QQ群聊

• CN 62-1112/TF
• ISSN 1005-2518
• 创刊于1988年

## 基于立方定律的断层流—热耦合数值计算方法

1.昆明理工大学国土资源工程学院，云南 昆明 650031

2.昆明理工大学城市学院，云南 昆明 650051

## Numerical Calculation Method of Fault Flow-Thermal Coupling Based on Cubic Law

CHEN Gang,1, MA Ling,2, GONG Hongsheng1

1.Faculty of Land Resources Engineering，Kunming University of Science and Technology，Kunming 650031，Yunnan，China

2.City College，Kunming University of Science and Technology，Kunming 650051，Yunnan，China

 基金资助: 国家自然科学基金项目“基于裂隙三维空间分布的矿区地下水流动模拟研究”.  41562017

Received: 2020-07-09   Revised: 2020-08-03   Online: 2021-01-29

Abstract

For the ore-forming process of hydrothermal deposits，the seepage of fluids in the rock matrix and fissures （faults） produces material and energy transmission，and forms orebodies at specific locations with changes in temperature and pressure.Because the width of the fault is much smaller than the dimension of its extension direction，it causes problems such as difficulty in modeling numerical models and low calculation efficiency.According to the geometric characteristics of the fault，it can be generalized into a space surface to reduce the difficulty of modeling.The generalized fault uses the cubic law in rock mass fracture seepage theory to calculate the fault seepage problem.The seepage of hydrothermal fluid is not limited to faults，but also occurs in bedrock，and this process is calculated using Darcy’s law. The fracture flow in the fault and the Darcy flow in the bedrock interact with each other.In order to ensure the continuity of the pressure，velocity，mass，and energy of the seepage field in the numerical model calculation domain，the flow-heat coupling calculation is required.The purpose of this study is to verify the feasibility and rationality of the generalization method of the fault space surface，and to solve the problem of flow-heat coupling between the fissure flow in the fault and the Darcy flow in the bedrock.The viscosity of fluid has the property of changing with temperature.This article will discuss whether the change of viscosity has an effect on the calculation result of the numerical model initially.Based on the theoretical formula of cubic law，the formula is derived according to the characteristics of small fault thickness，and the flow-heat coupling control equation of fracture flow and Darcy flow is obtained. In order to verify the rationality of the control equations，numerical model experiments are used for verification and analysis.After analysis，it is considered that the method of calculating the seepage of the fault using cubic law is feasible when the internal structure of the fault is not taken into consideration，which can reduce the difficulty of modeling the numerical model.Because the fault uses a spatial surface，the reduction of the dimension compared to the overall model also brings increased computing efficiency. After analyzing the results of the numerical experiments，it is considered that the coupling control equation is reasonable and effective for the calculation of the flow-heat coupling between the bedrock and the fault，which is in accordance with the laws of seepage and heat conduction.Based on the original experimental numerical model，a model in which the viscosity coefficient of the fluid does not change with temperature is established，and the change curves of mass and heat conduction flux are compared.It is found whether the change of the fluid viscosity is considered to have a significant effect on the calculation result of the numerical model.

Keywords： cubic law ; fault ; flow-thermal coupling ; numerical calculation ; hydrothermal deposit ; fissure seepage

CHEN Gang, MA Ling, GONG Hongsheng. Numerical Calculation Method of Fault Flow-Thermal Coupling Based on Cubic Law[J]. Gold Science and Technology, 2020, 28(6): 846-858 doi:10.11872/j.issn.1005-2518.2020.06.122

## 2个光滑平板之间的隙宽为常数$a$，则通过等宽缝隙的流量q与隙宽$a$之间的关系［14］表示为

$q=ga312νJ$

$κ=a212$

$κ=a212f$

## 2 达西流与裂隙流耦合控制方程

$qF=-κFμa∇TPF+ρg∇TD$

$a∂∂tεFρ+∇T⋅ρqF=aQFm$

### 图1

Fig.1   Conceptual model of fracture flow-Darcy flow coupling

$a∂∂tεFρ+∇T⋅ρqF=aQFmup+aQFmdown$

Pup=Pdown =PF （8）

$a∂CFTF∂t+aρwcwu⃗⃗w∇TTF-a∇TλF∇TTF=fe$

$uw=-κFμ∇TPF+ρg∇TD$

$fupe=ρwcw-κFμ∂Pup∂nupTup-λeq∂Tup∂nupfdowne=ρwcw-κFμ∂Pdown∂ndonwTdown-λeq∂Tdown∂ndown$

Tup=Tdown=TF （12）

### 图2

Fig.2   Variation curve of water viscosity coefficient[30]

