Experimental study of the transition from non-Darcian to Darcy behavior for flow through a single fracture*

QIAN Jia-zhong （錢家忠）， WANG Mu （王沐）， ZHANG Yong， YAN Xiao-san （嚴小三），ZHAO Wei-dong （趙衛(wèi)東）

1. School of Resources and Environmental Engineering， Hefei University of Technology， Hefei 230009， China，E-mail: qianjiazhong@hfut.edu.cn

2. Department of Geological Sciences， University of Alabama， Tuscaloosa， USA

Experimental study of the transition from non-Darcian to Darcy behavior for flow through a single fracture*

QIAN Jia-zhong （錢家忠）1， WANG Mu （王沐）1， ZHANG Yong2， YAN Xiao-san （嚴小三）1，ZHAO Wei-dong （趙衛(wèi)東）1

1. School of Resources and Environmental Engineering， Hefei University of Technology， Hefei 230009， China，E-mail: qianjiazhong@hfut.edu.cn

2. Department of Geological Sciences， University of Alabama， Tuscaloosa， USA

2015，27（5）:679-688

Laboratory experiments are designed in this paper using single fractures made of cement and coarse sand for a series of hydraulic tests under the conditions of different fracture apertures， and for the simulation of the evolution of the flow pattern at places far from the outlet. The relationship between the hydraulic gradient and the flow velocity at different points， and the proportion evolution of the linear and nonlinear portions in the Forchheimer formula are then discussed. Three major conclusions are obtained. First， the non-Darcian flow exists in a single fracture in different laboratory tests. Better fitting accuracy is obtained by using the Forchheimer formula than by using the Darcy law. Second， the proportion of the Darcy flow increases with the increase of the observation scale. In places far enough， the Darcy flow prevails， and the critical velocity between the non-Darcian flow and the Darcy flow decreases as the fracture aperture increases. Third， when the fracture aperture increases， the critical Reynolds number between the non-Darcian flow and the Darcy flow decreases.

non-Darcian flow， Darcy behavior， transition， single fracture， experimental study

Introduction

Fracture formations are ubiquitous in nature， and are related with a number of geological activities such as the mineralization processes. Fracture formations may be in the form of a single fracture and in a fracture network， in which the single fracture is the fundamental ingredient， so the water flow and the solute transport in single fractured media are the main concern. Traditionally， the fractured flow was treated as the Darcy flow. Gray et al.［1］presented an analytical solution for the steady Darcy flow of an incompressible fluid through a homogeneous， isotropic porous medium filling a channel bounded by symmetric wavy walls. Marusic-Paloka et al.［2］focused on the Brinkman and Darcy law， derived from microscopic equations by up scaling， and compared the results and estimated the error in applications.

More studies show that the non-Darcian flow is a significant existence［3-9］. For example， Qian et al.［6，7］used well-controlled laboratory experiments to investigate the flow and the transport in a fracture under non-Darcian flow conditions， and found that the Forchheimer equation fits the experimental v-Jrelationship nearly perfectly， whereas the Darcy equation is inadequate in this respect. Quinn et al.［8］quantified the non-Darcian flow by packer testing. They found that the flow is nonlinear but not quadratic in nature. Cherubini et al.［9］investigated the nonlinear flow by analyzing hydraulic tests on an artificially created fractured rock sample， and the experimental result shows that the relationship between the flow rate discharge and the head gradient matches the Forchheimer equation and describes a strong inertial regime well.

The equation or the numerical results for the non-Darcy flow attracted much interest［10］. Chen et al.［11］discussed the versatility of the non-Darcy flow equation. Qian et al.［12-14］and Brown［15］discussed the influence of the roughness and the Reynolds number onthe rough single fracture flow. It is concluded that the friction factor for the flow in single fractures is influenced by them. Thiruvengadam and Kumar［16］evaluated the effectiveness of the Forchheimer formula for the rough medium flow. Xu et al.［17］studied the non-Darcy flow inertia coefficient.

Only a few studies focused on relating the non-Darcian flow to the Darcian flow and their conversion in a single fracture. Quinn et al.［8］quantified the rate of flow versus the ambient head in the linear range through the transition to the nonlinear flow. Zhang et al.［18］investigated the fluid flow regimes through deformable rock fractures， by conducting water flow tests through both mated and non-mated sandstone fractures.

But， the theoretical analysis and a quantitative discussion of the Forchheimer formula remain to be an issue. So this study uses laboratory experimental data and the Forchheimer formula to explore the relationship between the flow rate and the hydraulic gradient， under conditions of rough surface， no pressure and different apertures. The purpose of this paper is to（1） experimentally investigate the non-Darcian flow evolution from the Darcian flow in a single fracture of different fracture apertures and hydraulic gradients，and （2） study the relationship between the hydraulic gradient and the flow rate at different points and the proportion evolution of the linear and nonlinear portions in the Forchheimer formula.

