Assessments of the effects of various fracture surface morphology on single fracture flow: A review

Natural rock joint permeability deviates from the classic fluid flow governing equations due to the inher-ent fracture surface roughness(i.e.,contact points,spatial correlation,matching,varying aperture,iso-lated voids,infilling material,tortuosity and channellings)and engineering disturbance such as excavations.To improve the accuracy of fracture permeability evaluation,many efforts have been made in analytical,experimental,and numerical methods.This study reviews the modified mathematical gov-erning equations of fluid flow and classifies them based on different influencing factors,such as friction factor,aperture,tortuosity,inertia,and various in situ stress effects.Various experimental and simulation techniques for the coupled normal-and shear-stress flow problems were assessed,and their advantages and disadvantages were also analysed.Furthermore,different surface roughness descriptions and their impacts on mechanical and hydraulic behaviours were discussed,followed by the potential research directions for fracture flow problems.

Theoretic expression of flow rate for a single fracture with linearly varying width

By means of reasonable assumption and mathematical derivation, a theoretic expression of flow rate for a single fracture with linearly varying width was obtained. The mathematical derivation was based on the cubic law and the new theoretic expression was an extention of traditional parallel plate model. This study may help to analyze seepage in fractured rock mass.

SIMULATION OF SOLUTE TRANSPORT IN A PARALLEL SINGLE FRACTURE WITH LBM/MMP MIXED METHOD

This article deals with the solute transport in a single fracture with the combination of the Lattice Boltzmann Method （LBM） and Modified Moment Propagation （MMP） method, and this mixed method is proved to have several advantages over the LBM and Moment Propagation （MP） mixed method which leads to negative concentrations under some conditions in computation. The disadvantage of LBM/MP has been overcome to a certain extent. Also, this work presents an LBM solution of modeling single fractures with uniformly or randomly distributed grains, which can provide a new path of applying the LBM in solute transport simulation in fractures.

EXPERIMENTAL STUDY OF THE NON-DARCY FLOW AND SOLUTE TRANSPORT IN A CHANNELED SINGLE FRACTURE

The characterization of fracture rocks is always a key issue in understanding the flow and solute transport in fractured media. This article studies the solute transport in a Channeled Single Fracture （CSF）, a single fracture with contact in certain areas. The flow in a CSF often has preferential pathways and the transport in a CSF often has Break Through Curves （BTCs） with long tails. The Surface Contact Ratio （SCR）, the ratio of the contact area to the total fracture area, is an important indicator for the fracture surface roughness. To study the flow and solute transport in a CSF, a controlled physical model is constructed and a series of flow and tracer test experiments are carried out. Under our experimental conditions, the flow in a CSF is found to follow the Forchheimer equation , where and are the hydraulic gradient and the average pore velocity, respectively and and are two parameters related to the viscous and inertial flow components, respectively. Furthermore, it is found that b decreases with the decrease of SCR. For the solute transport, it is found that the BTCs often deviate from the traditional Fickian behavior, by the early-arrival and the long tailing. More interestingly, the observed BTCs often have a double-peak or a multi-peak, that would be difficult to explain using the existing transport theory such as the Advection-Dispersion Equation （ADE）. In addition, the longitudinal dispersion coefficient is found to be scale-dependent in a CSF and the relationship is of exponential type.

FORMULAE FOR AVERAGE VELOCITY OF GROUNDWATER FLOW AND EXPERIMENTAL EVIDENCE OF NON-DARCY' S FLOW THROUGH A SINGLE FRACTURE

The formulae for average velocity of groundw-ater flow in a single fracturewere derived based on the characteristics of fracture properties and hydraulic methods. The resultsshow that the average velocity is proportional to the square root of the hydraulic gradient. Inorder to verify the results, a laboratory model was established, and the experimental data wereanalyzed. Experimental results indicate that the relation between the average velocity and hydraulicgradient is nonlinear, and can he filled with power functions. And for both the unconfined andconfined flows, the value of the exponent of power functions are close to 0. 5. Thus the experimental results agree well with those from the theoretical analysis. By comparing the calculated andmeasured values of the average velocity under the same conditions, the formulae presented herein aremore effective than the traditional formula based on Darcy' s Law. These results provide theevidences of non-Dar-cy's flow in single fracture.

