizes is given as: Where;

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H is the free surface elevation to sea level, Z is the topographic elevation for sea level, Ksis the sediment 164 type, lks, is the volumetric sediment concentration of a specific type (k).

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The diffusion process replicates sediment movement from a higher slope (source location) and deposition the type of sediment and "energy" of the depositional environment. In this contribution, the highest depth-dependent diffusion coefficient occurs near sea level, where the "energy" is highest over a geological time (Dashtgard et al. 2007).

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In GPM TM , sediment diffusion is calculated using a simplified expression: where z is topographic elevation, Di is the diffusion coefficient, t for time, and ∇²z is the laplacian of z, 178 and Sn is the sediment source term.

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Sediment diffusion (Di) is estimated by assuming that the grain size for each sediment component (coarse (9), with CD sediment grain coefficient.

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With the flow component in place, the diffusion coefficient (Di) is deduced from the Einstein equation.

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Using an assumption that the diffusion coefficient decreases with increasing grain size and rise in 190 temperature, and that the coefficient f is known, the expression for Di is: (10) 192 Meanwhile, f is a function of the dimension of the spherical particle involved at a particular time (t). In 6. .ղ . (11)

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The sediment accumulation process in GPM is designed to generate an arbitrary amount of sediment 197 representing the artificial vertical thickness of a lithology as interpreted in a well or outcrop data (Tetzlaff,198 D., personal communication, February 2021). The areal input rates for each sediment type (coarsegrained, fine-grained sediments) use the value of the map surface at each cell in the model and multiply Where;

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AT is the total sediment accumulated in a cell over a period, S is the sediment type, Mv is the map value 208 of sediment in each cell, and SC is the sediment supply curve as a function of topographic elevation.

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Realistic reproduction of stratigraphic patterns in the model area requires input parameters (initial 211 conditions), such as paleo-topography, sea-level curves, sediment source location, and distribution curve, 212 tectonic event maps (subsidence and uplift), and sediment mix velocity. The application of these input 213 parameters in GPM TM and their impact on the resultant stratigraphic framework is below.

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Hypothetical Paleo-Surface: The hypothetical paleo-topographic for the stratigraphic simulation is from the 215 seismic data (Figure 3), using the assumption that the present day stratigraphic surface (paleo shoreline in Figure  216 4a) occurred as a result of basin filling over geological time. Since the surface obtained from the seismic section 217 have undergone various phases of subsidence and uplifts, it is significant to note that the paleo topographic surface 218 used in this work does not represent an accurate description of the basin at the period of sediment deposition; thus 219 presenting another level of uncertainty in the simulation. To derive an appropriate paleo-topographic for this task, five paleo topographic surfaces (TPr) were generated, by adding or subtracting elevations from the where, Sbs is the base surface scenario (in this instance, scenario 6), and EM an elevation below and 224 above the base surface.

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The paleo-topographic surface in scenario 3 (figure 4d) is selected because it produced a stratigraphic 226 sequences that fit the depositional patterns interpreted from the seismic section (Figure 5d).   The sensitivity of input parameters in the forward stratigraphic simulation is notable when there is a 248 change of value in sediment diffusion, and tectonic rates or dimension of the hypothetical topography.  Table 3). The classification is driven by depositional depth, geologic flow    (2) Pillar gridding: building a "grid skeleton" made up of a top, middle and base architectures. foundation for each cell within the model. The prominent orientation of faults (I-direction) within 295 the model area was in an N-S and NE-SW direction, so the "I-direction" was set to NNE-SSW to 296 capture the general structural description of the area.

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(4) Upscaling: involves the substitution of smaller grid cells with coarser grid cells. Here, log data is 306 transformed from 1-dimensional to a 3-dimensional framework to evaluate which discrete value 307 suits selected data point in the model. One advantage of the upscaling procedure is to make the 308 modeling process faster.