Fig.1 Schematic diagram of experimental setup for groundwater flow in a single fracture

1. Basic formulas

The Darcy law is a well known basic equation for porous media saturated seepage. It is valid when the viscous force plays a dominant role and the inertia force can be ignored in the seepage field. The seepage behavior in nature mostly shows a linear relationship between the seepage velocity and the hydraulic gradient. Similarly， the Darcy’s law can be used to reflect the permeability characteristics and to determine the penetration parameters， usually expressed by the mathematical expression:

where V is the seepage velocity，K is the permeability coefficient， and J is the hydraulic gradient.

Fig.2 Schematic diagram of the dupuit assumption

Table 1 The function values of the slope angles between two adjacent measurement points at the largest water head difference of 0.506 m

But it is found that in some exceptional circumstances， the relationship between V and J will become nonlinear， and a non-Darcian relationship will be evident instead， when v is sufficiently large. Previous studies of non-Darcy flows revealed various non-Darcy flow formulas under various application conditions and scopes. Nevertheless， some formulas rely on many parameters， which limit their applications. The Forchheimer formula， on the other hand， has a clear theoretical basis of the NS equation of hydrodynamics，

Fig.3 Fitting curve of J and v for different equations and for rough single fracture of 0.0005 m aperture in sections

wherep，ρare the pressure， density of the fluid，respectively. In the percolation porous structures with the void-space-occupied probability p， the system sizeLand the pressure drop ?p， the relationship between the?pand theV can be described by

whereβis the inertial parameter. The above relationship is then generalized into the familiar form

wherea，b are parameters.

The linear term on the right-hand side of the formula （4） represents the viscous force， while the quadratic term represents the inertial force. For engineering applications， the form of the Forchheimer formula is simple and convenient. With a few parameters， it reflects both the Darcy flow and the non-Darcy flow characteristics in fractured media. Therefore， it is a widely used non-Darcy flow calculation formula.

2. Experimental design

2.1 Experimental setup

Figure 1 is a schematic diagram of the experimental setup for the groundwater flow in a single fracture.The model is made of cement and coarse sand （with diameters between 0.001 m and 0.002 m）， and is 3.8 m long， 0.55 m high， and with adjustable width. A single fracture is formed by two parallel vertical planes of several centimeters thick. The aperture of the single fracture is adjustable. Eight manometers are installed inside of the fracture planes to measure the hydraulic heads. The measurement points of the manometers are on the plane surfaces， approximately 0.05 m above the bottom of the plane. In this way， the manometers’ disturbance to the flow field could be minimized. The water level in the fracture is below the upper cover plate during this experiment， so the flow is unconfined. Of course， the artificial single fracture used in this laboratory study is much simpler than the natural single fractures， as the natural walls are rarely parallel. Even so， the results of this study could help us to gain some insight into the scaling behavior of the hydraulic conductivity under well-controlled flow conditions.

2.2 Testing method

The cylinder is used to measure the steady flow rate， and the error is less than 2%. The water level is measured by piezometric tubes with error of ±0.0005m.

Fig.4 Fitting curve of the hydraulic gradient and the flow velocity for different equations and for rough single fracture of 0.0015 m aperture in sections

3. Experimental results and discussions

The original first-hand experimental data include the water level and flux data in the single rough fractures （with the apertures of 0.0005 m， 0.0015 m and0.002 m， respectively）. Points 7 to 1 are marked from left to right according to the water level measurement points， with the last discharge end， Point 1， as the outlet （as is proposed by this study）， the distances from the measurement Points 2 to 7 to the outlet are 0.096 m， 0.688 m， 1.365 m， 2.122 m， 2.93 m and 3.54 m. Figure 1 shows the position of each water level measurement point.

Fig.5 Fitting curve of the hydraulic gradient and the flow velocity for different equations and for rough single fracture of 0.002 m aperture in sections

3.1 Disscussions for the effect of dupuit assumption

As shown in Fig.2， at pointp on the phreatic surface，J=-dH/ds=-dz/ds=-sinφ， its velocity isVp=-KJ=-Ksinφ. The Dupuit assumption says that we can replace the sine value of the slope angle with the tangent value when the slope angle is very small. Under this condition， the phreatic surface is flat，the water flow is horizontal mainly， and the vertical component of the seepage velocity can be ignored. In our experiment， when the hydraulic gradient is the largest， the tangent value of the slope angle is 0.133， and the sine value is 0.132， with a relative error of 0.0076，much less than 1. The function values of the slope angles between two adjacent measurement points are calculated， as shown in Table 1， with relative errors much less than 1， too. Therefore the Dupuit assumption is valid for our experiments， and the seepage could be seen as one-dimensional Darcian/non-Darcian flow.

