^{*}

Edited by: Renaud Toussaint, University of Strasbourg, France

Reviewed by: Einat Aharonov, Hebrew University of Jerusalem, Israel; Marcus Ebner, OMV, Austria

*Correspondence: Daniel Keszthelyi

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The Ekofisk field, Norwegian North sea, is an example of a compacting chalk reservoir with considerable subsequent seafloor subsidence due to petroleum production. Previously, a number of models were created to predict the compaction using different phenomenological approaches. Here we present a different approach which includes a new creep model based on microscopic mechanisms with no fitting parameters to predict the strain rate at reservoir scale. The model is able to reproduce the magnitude of the observed subsidence making it the first microstructural model which can explain the Ekofisk compaction.

The Ekofisk field is one of the largest petroleum fields in the Norwegian North Sea and the largest where oil is produced from chalk formations (map on Figure

After the identification of the subsidence, a new production scheme started by waterflooding the reservoir and thus increasing the pore pressure and decreasing the effective stress inside the reservoir. However, the subsidence continued even after the new production scheme was introduced and pore pressure was raised to the initial values. Currently, the total subsidence observed since the beginning of oil production at the Ekofisk field is 9 m at the center of the field [

Several compaction models were created in an attempt to predict reservoir compaction and seafloor subsidence (e.g., [

_{t} critical tensile strength, the shear surface by φ friction angle and _{c} critical stress and the shape of the ellipse. Some models introduce smooth changes close to the intersections while the model of Papamichos et al. [

Here we present a different approach, based on microscopic mechanisms with no fitting parameters using a universal creep model which combines microscopic fracturing and pressure solution where the local fracture stress is related to pore size, and then use a statistical mechanical approach to scale it up and predict strain rate at reservoir scale. We apply the model to field data and find that it reproduces the observed magnitude of seafloor subsidence. An advatage of this model over previous models is that it follows a bottom-up physics-based approach and therefore it provides a thorough insight into the underlying physical phenomena giving a higher predictive value.

The Ekofisk field is an elongated anticlinal structure in the Southern part of the Norwegian North Sea with an aerial extent of about 6.8 by 9.3km. The thickness of the overlying sediments is 2840m in the central part of the field and increases toward the flanks [

The reservoir rocks are high-porosity fine-grade chalk, a limestone composed of coccolith fragments, the skeletal debris of unicellar algae (Coccolithophorids). The reservoir rock's porosity ranges between 30 and 48%, while the Ekofisk Tight Zone has porosity between 10 and 20%. Even the high porosity chalk has relatively low matrix permeability (i.e., the permeability of the matrix itself) between 1 and 5mD (1 – 5 · 10^{−15}m^{2}). Natural fractures give a high fracture permeability (i.e., the permeability caused by the macroscopic fractures) resulting in total chalk permeability between 10^{−13} and 10^{−14} m^{2} (10 − 100mD) [

The overburden is mainly composed of clays and shales with thin interlayered limestone or silty layers [^{−18} – 10^{−21} m^{2}, 10^{−3} – 10^{−6} mD) [

In the early stage of oil production, pore fluid pressure dropped and the reservoir compacted, leading to a seafloor subsidence of up to 0.4 m per year which corresponds to a mean strain rate of 5 · 10^{−11}

In this paper we predict subsidence history for the production phase before water injection using a very simple creep model (detailed in Keszthelyi et al. [

Prior to production, the reservoir rock is a non-reactive, elastic solid with a collection of pores with a probability distribution,

The reservoir rock is subject to confining stress σ of overlying sediments and a pore fluid pressure, _{e} = σ −

When the effective stress exceeds a threshold, microscopic fractures will start to propagate from all pores with radius larger than a threshold value _{max}. This threshold is defined by linear elastic fracture mechanics:

The number of fractures created is proportional to the number of pores involved in the fracturing process. The fracture density ρ_{f} describes the abundance of the microscopic fractures and has the unit m∕_{0}.

Fracturing is instantaneous and we neglect the strain in the solid due to the formation of these fractures and the poroelastic response of the rock.

The new microscopic fractures are reactive sites where pressure solution takes place if there is water present. The fraction of water-wet fractures equals the water saturation _{w} of the rock.

The rate _{0}, temperature _{e}. In the other approach pressure solution rate is calculated from long-term strain rates measured in creep experiments.

The equations of the compaction model are presented in Data Sheet

The change in rock volume _{p} + _{s}, where _{s} is the solid volume and _{p} the pore volume, with time

The input parameters of the creep model are: effective stress σ_{e}, initial porosity Φ_{0}, pore size distribution _{w}. Except for the effective stress we use constants obtained from literature (see Table _{mean} = 2.2μm as found in Japsen et al. [

Initial porosity, Φ_{0} |
Sulak and Danielsen [ |
37.5% |

Reservoir temperature, |
Sulak and Danielsen [ |
150°C |

Water saturation, _{w} |
Sulak and Danielsen [ |
4.5% |

Mean pore size, _{mean} |
Japsen et al. [ |
2.2μm |

This is a much simpler definition for effective stress than used in some previous chalk compaction papers. In the following subsections we show that choosing this relation introduces only a small error in the result while the model is kept simple and physical. Then we present simple calculations to illustrate why the reservoir pressure

According to Coussy [

where _{ij} is the element of the confining stress tensor, δ_{ij} is the Kronecker-delta function, _{p} is a plastic compressibility parameter.