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The Volve field porosity and permeability model from Equinor are adopted as the base (reference) model. 311 The model, which covers 17.9 km 2 was generated with the reservoir management software (RMS),  In Option 1, the porosity and permeability values were assigned to the synthetic lithofacies wells that 319 correlate with known facies-association in the study area (see Table 4).
The pseudo wells comprising porosity and permeability are situated in-between well locations to guide 321 porosity and permeability simulation in the model. For option 2, the best-fit forward stratigraphic model 322 changes by assigning porosity and permeability attribute using the general stratigraphic orientation 323 captured in the seismic data (NE-SW; 240⁰). Porosity and permeability pseudo (synthetic) logs were then 324 extracted from the forward stratigraphic output to build the porosity and permeability models (Figure 8). 325 Porosity modeling is through normal distribution, whiles the permeability models were produced using a 326 log-normal distribution and the corresponding porosity property for collocated co-kriging.

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Considering that vertical trends in options 1 and 2 will be similar within a sampled interval, option 2

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The stratigraphic model in stage 4 (Figure 5d iv) shows the final geometry after 700,000 years of 348 simulation time. The initial stratigraphic simulation produced a progradation sequence with foreset-like 349 features (Figure 5d i) and a sequence boundary, which separates the initial simulated output from the 350 next prograding phase (Figure 5d ii). An aggradational stacking pattern commences and becomes 351 prominent in stage 3 (Figure 5d iii). These aggradational sequences observed in the forward stratigraphic the Volve field petrophysical model (Figure 11a) went through various phases of history matching to 358 obtain a model to improve well planning and production strategies, it is reasonable to assume that porosity 359 and permeability distribution in the petrophysical model will be geologically realistic and less uncertain.

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This view formed the basis for using the porosity and permeability models developed by Equinor as a 361 reference for comparing outputs in the stratigraphic model. Table 5a shows an almost good match in 362 porosity at different intervals in the forward stratigraphic-based models (i.e. R14, R20, R26, R36, R45, 363 and R49). An analysis of the well logs in the model area shows that a large proportion of reservoir porosity 364 is between 0.18 -0.24. Also, the analysis of the forward stratigraphic-based porosity model is consistent 365 with the porosity range in the Volve field model (see Figure 12).

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A notable limitation with this approach is the assumption that variogram parameters and stratigraphic 367 inclination within zones remained constant throughout the simulation. The difference in permeability 368 attributes between the original permeability model and the forward stratigraphic-based type is the 369 application of other measured parameters in the original model (Table 5b). Typically, a petrophysical 370 model like the Sleipner Øst and Volve field model will factor in other datasets such as special core analysis 371 (SCAL) and level of cementation, which enhances reservoir petrophysics assessment. Bearing in mind that the forward stratigraphic model did not involve some of this additional information from the 373 reservoir, it is practicable to suggest that results obtained in the forward stratigraphic-based porosity and 374 permeability models have adequately conditioned to known subsurface data.

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Results show the influence of sediment transport rate (or diffusion rate), initial basin topography, and

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In this paper, variogram parameters from a forward stratigraphic simulation are combined with subsurface 424 data to constrain porosity and permeability distribution in the Volve field model. The caution for 425 subsurface modeling practitioners is that the stratigraphic simulation scenarios presented in this 426 contribution do not prove that spatial and geometrical data derived from forward stratigraphic models are 427 absolute input parameters for a real-world reservoir modeling task. Uncertainties in the choice of improve property prediction in inter-well zones. This suggestion supports the idea that more conditioning 434 data (well data) will increase the chance of producing realistic property distribution in the model area.

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This work also made some key findings:    Stratigraphic simulation scenarios depicting sediment deposition in a shallow marine framework. a. scenario 1 involves equal proportions of sediment input, a relatively low subsidence rate and low water depth, b. scenario 10 uses high proportions of fine sand and silt (70%) in the sediment mix, abrupt changes in subsidence rate, and a relatively high water depth, c. scenario 15 involves very high proportions of fine sand and silt (80%), steady rate of subsidence and uplift in the sediment source area, and a relatively low water depth.          Table 3. Lithofacies classification in the forward stratigraphic model in the property calculator tool in Petrel TM . Table 4. Porosity and Permeability estimates of lithofacies packages in the model area. Table 5. A comparison of a) porosity, and b) permeability estimates from selected intervals in the original porosity/permeability models and forward modeling-based porosity and permeability models.