Numerical modelling and analysis of reactive flow and wormhole formation in fractured carbonate rocks
Introduction
Matrix acidizing is a common stimulation treatment for improving the flow characteristics of the near-wellbore region in carbonate reservoirs. It consists of injecting acid into the formation around some interval of the wellbore at pressures below the fracturing pressure. During the process, acid penetrates into the pores of the formation and dissolves some rock components such as cements or grains, and usually, if successful, creates wormholes, which can bypass a potentially-damaged zone around the wellbore, providing some highly permeable channels for hydrocarbon flow into the well. The productivity of a well can be enhanced significantly by acidizing, especially when near-wellbore damage is present and has motivated the decision to undertake this treatment. If a well does not have drilling-caused damage, the economic benefit of performing acidizing stimulation is less obvious, so matrix acidizing is generally applied only to a well that has a high skin factor which cannot be attributed to the mechanical aspects of the completion, such as partial penetration, perforation efficiency and so on (Economides and Nolte, 2000). Nevertheless, this rule is commonly broken when natural fractures are present in the formation. In naturally fractured reservoirs, injected acid may penetrate to a sufficient distance to yield a productivity enhancement greater than that normally expected from an acidizing treatment in un-fractured carbonate reservoir (Economides and Nolte, 2000, Liu et al., 2017). For this reason, acidizing is a widely conducted stimulation treatment in fractured carbonate reservoirs to increase the production (Bratton et al., 2006). Since the details of acid flowing through the fractured reservoir are different from the case of general matrix acidizing in a comparable un-fractured reservoir, a good understanding of the effect of natural fractures on the acidization process is needed to improve the design of the operation.
Numerous experimental studies have been conducted to understand the dissolution process in fractured cores (Detwiler, 2008, Detwiler et al., 2003, Dijk et al., 2002, Dong et al., 1999, Durham et al., 2001, Gouze et al., 2003, Polak et al., 2004). In laboratory experiments, artificial fractures are designed by contacting two rock samples together along a quasi-planar interface, and the fracture aperture is represented by the gap between the samples (Dong et al., 1999). The surfaces of the core samples, which compose the “fracture” walls, are either smooth or rough. Before and after the dissolution experiments, the aperture fields are quantified with digital reconstruction techniques, such as high-resolution X-ray computed tomography (CT) (Gouze et al., 2003) and nuclear magnetic resonance imaging (NMRI) (Dijk et al., 2002). The influence of parameters, such as the roughness of the fracture wall, the reaction kinetics, and the mineral dissolution rate, on the dissolution pattern can therefore be investigated. These experiments provide a direct observation of the consequences of dynamic acid flow and dissolution in rock fractures, and are fundamental for the mathematical model development to predict the reactive flow process in fractures. For example, Detwiler et al. (2003) conducted a series of reactive flow experiments on artificial fractures and concluded that the dissolution-induced evolution of aperture variability is determined by the relative magnitude of the diffusion and advection of the reactants as well as the mineral dissolution rate. In addition, two types of geometrical changes in fractures after being exposed to reactive fluid are observed (Deng et al., 2016). One is the enlargement or reduction of the fracture aperture, resulting from dissolution or precipitation of minerals around the fractures (Deng et al., 2016, Detwiler, 2008, Durham et al., 2001). The other one is the formation of a porous altered layer in the near-fracture region, resulting from the reactant dispersion into the rock matrix and the fast reaction rate (Deng et al., 2016, Ellis et al., 2013, Noiriel et al., 2007). However, the experimental works have so far only investigated the dissolution process in a single fracture. The effect of the presence of multiple fractures, especially those which exist in a complex fracture network, relative to the dissolution process, have not yet been investigated through experiment, and hence a numerical method is needed.
The numerical models developed to investigate the reactive flow in fractured rocks can be classified mainly into three types: (1) single fracture model (Deng et al., 2016, Detwiler and Rajaram, 2007, Dong et al., 2002b, Hanna and Rajaram, 1998, Hill et al., 2001, O’Brien et al., 2003, Szymczak and Ladd, 2009, Upadhyay et al., 2015), (2) fracture network model (Dong et al., 2001, Dong et al., 2002a), and (3) pseudo-fracture model (Kalia and Balakotaiah, 2009, Yuan et al., 2016). Here, we give a brief review of these models. In the single fracture model, only one fracture is considered, which is generated by contacting two surfaces together with a gap between them. The two surfaces can be either smooth or rough, and the resulting aperture is assumed to depend on a spatial variable. The model is based on mass conservation for fluid flow and reactant transport, and equations for chemical kinetics within the fracture space. The matrix leakoff is considered by introducing a source/sink item. The pressure, acid concentration, and fracture aperture as functions of space and time can be calculated by numerical simulation. Different dissolution patterns, which depend on the physical and chemical characteristics of the fracture-fluid system, are obtained, and the results have a good agreement with those observed in the experiments. These models provide a useful starting point for numerical analysis of reactive flow in fractured porous media. However, a single-fracture model cannot explain the dissolution process in real fractured rock, in which the fracture distribution is complex.
In the fracture network model, the matrix is ignored and the fractured medium is represented by a system of intersecting fractures, which provides the pathway for acid transport and dissolution. The fracture network model is an extension of the single fracture model and is based on the assumption that acid always creates a channel in only one main flow path in the fracture network. Acid flowing into tail fractures (fractures that are not on the main flow path) and matrix is calculated by using a leakoff coefficient. Subject to these assumptions, the flow and dissolution in those fractures that are not connected with the wellbore cannot be characterized in this model. In addition, as the main flow path should be pre-determined and the acid can only flow along this main flow path, the branching characteristic of the dissolution pattern cannot be described. However, this model proposes a field-scale design method for well treatments in a fractured reservoir and demonstrates that acidizing in a fractured reservoir is more efficient than in un-fractured reservoir, as observed in many carbonate acidizing treatments.
In pseudo-fracture models, the fractures are treated as a type of matrix that has anomalously-high porosity, and the fracture is represented by one mesh cell whose thickness is greater than the actual fracture aperture. These models are actually the same as those matrix acidizing models in terms of the mathematical equations. Therefore, they can naturally couple the acid flow and reaction in matrix with that in the fractures. Pseudo-fracture models provide an easy way to investigate the effect of the presence of fractures and the fracture location and orientation on dissolution dynamic, and the results are convenient for comparison with the results of matrix acidizing simulations. However, these models just set up some high-porosity channels in a domain of matrix to represent the fractures, and they lead to the need to create a finer grid to represent the fracture with one mesh cell, which will undoubtedly increase the computational time. In addition, this sort of model is particularly unsuited for use if one wished to extend the analysis to include concurrent geomechanical effects related to the fracture system changes caused by the acidizing (this will be addressed in our future work). All of these shortcomings make pseudo-fracture models inappropriate for simulating the reactive flow in porous media that have complex fracture arrays.
The main aim of this paper is to develop a reactive-transport simulation model that permits an evaluation of the effect of multiple, intersecting fractures on reactive flow in carbonate rocks. This can be accomplished by the combination of the two-scale continuum model developed by Panga et al. (2005) with the methods of a discrete fracture model. For the sake of convenience, we denote the combined new model, described below, as the two-scale discrete-fracture continuum model (TSDFC). The locations of fractures in the TSDFC model are explicitly defined, and hence the effect of each individual fracture on fluid flow and solute transport can be accounted explicitly. In comparison to the single-fracture model, the fracture-network model, and the pseudo-fracture model, the TSDFC model is more effective for a systematic investigation of the acidization process in fractured carbonate rock masses.
This paper is organized as follows. In Section 2, the mathematical model is presented, to describe the fluid flow, solute transport, and chemical reaction in both the matrix and the fracture system. In Section 3, a detailed numerical solution method is given, which discretizes the physical domain with Delaunay triangulation and discretizes the governing equations with the finite volume method. In Section 4 we compare the simulation results obtained from a degenerate model with a previous computational study to verify our work. In Section 5, the 2-D simulation results of the reactive flow in fractured carbonate media, with simple or complex fracture networks, under linear or radial flow conditions, are presented. The sensitivity of the acidization process with respect to fracture aperture, fracture distribution, and the geometry of the domain are also analyzed in this section. Finally, the paper is summarized by conclusions in Section 6.
Section snippets
Mathematical model
In this section, by adding the equations describing fluid flow, solute transport, and rock dissolution in fracture system, the two-scale continuum model presented by Panga et al. (2005) is extended to simulate the reactive dissolution of fractured carbonate rocks. The TSDFC model describes the phenomenon of solute transport and reaction in porous media at the Darcy (i.e. continuum) scale, and couples the dissolution processes occurring at the pore scale through structure–property relationships,
Numerical solution
In this section we detail the numerical solution procedure on a 2-D fractured medium by using the finite volume method, which is locally conservative and has a clear physical interpretation (Eymard et al., 2000, Eymard et al., 2006, Gallouët et al., 2000, Karimi-Fard et al., 2003, Moukalled et al., 2016, Versteeg and Malalasekera, 2007). This procedure is comprised of the discretization of the domain, a process known as meshing, and the discretization of the partial differential equations (
Model testing
For the reason that no numerical or experimental result for wormhole formation in fractured masses is available in the literature, the model described in this work is tested by comparing it with the reactive flow problem (described in previous works) without considering fractures. This demonstrates that the discretization method used in the unstructured grid formulation is suitable for the reactive flow problem. Furthermore, we must ensure that the method to characterize the fractured medium,
Simulation results and discussion
This section presents 2D simulation results describing the sensitivity of the acidization process in fractured carbonate media, with respect to fracture aperture, fracture distribution, and the geometry of the domain. Acid is injected at one end using the boundary and initial conditions in Eqs. (3), (4), and the model is calculated until breakthrough occurs. The values of parameters and dimensionless numbers used in the simulations, are shown in Table 1. All the values remain fixed throughout
Conclusions
The main contribution of this work is the development of a continuum model (TSDFC) that can calculate the reactive flow of acid in fractured carbonate rock with a complex fracture network. The TSDFC model is the combination of the two-scale continuum model, which is developed by Panga et al. (2005) for simulating reactive flow in un-fractured carbonate rock, and a discrete fracture model. The model retains the inherent features of two original models and calculates the dissolution of rock mass,
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 51504276 and 51504277); and the Fundamental Research Funds for the Central Universities (No. 17CX02008A).
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2023, Journal of Petroleum Science and EngineeringCitation Excerpt :As we can see in the contours of Si(OH)4 concentration distribution in Fig. 17, the maximum concentration of Si(OH)4 was only about 0.6% at a low injection rate while it was 1.8% at a high injection rate. The competing reaction rates between the advection-dispersion transport and reaction rate in carbonate caused different dissolution patterns formed with the different injection rates (Liu et al., 2017a, 2017b; Qi et al., 2019, 2021; Khoei et al., 2020; Luo et al., 2021). However, all the dissolution patterns of sandstone with varying injection rate are similar to the face dissolution patterns in carbonate (Al-Harthy et al., 2008), mainly because the reaction rate between acid and sandstone was relatively low compared with carbonate, and the surface reaction rate dominated the acid reaction rate in sandstone rocks.