### 3.1 模型求解方法及验证

Lauwerier16在研究油层传热问题时，为便于开展研究，提出以下假设条件：

（1）水只在裂隙中进行层流流动，且在裂隙中的水温沿y轴方向保持相同，即裂隙水的温度仅仅与x坐标相关；

（2）裂隙宽度为2b，裂隙内水的流速恒定；

（3）裂隙及上下两层岩石初始温度为T0，注入水的温度为Tin

（4）忽略岩石基质中平行于裂隙方向的热传导；

（5）裂隙周围岩石基质为无限厚。

$Tx,t=T0+(Tin-T0)erfcλxx/ρwCwdf2λs/ρsCsufuft-xUt-xuf$

### 图3

Fig.3   Temperature change curve inside the fracture

### 图4

（a）模型结构 （b）模型中测点分布位置

Fig.4   Geological model structure and spatial location of measuring points

Table 1  List of formation parameters

### 图5

Fig.5   Temperature and pressure curve applied by the model

### 图6

Fig.6   Flux curves of fault plane of numerical model

### 图7

Fig.7   Multi-point temperature and pressure curves in the second layer of bedrock

### 图8

Fig.8   Slices of temperature distribution in different time models（temperature unit：℃）

### 图9

Fig.9   Ratio curves of flux and heat conduction flux

## 4 结论

（1）裂隙和岩石基岩中的渗流有着不同的渗流理论，本文得出的裂隙流和达西流耦合控制方程，保证了数值模拟过程中需要保证数值模型计算域内渗流场压力、速度、质量和能量的连续性；通过对数值模型的分析，认为耦合控制方程可以合理、准确地实现裂隙流和达西流的耦合。

（2）由于裂隙和断层的空间形态具有宽度远小于延伸尺寸的特性，本文将断层简化为曲面，基于热—流耦合控制方程，使用数值方法计算了涵盖达西渗流、裂隙流和热传导的成矿动力学过程的数值模型；由于裂隙流内渗流速度快、热传导效率高，对模型最终的温度场和能量传导影响十分显著。

（3）经过数值模型的验证认为，在不考虑断层内部结构时，成矿动力学模拟中将断层概化为无几何厚度的空间曲面，使用裂隙渗流理论进行计算是可行的。

http://www.goldsci.ac.cn/article/2020/1005-2518/1005-2518-2020-28-6-846.shtml

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Liu L MZhao Y LZhao C B.

Coupled geodynamics in the formation of Cu skarn deposits in the Tongling-Anqing district，China：Computational modeling and implications for exploration

［J］.Journal of Geochemical Exploration，20101061/2/3）：146-155.

［J］.大地构造与成矿学，2011351）：128-136.

Zhao YilaiLiu Mingliang.

3D-numerical modeling of coupled geodynamic processes and mineralization at the contact zones of complex plutons：Example form the Anqing deposit，Anhui Province，China

［J］. Geotectonica et Metallogenia，2011351）：128-136.

［J］.地质学刊，2019433）：506-512.

Zhu JingChen Jiangping.

Research status of FLAC3D-based mineralization process simulation

［J］.Journal of Geology，2019433）：506-512.

［J］.地质力学学报，2019255）：163-169.

Liu Xiangchong.

Finite-element simulations of structure-fluid coupling：A case study in vein-type tungsten deposits

［J］.Journal of Geomechanics，2019255）：163-169.

［J］.合肥工业大学学报（自然科学版），2019423）：346-354.

Dai WenqiangLi XiaohuiYuan Fenget al.

Numerical simulation of formation process of typical skarn minerals in Anqing copper deposit

［J］.Journal of Hefei University of Technology（Natural Science），2019423）：346-354.

［J］.中国科学（D辑：地球科学），2008385）：646-652

Zhao ChongbinHobbs B EOrd A.

Investigating dynamic mechanisms of geological phenomena using methodology of computational geosciences：An example of equal-distant mineralization in a fault

［J］.Science in China （Series D：Earth Sciences），2008517）：947-954

［J］.地学前缘，2011189）：1-18.

Chi GuoxiangXue Chunji.

Principles，methods and applications of hydrodynamic studies of mineralization

［J］.Earth Science Frontiers，2011189）：1-18.

Guilert J MPark C F.The Geology of Ore Deposites［M］.New YorkFreeman and Company1986985.

［J］.大地构造与成矿学，2019432）：33-45.

Li RuihongAn PingYun Mengheet al.

3D ore-controlling structural model and numerical simulation of Jiaojia fault zone：A case study of the Xincheng gold deposit

［J］. Geotectonica et Metallogenian，2019432）：33-45.

［J］.西北地质，2018511）：88-103.

Yang YongchunLiu JiajunWang Xueyinet al.

Geochemical characteristics and structural ore-control mechanism about different structural-lithofacies zones of the Dishuishan gold deposit in Gansu Province

［J］.Northwestern Geology，2018511）：88-103.

