Upscaling capillary pressure curves for numerical modeling of gravity-capillary driven flow

https://doi.org/10.1016/j.advwatres.2020.103639Get rights and content

Highlights

  • Existing Pc upscaling methods are evaluated in gravity-capillary dominated flow

  • All existing methods have significant errors in high capillary heterogeneity cases

  • A new optimization upscaling approach is introduced showing sufficient accuracy

  • Same upscaled curves can be used in ensembles of different permeability realizations

Abstract

This work investigates upscaling of capillary pressure curves for modeling gravity segregation under the influence of capillary heterogeneity. We consider two flowing phases driven by gravity and capillary forces and seek the saturation spatial and temporal variation until equilibrium is reached. Existing upscaling methods, found in the literature, are applied to a number of cases. Resulting saturation solutions are compared to fine-scale simulations and the different methods are evaluated, showing in general that the popular capillary limit method produces the best results. However, a large number of cases are found to have significant errors. This leads to the conclusion that all existing methods are often inadequate and developing new methods should be considered. We therefore propose a new optimization-based upscaling method. Using this approach it is shown that capillary pressure can be upscaled to Brooks-Corey type functions and produce accurate upscaled simulations matching the fine-scale solutions. Optimization upscaling is computationally demanding and requires the fine-scale simulations for minimizing an objective function. However, it is shown that the upscaled curves can be applied to different permeability realizations to calculate ensemble average saturation solutions.

Introduction

Numerical modeling of field scale multiphase flow processes is essential for many applications, e.g., aquifer contaminant remediation, oil recovery, natural gas storage and CO2 geological sequestration. In some cases, when there are no sources or sinks of flow in the model, the process is dominated by gravity and capillary forces, i.e., immiscible phases will migrate under the influence of buoyancy and capillary pressure. This occurs, for example, in gravity drainage enhanced oil recovery and CO2 storage in aquifers, when gas or light fluids are injected into a reservoir saturated with oil or water. After injection is stopped the phases will migrate mainly due to gravity and capillary effects. Accurate numerical modeling of these processes requires a fine mesh because of the large impact of heterogeneity on capillary trapping. However, these fine-scale models are not feasible due to computational limitations and thus upscaling methods must be applied. In this work, we show that upscaling of capillary pressure functions with existing methods leads to significant simulation errors in gravity-capillary driven flows and an upscaling method based on optimization is proposed.

Upscaling of multiphase flow has been an important topic of research in the past four decades and developments have been summarized in the literature, e.g., Christie (2001); Durlofsky (2003); Das and Hassanizadeh (2005). The general procedure begins with single-phase flow upscaling to obtain upscaled permeability (Farmer, 2002, Durlofsky, Chen, 2012) and these methods are well established. In many cases, upscaling of multiphase flow properties, i.e., relative permeability and capillary pressure is also necessary. Pseudo-relative permeabilities for upscaling have been discussed in detail in the literature (Kyte, Berry, et al., 1975, Barker, Thibeau, et al., 1997) and various methods for calculation have been proposed, e.g., analytical (Saad, Cullick, Honarpour, et al., 1995, Rabinovich, Li, Durlofsky, 2016, Rabinovich, 2018), steady state methods (Ekrann, Aasen, 2000, Virnovsky, Friis, Lohne, 2004) and dynamic methods (Darman et al., 2002). Much less literature deals with upscaling of capillary pressure-saturation curves, denoted Pc(Sw). This is mostly due to the fact that for many applications capillary effects are minor and Pc(Sw) can be neglected or modeled simplistically without considering spatial variations. However, in low-rate flow such as gravity-capillary driven flow, spatial variations of Pc curves, known as capillary heterogeneity, are extremely important. We note that the upscaling literature discussed above is limited to Darcy-scale flow and an additional body of literature exists, dealing with upscaling of pore-scale flow to obtain permeability (Mostaghimi, Blunt, Bijeljic, 2013, Starnoni, Pokrajac, Neilson, 2017), relative permeability (Raeini, Blunt, Bijeljic, 2014, Raeini, Bijeljic, Blunt, 2015) and capillary pressure (Ferrari, Lunati, 2013, Starnoni, Pokrajac, 2020).

Capillary heterogeneity is usually modeled by a relationship between Pc and permeability (k), e.g., the Leverett J-function scaling. Flow with capillary heterogeneity is complex and computationally demanding to model, particularly considering capillary entry pressure trapping, a well-known mechanism in pore-scale flow (Lenormand et al., 1983). This phenomenon is expressed in reservoir simulations by the constraint that a fully wetting-phase saturated grid block (i) will not be invaded by a nonwetting phase from an adjacent block (i+1) unless the capillary pressure of block (i+1) exceeds the entry pressure of block i (Li, 2011). This trapping mechanism constitutes a major difficulty in Pc(Sw) upscaling.

Existing methods of capillary pressure upscaling generally consist of two different approaches. The first, is based on the assumption that capillary pressure representing a region, similar to phase pressure, can be be calculated simply by arithmetic averaging. This is done either using a dynamic method (Pickup and Sorbie, 1996) or a steady state method (Virnovsky et al., 2004). The second approach is to assume capillary limit conditions in which phases are in equilibrium and capillary pressure is constant in each upscaling volume. This is usually referred to as the capillary limit (CL) upscaling method and applied in the majority of existing literature, e.g., Desbarats (1995); Pickup et al. (2000); Lohne et al. (2006); Mouche et al. (2010). The dynamic method is computationally expensive and has not been implemented in many studies with capillary heterogeneity. The CL method, however, is very simple and computationally efficient and has been shown to be accurate for cases with little or no entry pressure effects. To extend its application to cases with entry pressure trapping, the CL method was combined with percolation considerations by a number of authors (Braun, Helmig, Manthey, 2005, Behzadi, Alvarado, 2012, Yang, Tian, Niemi, Fagerlund, 2013, Wolff, Flemisch, Helmig, 2013) and used to model gravity-capillary driven flow. Another extension of the CL method was introduced in Rabinovich et al. (2015), where coarse-scale simulations were conducted iteratively to improve Pc upscaling accuracy.

Altogether, the studies on capillary pressure upscaling discussed above have a number of shortcomings. First, all studies but one consider two-dimensional flow problems, a significant simplification in comparison to realistic three-dimensional problems which often exhibit major differences in upscaling (Jankovic et al., 2017). Second, the upscaling methods are tested on a limited number of examples and not necessarily choosing the most challenging conditions, e.g., simple heterogeneity structures. Finally, and most importantly, many of the studies show that their method does not achieve sufficiently good results.

In this work we first show that all previous upscaling methods fail to reproduce fine-scale simulations for a number of cases of gravity-capillary driven flow. Various examples are presented, with different problem parameters and cases pertaining to applications of gravity drainage oil recovery and CO2 storage in aquifers. The three-dimensional, capillary heterogeneous Pc curves are upscaled to vary only in the vertical direction, i.e., Pc(Sw,x,y,z)Pc*(Sw,z), where * denotes an upscaled property. The first conclusion is that upscaling in these types of gravity segregation flows with capillary heterogeneity is an issue that has not been fully resolved, despite the impression that may be created by some of the previous studies. We also evaluate the different methods and discuss their accuracy, showing, for example, that adding percolation considerations to CL upscaling does not have a significant impact on the upscaled models. Then we apply an optimization approach in which we seek Pc* such that upscaled simulations reproduce the fine-scale solution accurately. We find that assuming Pc* curves of Brooks-Corey type (Brooks and Corey, 1966) and optimizing 100 parameters leads to sufficient accuracy of the upscaled simulations. While the optimization upscaling method is computationally demanding, it is the only method we found that can adequately model all the considered cases.

Upscaling methods are evaluated not only by their accuracy but also by their computational efficiency. Global methods such as dynamic upscaling, are computationally intense, requiring a full fine-scale simulation to calculate upscaled properties. Despite having acquired the fine-scale solution, the upscaling may still be a practical tool if it applies to other cases, e.g., different realizations of the permeability. Other, non-global approaches, such as analytical or semi-analytical upscaling methods (e.g., CL method with/without percolation), are much less computationally demanding. In this work, we propose a new upscaling approach using optimization. This method is computationally expensive since in addition to being a global method, it also requires many coarse-scale simulations to optimize the upscaled capillary pressure curves. Nevertheless, the purpose of applying this approach here is mainly to investigate whether we can find Pc* curves that accurately reproduce the fine-scale saturation distribution, having shown that existing upscaling methods fail. Optimization upscaling could, nonetheless, be a practical tool if it is found to be robust, and here we show one example in which the same Pc* curves are used for different realizations in Monte Carlo simulations. We emphasize that this work is only a first step and it is not our goal yet to perfect a new method or prove its applicability. In the future, we expect that optimization upscaling can be fine tuned and developed to be a powerful tool incorporating state of the art optimization algorithms. We note that other model-complexity reduction approaches such as vertical equilibrium (Yortsos, 1995, Gasda, Nilsen, Dahle, 2013) or invasion percolation (Ioannidis, Chatzis, Dullien, 1996, Nooruddin, Blunt, 2018) simulators are outside the scope of this work.

