Abstract
Ensembles of geomodels provide an opportunity to investigate a range of parameters and possible operational outcomes for a reservoir. Full-featured dynamic modelling of all ensemble members is often computationally unfeasible, however some form of modelling, allowing us to discriminate between ensemble members based on their flow characteristics, is required. Flow diagnostics (based on a single-phase, steady-state simulation) can provide tools for analysing flow patterns in reservoir models but can be calculated in a much shorter time than a full-physics simulation. Heterogeneity measures derived from flow diagnostics can be used as proxies for oil recovery. More advanced flow diagnostic techniques can also be used to estimate recovery. With these tools we can rank ensemble members and choose a subset of models, representing a range of possible outcomes, which can then be simulated further. We demonstrate two types of flow diagnostics. The first are based on volume-averaged travel times, calculated on a cell by cell basis from a given flow field. The second use residence time distributions, which take longer to calculate but are more accurate and allow for direct estimation of recovery volumes. Additionally we have developed new metrics which work better for situations where we have a non-uniform initial saturation, e.g., a reservoir with an oil cap. Three different ensembles are analysed: Egg, Norne, and Brugge. Very good correlation, in terms of model ranking and recovery estimates, is found between flow diagnostics and full simulations for all three ensembles using both the cell-averaged and residence time based diagnostics.
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Data Availability
Some of the models (data) used in this manuscript are available online:
• The Egg model: https://data.4tu.nl/articles/dataset/TheEggModel-datafiles/12707642
• The Norne model: Available via the OPM initiative https://opm-project.org/?pageid=559 with automatic download and setup available in MRST.
• The Brugge model: Porosity / permeability realisations used here are not publicly available. However, a single realisation of the model has been made available by TNO https://github.com/TNO/Brugge
Code Availability
The simulations for the numerical examples are run in MRST, freely available at https://www.sintef.no/mrst/. The new code developed for this manuscript is built upon MRST capabilities which are publicly available in MRST release 2020a and subsequent releases
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Acknowledgements
The research reported in this paper was funded in part by the Research Council of Norway through grant no. 280950 and in part by Equinor Energy AS, Total E&P Norge AS, and Wintershall DEA Norge AS. We thank Yan Chen for help with the Brugge ensemble case.
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Open access funding provided by SINTEF AS. In part by the Research Council of Norway through grant no. 280950 and in part by Equinor Energy AS, Total E&P Norge AS, and Wintershall DEA Norge AS.
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Appendix
Appendix
In this appendix, we will derive some basic properties of the pulse Eq. 5 and its relation to flow/storage capacities and sweep. The k-th moment of c is defined as
By integrating Eq. 5 over time and using that \(c(\mathbf {x}, 0) = \lim _{t\rightarrow \infty }c(\mathbf {x}, t) = 0\) for all x ∈Ω, we obtain
Hence, m0 satisfies the tracer Eq. 2 and it also follows that
Similarly, by integrating the product of Eq. 5 with t, we obtain
In other words, m1 satisfies the time-of-flight Eq. 1, which we can interpret as follows
To represent the solution of Eq. 5, we first observe that if we disregard domain boundary conditions, then any function of the form \(c(\mathbf {x}, t) = g\bigl (t-\tau _{f}(\mathbf {x})\bigr )\), where \(g:\mathbb {R}\rightarrow \mathbb {R}\) is a differentiable function, is a solution of Eq. 5. By substitution, we have
Since we can regard the delta function as the limit of a sequence of differentiable functions, it follows that the solution of Eq. 5 can be represented as \(c(\mathbf {x}, t) = \delta \left (t-\tau (\mathbf {x})\right )\).
By conservation of mass, we have
which leads directly to the following relation between flow capacity and sweep efficiency:
From this relation and the definitions of flow and storage capacities Eq. 8, we get the following alternative RTD-representation:
Furthermore, we obtain the relation [15]
By observing that \({{\varPhi }}_{o}\frac {d{{\varPhi }}}{dt}= tF_{o}\frac {dF}{dt}\), we obtain the relation \(\frac {d{{\varPhi }}}{dF}=\frac {F_{o}}{{{\varPhi }}_{o}}t\), and hence
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Watson, F., Krogstad, S. & Lie, KA. The use of flow diagnostics to rank model ensembles. Comput Geosci 26, 803–822 (2022). https://doi.org/10.1007/s10596-021-10087-6
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DOI: https://doi.org/10.1007/s10596-021-10087-6