The two constants (α and β_{p}) in the equations can vary between 0 and 1 and they describe how strain is distributed in the porous media between the solid matrix and the pore volume. α = 1 means an elastically incompressible matrix and 0 corresponds to the case when the pore volume is incompressible and all the elastic strains originate from the elastic deformation of the matrix. Similarly, β_{p} = 1 means a plastically incompressible matrix and 0 corresponds to an incompressible pore volume and all plastic stains originating from the plastic deformation of the matrix.

There have been several studies on the determination of the Biot-coefficient of chalk. While some of them claim a coefficient as low as 0.7–0.8 [

As the determination of the exact Biot-coefficient and compressibility parameter is beyond the scope of this article we assume that both are 1.

Furthermore, we assume that compaction is conrolled by the compressive stress:

With these assumptions we can apply the simplified definition of the effective stress:

Oil is produced from the reservoir layer through boreholes penetrating the layer. Inside the boreholes they introduce a lower pressure than in the surrounding reservoir layer to facilitate the flow of pore fluids toward the well. To characterize the pressure changes inside the reservoir layer away from boreholes we treat the problem as the boreholes were uniformly distributed and calculate the fluid density function ρ(

The reservoir layer is treated as an axisymmetric layer with a finite thickness and with relatively high permeability compared to the surrounding (i.e., no fluid flow into or out from the layer). Its horizontal dimensions exceed the diameter of the reservoir field, the outer part is filled with water.

For the calculations we follow Muskat [

where Φ is the porosity,

Furthermore, we assume that the reservoir fluid is compressible and it has the pressure-density relation ρ = ρ_{0} · _{0} is a reference density at _{0} throughout the reservoir, and that there exists an outer boundary at distance _{2} where the fluid pressure remains constant and a borehole radius _{w} inside which the pressure equals to well pressure _{w} we obtain:

and α_{n} is a constant depending on the boundary conditions. For details see Data Sheet

We perform calculations with material parameters relevant to the Ekofisk field (see Table _{w} and reservoir outer boundary values _{2} (Figure _{w} as unknown since this radius corresponds to an extremely highly fractured region around the physical borehole. Figure

Matrix permeability (Sulak and Danielsen [ |
_{m} |
1 − 5mD |

Total permeability (Sulak and Danielsen [ |
100mD | |

Compressibility of reservoir fluid (Mackay [ |
β | 10^{−10}m^{2}N^{−1} |

Viscosity of reservoir fluid (Mackay [ |
μ | 10cP |

Porosity (Sulak and Danielsen [ |
Φ | 35% |

Pressure dependence of porosity | 0 |

_{well}) is constant—the well radius _{w}—and an outer finite radius _{2} outside which pressure is constant and equals to the initial pressure.

Based on this we calculated the pressure evolution during time from production data. The calculation assumes that the ratio between the produced gas and liquid phases at the surface depends on the gas and liquid fraction in the reservoir which is a function of the pressure inside the reservoir.

The stationary solution of the pressure propagation problem around a well is a conical pressure depression around the well where the center of the cone is the borehole where pressure is decreased. However, in the Ekofisk field hydrocarbons are produced from numerous wells penetrating into the reservoir and pressure is decreased in all of these wells. By 1980 forty wells were already producing from the reservoir. Given that the extent of the field is ~9 by 4 km, this implies that boreholes are placed closer than 1 km to each other. Hence the pressure depression cones are overlapping and the pressure changes inside the reservoir between two boreholes are small. Therefore, if we neglect the permeability inhomogeneites in the reservoir on the large scale we can assume a constant pressure throughout the reservoir as in [

To estimate how our model performs in terms of predicting the subsidence during the years of production, historical reservoir pressure data were needed. We use publicly available production and crude oil property data to make an estimae of the reservoir pressure history. We use the widely accepted concept of reservoir engineering (e.g., [

Initially, when the reservoir was highly pressurized all hydrocarbons were in liquid phase. As pore pressure was decreased the reservoir became more saturated until it reached its bubble point (5990 psi, 41.3 MPa) in 1976 [

The increasing amount of gas inside the reservoir was clearly reflected in production data. Initially, before reaching the bubble point, the ratio of produced gas and oil—commonly referred as GOR, gas-oil-ratio—was constant and reflected the amount of gas deliberated from the fluid phase as the hydrocarbons were exposed to surface pressure. However, as pressure declined below the bubble point, gas bubbles appeared in the liquid phase and therefore the produced hydrocarbon contained considerably larger amount of gas which now were partly due to degassing from the liquid phase and partly due to the expansion of the gas bubbles originally present in the reservoir when they became exposed to surface conditions (see Figure

_{1} of oil produced at the surface_{A} volume unsaturated oil. _{B}. _{C} volume which largely depends on the reservoir pressure. Meanwhile the saturated oil phase also expands to _{1} in all cases. _{2} volume is formed by the degassing due to pressure drop from reservoir to surface conditions. Note that the gas-oil ratio is the same in both cases since the degasing starts only at the bubble point pressure. _{C}). _{2} since _{B}, however

Knowing the amount of dissolved and free gas at different pressure and the relative volume of the phases at different pressure [

^{3} ∕ barrel oil = 0.1781 sm^{3} ∕ sm^{3} oil. sm^{3}: m^{3} at surface conditions.

