Geofluids

Volume 2018, Article ID 4209197, 21 pages

https://doi.org/10.1155/2018/4209197

## Experimental Investigation of Forchheimer Coefficients for Non-Darcy Flow in Conglomerate-Confined Aquifer

^{1}School of Resource and Safety Engineering, China University of Mining & Technology, Beijing 100083, China^{2}State Key Laboratory of Mining Response and Disaster Prevention and Control in Deep Coal Mines, Anhui University of Science and Technology, Huainan, Anhui 232001, China^{3}State Key Laboratory of Coal and Safe Mining, China University of Mining & Technology, Beijing 100083, China^{4}Department of Petroleum Geology & Geology, School of Geosciences, University of Aberdeen, UK^{5}College of Engineering Peking University, Beijing 100871, China

Correspondence should be addressed to Yixin Zhao; nc.ude.btmuc@xyoahz

Received 8 May 2018; Revised 15 July 2018; Accepted 12 August 2018; Published 3 December 2018

Academic Editor: John A. Mavrogenes

Copyright © 2018 Tong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A conglomerate-confined aquifer (CCA) plays an important role in affecting the safety and environmental protection during energy mining. In this study, the Forchheimer coefficients and associated seepage characteristics of the CCA were studied for different hydrogeological conditions via integrating theoretical analysis, hydromechanical experiment, and field investigation. Empirical models related to the intrinsic permeability () and inertial resistance () were developed based on theoretical and experimental solutions, governed by the parameters of particle size, initial porosity, and stress. The non-Darcy flow was obtained through experiments conducted with different ratios of the aggregate particle subjected to stress ranging from 1.43 MPa to 4.38 MPa, and a discharge model associated with and was proposed. The aggregate particle and filling material of the CCA presented positive and negative effects on the interconnected pores, respectively, accompanied by wedging and wall effects. Distribution state of the aggregate particle and the filling material was affected by the stress, resulting in the reduction of the hydraulic conductivity and the weakening of the wedging and wall effects in the CCA. In addition, the transportation effect and broken effect occurred for the lower and higher stress situations and contributed to the shrinkage of the interconnected pores. Finally, the models of the CCA were validated using a normalized objective function (NOF), a linear slope function, and field measurements.

#### 1. Introduction

The seepage response widely exists in the study of gases, liquids, and geothermal activities [1–8]. The theories of Darcy flow and non-Darcy flow [9–13], which describe the relationship between the discharge and the hydraulic gradient, are widely used to investigate seepage behaviors in various fields, such as hydrocarbon resource mining of coal, coalbed methane, shale gas, tight sandstone gas [14–19], tunnel excavation, slope reinforcement, and underground mining [20–25], and also in the fields of garbage disposal, nuclear waste treatment, and sewage control [26]. Therefore, implications from investigating the seepage behavior and corresponding characteristics can greatly contribute to the development of geological science, energy mining, and the prevention and control of geological disasters.

Darcy’s (1856) law has been widely applied in experiments and simulations, and it states that the discharge is proportional to the hydraulic gradient. However, this type of flow model is only appropriate for low velocity, steadiness, and laminar flow [27–31]. For a fluid seepage presented in high-velocity fluid and highly permeable porous media [32–35], the flow velocity and the hydraulic gradient do not have a linear relationship and was widely described by Forchheimer and Izbash laws [36, 37]. However, the stress effect for non-Darcy flows is seldom considered in the work efforts focused on fluid seepage in the porous media, such as unconsolidated porous media [38] and streambed packing [39]. It is widely accepted that geomechanical factors contribute significantly to the changes in the internal structure of porous media and the hydraulic conductivity. A conglomerate-confined aquifer (CCA) composed of conglomerate, coarse sandstone, and even uranium mine existed above the occurrence stratum of coal and oil plays an important role in the safety of coordinating mining of intergrown energy and resources and ecological protection; an example includes the area of the Ordos Basin [40, 41]. Therefore, by determining the Forchheimer coefficients and investigating corresponding seepage characteristics for CCA in various hydrogeological environments, this study contributes to the development of energy exploration and environmental protection.

The primary objective of this study is to determine the Forchheimer coefficients with the contribution of stress and investigate corresponding seepage characteristics for the CCA in various hydrogeological environments, including different components, hydraulic pressures, and stresses. Through theoretical derivation, a discharge model was proposed followed by the development of the models of and . Subsequently, five groups of hydromechanical experiments were performed using porous media with different particle sizes and volume fractions and subjected to different stresses corresponding to various hydraulic gradients. The experimental results provided insights into the seepage behavior and the related characteristics. The optimized parameters for the and models were obtained through nonlinear regression, and a specific discharge model for the CCA was developed based on the experiments. Finally, comparisons between the predicted values, experimental values, and field measurement data were conducted to verify the accuracy of the proposed model.

The remainder of this paper is organized as follows: In Section 2, the theories of the flow regime are presented and the models of , , and the discharge are developed. In Section 3, the experimental preparation and procedures are described and the results are presented in Section 4. Finally, a discussion and a summary of the findings are presented in Sections 5 and 6, respectively.