LABORATORY EXPERIMENTS ON SOLUTE TRANSPORT IN A PARTIAL TRANSFIXION SINGLE FRACTURE

In the study of solute transport in rough single fracture, the contact area is an important factor. The single fracture is defined as two categories in this article： the full transfixion single fracture and the partial transfixion single fracture. The purpose of this article is to research how the contact area affects the solute transport in partial transfixion single fracture. The contact area is generalized as square blocks with three sizes, and contact rate is variable, a series of experiments for solute transport were conducted in a simulation model which can simulate the two types of fractures in the laboratory. Based on the analysis of the breakthrough curves and the experiment phenomena, it is concluded that the difference of breakthrough curves of various contact rates is evident and increases with the increase of contact rate, the relative error curves reflect the difference of block sizes, and the maximum errors increase from smaller than 0.2 to about 0.8 with the increase of contact rate. These phenomena are also explained qualitatively in this article. It is concluded that the contact area strongly affects solute transport, and the research of channels formed by contact area is useful to further understand the rule of solute transport in partial transfixion single fracture.

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

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-LCL AND TRACER TEST FOR GROUNDWATER FLOW IN A SINGLE FRACTURE

The validity of Local Cubic Law （LCL） is an important issue to study groundwater flow and transport in fractured media. According to laboratory simulaion tests, the average velocity with a lower gradient in a single fracture is calculated by the LCL, which is compared with the measured average velocity. Then dye tracer test is designed and completed. The evidence for non-LCL, is drawn from the results of the simulation tests and the dye tracer tests. Then the Reynolds number of groundwater is calculated, the critical value of Re for laminar flow is discussed in a single fracture under different conditions. The motion types for groundwater flow have been discussed.

Lattice Boltzmann simulation of solute transport in a single rough fracture

In this study, the lattice Boltzmann method （LBM） was used to simulate the solute transport in a single rough fracture. The self-affine rough fracture wall was generated with the successive random addition method. The ability of the developed LBM to simulate the solute transport was validated by Taylor dispersion. The effect of fluid velocity on the solute transport in a single rough fracture was investigated using the LBM. The breakthrough curves （BTCs） for continuous injection sources in rough fractures were analyzed and discussed with different Reynolds numbers （Re）. The results show that the rough frac~＇e wall leads to a large fluid velocity gradient across the aperture. Consequently, there is a broad distribution of the immobile region along the rough fracture wall. This distribution of the immobile region is very sensitive to the Re and fracture geometry, and the immobile region is enlarged with the increase of Re and roughness. The concentration of the solute front in the mobile region increases with the Re. Furthermore, the Re and roughness have significant effects on BTCs, and the slow solute molecule exchange between the mobile and immobile regions results in a long breakthrough tail for the rough fracture. This study also demonstrates that the developed LBM can be effective in studying the solute transport in a rough fracture.

Investigations of Water Flow Behaviors Induced by Local Temperature Variations through a Single Rough Fracture for the Enhanced Geothermal Systems

In recent years,Enhanced Geothermal System(EGS)technologies have been applied to the geothermal resources production in the Hot Dry Rock(HDR).The core of EGS technologies is to adopt hydraulic fracturing in the reservoir to create a connected network of discrete fractures with the consideration of water as a working fluid for hydraulic fracturing and heat production.This paper investigates the characteristics of water flow behaviors through a single rough fracture under different temperature and pressure conditions.A single fracture model with rough fracture surfaces is constructed and then characterized,and influences of the anisotropic factor on the average tortuosity and frictional resistance coefficient of water flow through a single fracture with rough surfaces have been compared and analyzed.With consideration of other impacting factors(temperature,pressure,fracture roughness),the impact of mass flow rate has also been presented.Numerical simulation results present that changes of average tortuosity for water flow through a single rough facture are mainly affected by temperature.It can be observed that higher temperature leads to larger average tortuosity but the frictional resistance coefficient shows an opposite trend.As for pressure conditions,it is found that effects of pressure on average tortuosity and frictional resistance coefficient is very small,which can be neglected under high pressure conditions.Furthermore,the average tortuosity shows a progressively linear relationship with the mass flow rate.On the contrary,the frictional resistance coefficient has a negative relationship with the mass flow rate.It is revealed that when the mass flow rate reaches a critical point,the influences of temperature on the frictional resistance coefficient will be negligible.Comparisons of single rough fractures with different anisotropic factors show that values of average tortuosity and frictional resistance coefficient have positive relationships with the increase of anisotropic factors.