A schematic view of the Ekofisk reservoir during production can be seen in Figure

^{−10} m^{2} = 1 mD.

However, it has been pointed out that the overburden has a certain rigidity compared to the reservoir [

The existence and nature of fractures inside the overburden are not yet completely understood. Observations show that casing deformations inside the overburden are most pronounced inside those wells which have a horizontal position between the central and the peripheral part of the reservoir [

For simplicity, we assume that the overburden is completely soft and follows the motion of the compacting reservoir layer and we neglect the effect of shear. Therefore, the subsidence rate,

The model of Keszthelyi et al. [^{5}m^{−1}.

Due to the uncertainty of the pore size distribution data we calculate fracture densities for a range of mean pore sizes relevant to the Ekofisk field, while preserving the shape of the pore size distribution. We plotted the fracture density as a function of two variables: effective stress σ_{e} and mean pore size _{mean} (see Figure

We also calculate the resulting strain rate in the same range of the variables by using both approaches for the pressure solution rate calculation (theoretical model: Figure ^{−11}s^{−1} corresponding to the average measured strain rate is also shown. In the theoretical model the dissolution rate depends on effective stress while in the long-term model dissolution rate remains constant and fracture density has the only pressure dependence.

In both cases, the initial effective stress corresponds to a negligible amount of compaction, while at maximum effective values we get realistic predicted strain rates, the average measured strain rate is between the initial and maximal values.

In order to compare model prediction with field data we calculate the subsidence rate and resulting subsidence for a 20 year long period (see Figures

We were able to apply a micromechanical model of carbonate compaction which combines microscopic fracturing (pore failure) with creep (pressure solution) using upscaling to reservoir scale through the concept of fracture density. This model predicted surprisingly well the observed compaction and subsidence making it the first microstructural model which can explain the Ekofisk subsidence. The model contains a very small number of internal parameters: Young's modulus of chalk, the water-wet interfacial energy of calcite and reaction constants describing the dissolution-precipitation kinetics and diffusion of calcite all of which can be meausured with simple physical experiments independently. The input parameters are pore size distribution, water saturation, porosity and pressure history. Furthermore, the model is based on physical assumptions, eliminating the need of unphysical fitting parameters. We believe this results in a higher predictive power than previously used models with a large number of fine-tuned parameters.

The discrepancy between the model prediction and field observations is due to the uncertainty of input parameters and the simplifying assumptions during the application or inside the model.

The model is very sensitive to the pore size distribution. Application of this model depends on reliable data for the actual situation to estimate the pore size distribution. Pore size distribution data can be obtained by accurate measurements: currently the most accurate being X-ray microtomography data, although only a few measurements exist currently. Mercury injection data are considerably more common and can be used as a less reliable source of pore size distribution measurements since it does not measure pore size distribution directly but pore throat distribution.

A previous study [

Investigating the effect of water saturation can presumably help to address the question of ongoing subsidence during production with water injection when the pore pressure was increased to the original values. As water is pumped in and water saturation increases inside the reservoir the newly entering water can also flow into the initially oil-filled recent microscopic fractures triggering pressure solution there. This may cause the reservoir to compact after the start of injection, but to settle down as water saturation levels to equilibrium. The model also shows that there is a large potential of further subsidence if water saturations increase in the reservoir and thus it can serve as an explanation for “water weakening” effect of chalk.

Inhomogeneities inside the reservoir are present in every scale. Apart from the variability of material parameters (porosity, pore size distribution, and water saturation) a complicated network of macroscopic fractures makes the modeling difficult. While some part of the reservoir are extensively fractured and pressure changes can happen rapidly, other parts of the reservoir contain less macroscopic fractures slowing down pressure propagation and the whole compaction process. In order to characterize the compaction and subsidence in detail these variations should be considered.

The current micromechanical model keeps the model of the microscopic fracturing process simple and claims that compaction is mainly driven by vertical stress. This approach neglects the modifying effect of horizontal stresses which are present as far-field tectonic stresses, while laterial variations in the compaction process due to different reservoir parameters can also cause local build-up of horizontal stresses. Therefore, a 3-dimensional reservoir-scale model of compaction should take into account the effect of horizontal stresses and should be tested against a detailed data on the compaction process.

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We gratefully acknowledge the support of the FlowTrans Marie-Curie ITN for the funding of the PhD grant of DK (under grant agreement no. 31688 in the European Seventh Framework Programme). Thanks to Anja Røyne and Amélie Neuville for their valuable comments and for the employees of ConocoPhilips Norway for the thoughtful discussions.

The Supplementary Material for this article can be found online at: