The principal permeability tensor of inclined coalbeds during pore pressure depletion under uniaxial strain conditions: Developing a mathematical model, evaluating the influences of featured parameters, and upscaling for CBM recovery

https://doi.org/10.1016/j.jngse.2019.103099Get rights and content

Highlights

  • A mathematical model for the principal permeability tensor of inclined coalbeds under uniaxial strain conditions.

  • Influences of two featured parameters on the principal permeabilities during pore pressure depletion.

  • A tensor-based index system for coal permeability evaluation.

  • A preliminary prospect on the usage of the tensor-based index system in CBM recovery.

Abstract

In situ coalbeds are commonly inclined to one or multiple directions. The principal permeability tensor of an inclined coalbed may thus be non-coaxial to the uniaxial strain conditions, which are composed of two underlying assumptions: invariant vertical stress and confined horizontal boundaries. This paper developed a mathematical model to represent the principal permeability tensor of inclined coalbeds during pore pressure depletion under uniaxial strain conditions. This model adopted two geological properties to correlate the orientations of principal permeabilities and uniaxial strain conditions. One is the dip angle of inclined coalbeds, and the other is the pitch angle of cleats. The developed model was verified by field permeability data of the coalbeds in the San Juan Basin. Dip angle and pitch angle are the featured parameters of the developed model. This paper thus evaluated how the two angles influenced the magnitudes and evolution behaviors of principal permeabilities during pore pressure depletion under uniaxial strain conditions. The influences of dip angle and pitch angle on principal permeabilities can be attributed to their influences on the effective stresses normal to principal permeabilities. In order to upscale the principal permeability tensor for CBM recovery, this paper formulated a tensor-based index system for coal permeability evaluation. The tensor-based index system is composed of three indexes: the principal permeability tensor, the equivalent permeability tensors coaxial to engineering issues, and the geometric mean (GM) permeability of the three principal permeabilities. The principal permeability tensor can be used to optimize the pattern of vertical CBM wells, to determine the optimal directions of lateral wells, and to predict the propagating orientation of hydraulic fractures. The equivalent permeability tensors coaxial to engineering issues can be implemented into numerical simulators to improve the precision when evaluating and predicting the production performance of CBM wells. The GM permeability represents the ensemble transport capacity of coalbeds and can be used as an indicator to represent the productivity of coalbed reservoirs. This paper also preliminarily discussed how to use the tensor-based index system in CBM recovery.

Introduction

Permeability represents the transport capacity of coalbeds for gas and water flow (Clarkson and Bustin, 1997; Moore, 2012; Pan and Connell, 2012). This property thus determines whether a coalbed methane (CBM) reservoir can be commercially produced (Moore, 2012; Rogers, 1994). Permeability is considered as the single greatest controlling property on CBM production in several regions of the San Juan Basin (Clarkson and Bustin, 1997), which is a major producer of CBM in the world (Moore, 2012). Because of the significance of permeability for CBM recovery, this property has attracted worldwide attention and has been investigated comprehensively through experimental measurements, field observations, mathematical models, and numerical simulations (Liu et al., 2011; Pan and Connell, 2012; Zhang et al., 2008).

Although the transport conduits in coalbeds are composed of both micro-pores and macro-fractures, permeability is mainly dependent on fractures, and the contribution from pores is usually neglected (Pan and Connell, 2012; Wang et al., 2014). The fractures in coalbeds are typically composed of three orthogonal cleat sets: face cleats, butt cleats, and bedding planes (Laubach et al., 1998; Wang et al., 2014). Because of this orthogonal fracture structure, coal permeability is intrinsically anisotropic (Wang et al., 2014). The anisotropic permeabilities corresponding to this cleat structure can be represented by a second-order diagonal tensor, normally referred to as ‘principal permeability tensor’ (Fanchi, 2006; Lang et al., 2014). The diagonal elements in this tensor are referred to as ‘principal permeabilities’. The principal permeability tensor is defined by the orientations and magnitudes of principal permeabilities. This study assumes that the orientations of principal permeabilities do not change during pore pressure depletion under uniaxial strain conditions. The orientations of principal permeabilities are coaxial to the intersection lines of the three cleat sets (Lang et al., 2014). The principal permeabilities can be classified as the maximum principal permeability, the medium principal permeability, and the minimum principal permeability in terms of their magnitudes.

Coal permeability anisotropy has been investigated through experimental measurements. The experimental report on coal permeability anisotropy can be traced back to Gash et al. (1992), who evaluated the permeability anisotropy of Fruitland formation coals by using cylindrical cores. Besides cylindrical cores, cuboidal samples were used for experimental evaluation on coal permeability anisotropy in recent years (Liu et al., 2018; Massarotto et al., 2003a, 2003b; Niu et al., 2018; Tan et al., 2018; Wang et al., 2018a). These experimental measurements showed that anisotropy exists regarding the magnitudes of principal permeabilities. Generally, the maximum and medium principal permeabilities are along the bedding plane. The maximum principal permeability aligns with the face cleat direction and the medium principal permeability with the butt cleat direction. The minimum principal permeability is perpendicular to the bedding plane. The experimental results of Gash et al. (1992) showed that the maximum principal permeability could be 80 to 200 times the minimum principal permeability for Fruitland formation coals.