［J］.西北地质，2019521）：228-238.

Zhao ShaopanXu ShukuiXu Zongjiaoet al.

Stage division and ore-bearing evaluation of fault tectonic in the Longmendian silver deposit，Luoning，Henan Province

［J］.Northwestern Geology，2019521）：228-238.

Liu ChunxueNi Chunzhong Leiet al.Multi-Scale Spatial Distribution Simulation of Directional Variables in Geosciences［M］.BeijingScience Press2017.

Juanes RSamper JMolinero J.

A general and efficient formulation of fractures and boundary conditions in the finite element method

［J］.International Journal for Numerical Methods in Engineering，20025412）：1751-1774.

Zhang Youtian.Rock Hydraulics and Engineering［M］.BeijingChina Water Conservancy and Hydropower Press2004.

［J］.岩石力学与工程学报，2014331）：43-51.

Chen BiguangSong ErxiangCheng Xiaohui.

A numerical method for discreate fracture network model for flow and heat transfer in tow-dimensinal fractured rocks

［J］.Chinese Journal of Rock Mechanics and Engineering，2014331）：43-51.

Lauwerier H A.

The transport of heat in an oil layer caused by the injection of hot fluid

［J］.Applied Scientific Research，195552）：145-150.

Pruess KBodvasson G S.

Thermal effects of reinjection in geothermal reservoirs with major vertical fractures

［J］.Journal of Petroleum Technology，1984369）：1567-1578.

Cheng A H DGhassemi ADetournay E.

Integral equation solution of heat extraction from a fracture in hot dry rock

［J］.International Journal for Numerical and Analytical Methods in Geomechanics，20012513）：1327-1338.

［J］.岩石力学与工程学报，1999182）：119-123.

Zhao Jian.

Experiental study of flow-rock heat transfer in rock fractures

［J］.Chinese Journal of Rock Mechanics and Engineering，1999182）：119-123.

［J］.岩石力学与工程学报，20022112）：1751-1755.

Zhao YangshengWang RuifengHu Yaoqing.

3D numerical simulation for coupled THM of rock matrix-fractured media in heat extraction in HDR

［J］.Chinese Journal of Rock Mechanics and Engineering，20022112）：1751-1755.

［J］.岩土力学，2011328）：2507-2511.

Zhang ShuguangLi ZhijianXu Yihonget al.

Three-dimensional numerical simulation and analysis of fluid-heat coupling heat-transfer in fractured rock mass

［J］.Rock and Soil Mechanics，2011328）：2507-2511.

［J］.北京工业大学学报，20164210）：1560-1564.

Tang ZhiweiMi ChanghuaZhang Xuefenget al.

Numerical simulation and analysis of the coupled for heat-fluid-solid in enhanced geothermal systems

［J］.Journal of Beijing University of Technology，20164210）：1560-1564.

［J］.地球物理学进展，2017326）：2374-2382.

Qu ZhanqingZhang WeiGuo Tiankuiet al.

Research on the effect of geothermal reservoir and bedding fractures on geothermal deliverability based on COMSOL

［J］.Progress in Geophysics，2017326）：2374-2382.

［J］.地球物理学进展，2019342）：668-675.

Zhang WeiSun JiangQu Zhanqinget al.

Thermo-hydro-mechanical coupling model and comprehensive evaluation method of high temperature geothermal extraction

［J］.Progress in Geophysics，2019342） ：668-675.

David T S.

Anisotropic permeability of fractured media

［J］.Water Resources Research，1969512）：1273-1289.

Louis C.

A study of groundwater flow in jointed rock and its influence on the stability of rock mass

［R］. LondonImperial College of Science and Technology1969.

Wu Yanqing.Geotechnical Hydraulics［M］.BeijingScience Press2009.

［J］ .岩土力学，20103111）：3361-3366.

Su BaoyuZhang WenjieSheng Jinchanget al.

Study of permeability in single fracture under effects of coupled fluid flow and chemical dissolution

［J］.Rock and Soil Mechanics，20103111）：3361-3366.

［J］.水科学进展，2002131）：61-68.

Wang YuanSu Baoyu.

Research on the behavior of fluid flow in a single fracture ant its equivalent hydraulic aperture

Incropera P FDewitt P DBergman L Tet al. Fundamentals of Heat and Mass Transfer ［M］.7th eds. New YorkJohn Wiley & Sons2011.

［J］.岩石学报，2010269）：2869-2878.

Liu MingliangZhou RuichaoZhao Chongbin.

Constraints of tectonic stress regime on mineralization system related to the hypabyssal intrusion：Implication from the computational modeling experiments on the geodynamics during cooling process of the Yuenshan intrusion in Anqing district，China

［J］. Acta Petrologica Sinica，2010269）：2869-2878.

/

 〈 〉