The paper is planned as follows. In Section 2 we present the flow problem and simulation setup and in Section 3 the upscaling problem considered here. Section 4 is a brief description of existing upscaling methods and Section 5 presents results of applying these methods to a number of flow scenarios. Section 6 presents the new optimization upscaling method and discusses results of applying the method to various cases. Comparison to existing methods is carried out. Section 7 presents results for the new upscaling approach for ensemble averages of multiple realization simulations. Section 8 summarizes this work and highlights the main conclusions.

Section snippets

Problem statement

We consider injection of a nonwetting phase into an aquifer or oil reservoir, i.e., a wetting-phase saturated formation. For simplicity we assume that the nonwetting phase is CO2 and the wetting phase is water, however these could in fact be any other phases such as a gas injected into an oil reservoir. Injection is stopped after some time and the CO2 is distributed throughout the aquifer with some initial saturation SCO2init(x,y,z). This is the starting point for our problem. Then, CO2 will

Upscaling model

This work focuses on upscaling Pc curves and therefore fine-scale permeability and kr will be used in the upscaled simulations. Upscaling, in general, involves a transition to a coarser grid in which upscaled properties are assigned to each coarse grid block. The coarse grid simulations are associated with two types of error. The first is due to the inaccuracy of representing varying properties within a coarse block with a single uniform value. The second is a result of numerical dispersion,

Description of existing methods for Pc upscaling

In this section we will give a brief description of upscaling methods that have appeared in previous literature. For more details on each method, we refer the readers to citations given in Section 1.

Results for existing methods

The first step for initiating the gravity-capillary flow model is to generate initial saturation distributions SCO2init(x,y,z). This is done by simulating point injection of CO2 at the upper or lower boundaries of the domain, modeling situations related to gravity drainage oil recovery (top injection) or CO2 geological storage (bottom injection). The two initial saturation distributions used in this work are presented in Fig. 2. Results are for a vertical cross section through the center of the

Optimization upscaling approach

The idea for using optimization algorithms in order to calculate upscaled single phase or multiphase flow parameters has appeared in previous literature, e.g., Wang et al. (2009); McKee et al. (2013); Krogstad et al. (2014); Alpak (2015). However, this idea is far from having been fully explored, has gained very little attention and has not been seriously adopted in practice. We have shown in the previous section that the problem of gravity-capillary flow with capillary entry pressure effects

Upscaling of ensemble averaged solutions

Global upscaling methods, such as the optimization procedure presented above, have a significant disadvantage of requiring a full fine-scale simulation. Therefore, these methods are often only used for theoretical studies or as a stepping stone towards developing local methods which do not require fine-scale simulations. If, however, the upscaled properties calculated for one problem can be used in other problems, e.g., considering ensemble average solutions of different permeability

Summary and conclusions

This work presents a thorough investigation of capillary pressure upscaling for the purpose of reducing computational cost of gravity-capillary driven flow simulations. An initial saturation distribution is allowed to migrate due to gravity and capillary forces without any flow sources or sinks. All boundaries are considered impenetrable and two types of initial saturations are used in the examples, injection of CO2 at the top and bottom of the domain. Upscaling is carried out only for Pc and

CRediT authorship contribution statement

Kan Bun Cheng: Software, Data curation, Formal analysis, Validation, Writing - review & editing. Avinoam Rabinovich: Conceptualization, Methodology, Supervision, Writing - original draft.

Declaration of Competing Interest

The authors declare that they do not have any financial or nonfinancial conflict of interests.

Acknowledgments

This work was supported by the Israeli Ministry of Energy. The paper is part of a PhD study carried out by the first author at the School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University. Data presented in this work can be found in the website: http://doi.org/10.5281/zenodo.3706667.

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