Besides the magnitudes of principal permeabilities, the evolution behaviors of principal permeabilities are also anisotropic in response to effective stress change and gas sorption. Wang et al. (2018a) reported that the anisotropic ratios of the maximum principal permeability to the other two principal permeabilities tend to reduce with increasing effective stress for the cuboidal samples from the Junggar Basin. Tan et al. (2018) found that the anisotropic ratio of the maximum principal permeability to the minimum principal permeability fluctuates within a broad range from 3.68 to 12.93 for a cuboidal sample from the Bowen Basin.

The cylindrical samples used for coal permeability measurements are drilled along the orientations of principal permeabilities (Gash et al., 1992; Li et al., 2004; Zhu et al., 2018). The cuboidal samples are also prepared to correspond to these orientations (Liu et al., 2018, 2019; Massarotto et al., 2003a; Niu et al., 2018; Tan et al., 2018; Wang et al., 2018a, 2018b). These samples are then implemented into holders, and confining stresses are exerted normal to the orientations of principal permeabilities. Therefore, these laboratory measurements intrinsically investigate how the principal permeabilities evolve when their orientations are normal to boundary conditions. In addition, the current anisotropic permeability models also represent how the normal effective stresses influence the evolution behaviors of principal permeabilities. However, the principal permeabilities of in situ coalbeds may be oblique to rather than normal to their boundary conditions.

In situ coalbeds are generally assumed to behave under uniaxial strain conditions, which are composed of two underlying assumptions: invariant vertical stress and confined horizontal boundaries without deformation (Liu et al., 2011; Wang et al., 2014; Zang et al., 2015). The geological maps in the literature have shown that in situ coalbeds typically incline to one or multiple directions. (Cai et al., 2011; Li et al., 2011; Yao et al., 2014; Zhou and Yao, 2014). The bedding planes of inclined coalbeds may be oblique to rather than normal to the vertical and horizontal boundaries. The principal permeability normal to the bedding plane is thus oblique to the uniaxial strain conditions. Moreover, the orientations of face cleats and butt cleats may be not parallel to the strike direction of inclined coalbeds (Laubach et al., 1998). The two principal permeabilities along the bedding plane may be oblique to the uniaxial strain conditions as well. Although multiple mathematical models have been proposed to represent the evolution behaviors of coal permeability under uniaxial strain conditions, these models are either isotropic (Cui and Bustin, 2005; Palmer and Mansoori, 1998; Shi and Durucan, 2004) or only representative of the orthogonal case (Wang et al., 2014). These models may not be able to represent the evolution behaviors of principal permeabilities for an inclined coalbed under uniaxial strain conditions.

This paper will formulate two coordinate systems: one is coaxial to the three orthogonal cleat cleats, and the other coaxial to the uniaxial strain conditions. The principal permeability tensor and uniaxial strain conditions can thus be correlated through the rotation between the two coordinate systems. A mathematical model will be developed to represent the principal permeabilities during pore pressure depletion under uniaxial strain conditions. This paper will then evaluate how the featured parameters of the developed model (i.e., dip angle and pitch angle) influence the magnitudes and evolution behaviors of principal permeabilities during pore pressure depletion under uniaxial strain conditions. In order to upscale the principal permeability tensor for CBM recovery, this paper will formulate a tensor-based index system for coal permeability and discuss how to use this index system in CBM recovery.

Section snippets

Formulation of the coordinate systems coaxial to the principal permeability tensor and uniaxial strain conditions

This study assumes an inclined and flat coalbed under uniaxial strain conditions, which are considered representative of the boundary conditions of in situ coalbeds. The assumed coalbed is not influenced by tectonic stresses and only contains three orthogonal cleat sets: face cleats, butt cleats, and bedding planes. A rectangular coordinate system can thus be formulated to accommodate the three orthogonal cleats, as shown in Fig. 1. This coordinate system is referred to as the ‘cleat coordinate

Modeling configuration

The permeability model developed in Section 2 incorporates two featured parameters (i.e., dip angle and pitch angle) that have not been discussed in other permeability models. This section will thus focus on how the dip angle and pitch angle influence the magnitudes and evolution behaviors of principal permeabilities during pore pressure depletion under uniaxial strain conditions. Table 3 presents the parameter values used for modeling the influences of dip angle and pitch angle. Note that in

Formulation of a tensor-based index system for coal permeability evaluation

Coal permeability is conventionally represented by a single scalar (Liu et al., 2011; Pan and Connell, 2012). However, a universal anisotropy has been observed in the literature regarding the magnitudes of coal permeability at both laboratory and field scales (Gash et al., 1992; Koenig and Stubbs, 1986; Massarotto et al., 2003a, 2003b; Niu et al., 2018; Tan et al., 2018; Wang et al., 2018a, 2018b; Wold et al., 1995; Wold and Jeffrey, 1999). The modeling results in Section 3 show that anisotropy

Conclusions

This paper has developed a mathematical model to represent the principal permeability tensor of inclined coalbeds during pore pressure depletion under uniaxial strain conditions. This model incorporates two featured parameters that have physical meanings in geological geology. One parameter is the dip angle of inclined coalbeds, and the other parameter is the pitch angle of cleats. The developed model is verified by field permeability data of coalbeds in the San Juan Basin. Verification results

Acknowledgment

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (51804312; 51874314) and the State Key Laboratory Cultivation Base for Gas Geology and Gas Control (WS2019A03). The first author (Jie ZANG) also acknowledges Ms. Wenjuan SONG for her devotion in the past decade.

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