Fig.6 Fitting curve of the hydraulic gradient and the flow velocity for different equations and for rough single fracture of 0.0005 m aperture on the whole

3.2 Relationship between hydraulic gradient and flow velocity in sections

From the outlet to Point 2 and to Point 7， the Darcy law and the Forchheimer formula are used to fit the relationship between the hydraulic gradient and the flow velocity in sections， and the results are shown in Figs.3-5. It is found that the correlation coefficient of the Darcy law increases as the distance from the outlet increases. It indicates the flow transition into the Darcy flow from the non-Darcian flow as the flow velocity decreases.

3.3 Relationship between hydraulic gradient and flow velocity on the whole

On the other hand， the relationship between the hydraulic gradient and the flow velocity is fitted on the whole. The results are shown in Figs.6-8， and it is seen that the correlation coefficient of the Forchheimer formula is higher than that of the Darcy law， indicating that the quadratic relation between the hydraulic gradient and the flow velocity is more suitable for the actual seepage. That is because with the increase of the flow velocity， the inertial force and the viscous force reach the same order of magnitude， and then the Forchheimer formula could describe them both better than the Darcy law. We could also find that the coefficientsaandb of the Forchheimer formula decrease with the increase of the aperture， which is a conclusive phenomenon or needs not more study.

Fig.7 Fitting curve of the hydraulic gradient and the flow velocity for different equations and for rough single fracture of 0.0015 m aperture on the whole

3.4 Further analysis of Forchheimer formula

Denoting the linear termaV of the Forchheimer formula as Jand the nonlinear term bV2as J，viwhere “v ” stands for “viscous” and “i ” stands for“inertial”， the Forchheimer formula （4） becomes J= Jv+Ji. We could calculate the value of Jv/Jand Ji/J at the water level measurement points 2 to 7，and also obtain the proportions of linear and nonlinear terms. The results for the rough single fracture with aperture of 0.0005 m are shown in Table 2.

The relationship of Jv/Jand Ji/J versus the flow velocity is shown in Fig.9.

To identify the type of the fluid flow pattern， one may use a dimensionless parameter

whereRe is Reynolds number，d is the characteristic length of the fluid flow，n is the kinematic viscosity of the fluid （in this experiment， we let the kinematic viscosity be 1.2028×10-6m2/s）. Using formula（5）， in Fig.10 ， the flow velocity in Fig.9 is converted into the corresponding Reynolds number.

Fig.8 Fitting curve of the hydraulic gradient and the flow velocity for different equations and for rough single fracture of 0.002 m aperture on the whole

Table 2 The proportion of linear and nonlinear terms at different water level measurement points with aperture of 0.0005 m

As shown in Figs.9 and 10， the values of Jv/J and Ji/Jhave the same order of magnitude， meaning that both the inertial flow and the viscous flow are not negligible. The velocity and the Reynolds number decrease with the increase of the flow distance from the outlet， the proportion of the nonlinear term decreases as well. Due to the increased contribution of the linear term， when the velocity decreases，the water flow gradually evolves from the non-Darcian flow to the Darcy flow. We could not obtain the critical velocity and the critical Reynolds number between the non-Darcian flow and the Darcy flow according to the dotted lines in Fig.9 and Fig.10， however， one may see a decreased trend with the increaseof the fracture aperture. To obtain the specific values of them， further study is needed.

Fig.9 The relationship of Jv/J with Ji/Jand velocity for rough single fracture

Fig.10 The relationship of Jv/J with Ji/Jand Reynolds number for rough single fracture

4. Conclusions

From the above analysis， the following conclusions are drawn:

（1） The hydraulic gradient and the flow velocity assume a nonlinear relationship in the unconfined flow single fracture， and the relationship could be represented by J=aV+bV2. This finding is consistent with the theoretical derivation， which shows thatV∝J1/2under the turbulent flow assumption.

（2） When the observation distance from the outlet increases， the water flow becomes non-Darcian flow first， and then the proportion of the Darcy flow increases gradually. The farther from the outlet， the more evident the tend to the Darcy flow becomes. The critical velocity is decreased with the increase of the aperture.

（3） The critical Reynolds number between the non-Darcian flow and the Darcy flow decreases with the increase of the aperture under the same surface roughness condition.

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10.1016/S1001-6058（15）60530-3

（January 9， 2014， Revised May 9， 2015）

* Project supported by the National Natural Science Foundation of China （Grant Nos. 41272251， 41372245）.

Biography: QIAN Jia-zhong （1968-）， Male， Ph. D.， Professor