#### 2. Establishment of the Theoretical Model

For the fluid flow in porous media with complex structure, various flow regimes such as Darcy flow, weak inertial flow, non-Darcy flow, and turbulent flow can be identified. We focus on Forchheimer’s law to investigate the seepage behavior of the CCA because it can well describe the linear and nonlinear fluid flow. Moreover, the specific formation of Forchheimer’s law is presented as follows:
where [ML^{−2} T^{−2}] represents the hydraulic gradient, [LT^{−1}] is the flow velocity, [ML^{−3} T^{−1}] is the non-Darcy coefficient given by with [L^{2}] defined as the intrinsic permeability and [ML^{−1} T^{−1}] as the dynamic viscosity, and [ML^{−4}] is the coefficient expressed by with [L^{−1}] and [MT^{−3}] as the non-Darcy coefficient and the density of the fluid, respectively. The two terms on the right-hand side of (1) represent the viscous and inertial energy loss mechanisms, respectively.

It is evident that the coefficient in (1) depends on the effective stress based on the relationship between and the stress described by the power function providing a reasonable description of the permeability-stress relationship for relatively low stress [42–44] in Darcy’s law:
where [ML^{−1} T^{−2}] is the effective stress, [L^{2}] is the intrinsic permeability corresponding to , and and are material constants.

##### 2.1. The Forchheimer Coefficients and

For the purpose of using Forchheimer’s law in the field of analytical or numerical solutions, the determination of the Forchheimer coefficients and in (1) is necessary. Considerable research efforts have been devoted to determining the coefficients and for different particle sizes and porosities of porous media [45–48]. The coefficients and in Forchheimer’s law were estimated by Ergun [49] who proposed modified functions based on the classical Kozeny-Carman model that incorporates the particle diameter and porosity of porous media as follows: where is the porosity of the porous media and [L] is the diameter of the porous media. Subsequently, similar expressions were developed by [50, 51], all of whom took the effect of porosity into account.

##### 2.2. The Correlation between and the Stress

Considering the effect of the effective stress on the porosity, an exponential relationship between porosity and stress was developed based on numerous studies [52–54]: where is the stress sensitivity coefficient and is the initial porosity of the porous media. In subsequent studies, Huang et al. [55] developed a model for formulating as a function of porosity: where and are the material and exponent coefficients, respectively. Based on the above-mentioned research efforts, by combining the exponential relationship shown in (7) and (8) and eliminating the porosity , it is evident that can be expressed by stress in the form of an exponential equation: where is the attribute parameter of the porous media and is the stress sensitivity parameter.

##### 2.3. Empirical Models of , , and Discharge

Combining (1), (3a), and (3b), after unifying the dimensional units, we find that and depend on the particle diameter and the porosity. Consequently, by combining the results with (1), (2), (3a), (3b), and (6), the models of and can be obtained as follows: where [L] is the diameter of the aggregate particle; and are the initial attribute parameters representing the effects of the shape of the pores and particles, pore throat, and tortuosity on the porous media; and are the particle diameter coefficients; are the porosity coefficients; and is the stress parameter. Notably, the porosity is also affected by stress; however, due to the nonunique expressions for distinct materials and conditions [44, 53, 54], a specific correlation should be obtained through relevant experiments. For simplicity, hereafter, we just use the final porosity calculated by the porosity-stress law, such as (4), into consideration for a reanalysis of the experimental data.

Furthermore, a discharge model of the fluid flow is proposed based on a previous study [56] using the model ((7)) and the model ((8)).

##### 2.4. The Criteria of Linear and Nonlinear Flow

Because the determination of the transition from linear flow to nonlinear flow is critical for porous media, a large number of studies have been conducted on this subject [57]. Normally, the Reynolds number and the Forchheimer number have been widely used to describe the transition point [58]; the expression is defined as
where [L] is the characteristic length of the porous media, [ML^{−1} T^{−1}] is the dynamic viscosity, and [LT^{−1}] is the flow velocity. Ma and Ruth [59] defined the criterion of the Forchheimer number, , as the ratio of the inertial to the viscous losses:

Compared with the Reynolds number, the Forchheimer number possesses the advantage of a clear definition, an explicit physical meaning, and wide applicability in engineering.

The non-Darcy effect is the ratio of the hydraulic gradient induced by the inertial forces to the total hydraulic gradient, and it is defined as

Substituting (1) and (14) into (16), is formulated as a function of the Forchheimer number:

A large number of critical values of have been evaluated for porous media; Zeng and Grigg [36] suggested as a threshold for the nonlinear fluid flow effect, which corresponds to a critical of 0.11. Using a graphical evaluation of laboratory column experiments, Ghane et al. [26] estimated a high average critical value of 0.31. An even higher average critical value of 0.40 corresponding to for nonlinear fluid flow was discovered by Macini et al. [38] for natural sand.

#### 3. Experimental Preparation and Procedure

##### 3.1. Material Preparation

The CCA in the Ordos Basin is characterized by a mixture of coarse sandstone and fine sand in a loose state. The experimental materials are composed by sandstone (small), cobblestone (medium), and cobblestone (large) with ranges of diameter 10~20 mm, 20~30 mm, and 30~50 mm with corresponding densities of 2531.9 kg/m^{3}, 2744.7 kg/m^{3}, and 2580.4 kg/m^{3} as aggregates and fine sand with a density of 2090.4 kg/m^{3} as the filling material. During the experiment, different aggregates were mixed together at specific ratios. In addition, different quantities of fine sand were added to each group; the specific details of the mixture are shown in Table 1 and Figure 1.