Revisit of advection-dispersion equation model with velocity-dependent dispersion in capturing tracer dynamics in single empty fractures

An accurate quantification of the contaminant transport through fractured media is critical for dealing with water-quality related scientific and engineering issues, where the dispersion coefficient is an important and elusive parameter for the solute transport modeling. Many previous studies show that the dispersion coefficient(D) in the standard advection-dispersion equation(ADE) model can be approximated by D=avλ(where a is the dispersivity), a formula to be revisited systematically in this study by laboratory experiments and model analysis. First, a series of tracer transport experiments in single empty fractures are conducted in cases of different hydraulic gradients. Second, the tracer breakthrough curves are determined by simulations based on the ADE model, to obtain the dispersion coefficients corresponding to various fracture roughnesses and flow velocities. A varying trend of λ is analyzed under different flow conditions. Results show that although the standard ADE model cannot be used to characterize the late-time tailing of the tracer BTCs, likely due to the solute retention, this simple model can simulate most of the solute mass dynamics moving through fractures and may therefore provide information for estimating the dispersion in parsimonious models appropriate for the non-Fickian transport. The following three conclusions are drawn:(1) the peak of the breakthrough curves comes earlier with increasing the roughness, according to the ADE simulation,(2) the value of λ generally decreases as the relative roughness of the fracture increases,(3) the value of λ is approximately equal to 2.0 when the dispersion is dominated by the molecular diffusion in the smooth fracture.

Numerical simulations of seepage flow in rough single rock fractures

To investigate the relationship between the structural characteristics and seepage flow behavior of rough single rock fractures,a set of single fracture physical models were produced using the WeierstrasseMandelbrot functions to test the seepage flow performance.Six single fractures,with various surface roughnesses characterized by fractal dimensions,were built using COMSOL multiphysics software.The fluid flow behavior through the rough fractures and the influences of the rough surfaces on the fluid flow behavior was then monitored.The numerical simulation indicates that there is a linear relationship between the average flow velocity over the entire flow path and the fractal dimension of the rough surface.It is shown that there is good agreement between the numerical results and the experimental data in terms of the properties a of the fluid flowing through the rough single rock fractures.

NUMERICAL STUDY OF NON-AQUEOUS PHASE LIQUID TRANSPORT IN A SINGLE FILLED FRACTURE BY LATTICE BOLTZMANN METHOD

In this article,the Non-Aqueous Phase Liquid（NAPL）transport in the single filled fracture was studied with the Shan-Chen multi-component multiphase Lattice Boltzmann Method（LBM）with special consideration of wettability effects.With the help of the model,the contact angle of the non-wetting phase and wetting phase interface at a solid wall could be adjusted.By considering a set of appropriate boundary conditions,the fractured conductivity was investigated in condition that the NAPL blocks the channels in the single filled fracture.In order to study the wettability effects on the NAPL transport,a constant driving force was introduced in the Shan-Chen multi-component multiphase LBM.Flow regimes with different wettabilities were discussed.Simulated results show that the LBM is a very instrumental method for simulating and studying the immiscible multiphase flow problems in single filled fracture.

Optimization of acid fracturing for a tight carbonate reservoir

In this study,acid fracturing treatments were simulated for a tight limestone reservoir within a shale formation using FRACPRO software.The purpose was to investigate the optimum acid fracturing design that leads to a higher fracture etched length and width,and higher fracture conductivity.Moreover,the impact of the rock-acid contact time and whether to consider a post-flush or fluids flowback,on the acid fracturing outcomes were also investigated.A simple geological model was constructed which consists of different lithological layers.Different acid fracturing design scenarios were considered starting from a single stage of acid injection to multi-stage treatment.In multi-stage acid treatment,alternate acid-slickwater injection was considered.Plain HCl acids with different concentrations and other acids that are already included in FRACPRO database were used.The results showed that the acid loss during post-flush is among the main problems of the acid fracturing in tight carbonate shale reservoir.For the single stage of acid injection,it is recommended to flow back the acid after well shut-in instead of considering a post-flush stage.The multi-stage alternate acid slickwater injection reduces or even eliminated the acid loss.However,it is recommended to inject a slickwater before well shut-in to reduce the rock-acid contact time,thus reducing the formation damage.The results also showed that the created fracture etched width decreases and the fracture etched length increases as the fracturing stages increase.In this study,because of the low carbonate layer permeability and compressive strength,a two-stage alternate 28%HCl and slickwater injection with a post-flush stage is recommended.