Effects of fracture evolution and non-Darcy flow on the thermal performance of enhanced geothermal system in 3D complex fractured rock

In fractured geothermal reservoirs,the fracture networks and internal fluid flow behaviors can significantly impact the thermal performance.In this study,we proposed a non-Darcy rough discrete fracture network(NR-DFN)model that can simultaneously consider the fracture evolution and non-Darcy flow dynamics in studying the thermo-hydro-mechanical(THM)coupling processes for heat extraction in geothermal reservoir.We further employed the model on the Habanero enhanced geothermal systems(EGS)project located in Australia.First,our findings illustrate a clear spatial-temporal variation in the thermal stress and pressure perturbations,as well as uneven spatial distribution of shear failure in 3D fracture networks.Activated shear failure is mainly concentrated in the first fracture cluster.Secondly,channeling flow have also been observed in DFNs during heat extraction and are further intensified by the expansion of fractures driven by thermal stresses.Moreover,the combined effect of non-Darcy flow and fracture evolution triggers a rapid decline in the resulting heat rate and temperature.The NR-DFN model framework and the Habanero EGS's results illustrate the importance of both fracture evolution and non-Darcy flow on the efficiency of EGS production and have the potential to promote the development of more sustainable and efficient EGS operations for stakeholders.

Non-Darcy Flow in Molding Sands

Darcy’s law is widely used to describe the flow in porous media in which there is a linear relationship between fluid velocity and pressure gradient. However, it has been found that for high numbers of Reynolds this law ceases to be valid. In this work, the Ergun equation is employed to consider the non-linearity of air velocity with the pressure gradient in casting sands. The contribution of non-linearity to the total flow in terms of a variable defined as a non-Darcy flow fraction is numerically quantified. In addition, the influence of the shape factor of the sand grains on the non-linear flow fraction is analyzed. It is found that for values of the Reynolds number less or equal than 1, the contribution of non-linearity for spherical particles is around 1.15%.

Characteristics of non-Darcy flow in low-permeability reservoirs

Well testing is recognized as an effective means of accurately obtaining the formation parameters of low-permeability reservoirs and effectively analyzing the deliverability.Well test models must comply with the particular characteristics of flow in low-permeability reservoirs in order to obtain reasonable well test interpretation.At present,non-Darcy flow in low-permeability reservoirs is attracting much attention.In this study,displacement tests were conducted on typical cores taken from low-permeability reservoirs.Two dimensionless variables were introduced to analyze the collected experimental data.The results of the dimensionless analysis show whether non-Darcy flow happens or not depends on the properties of fluid and porous media and the pressure differential.The combination of the above three parameters was named as dimensionless criteria coefficient（DCC）.When the value of the DCC was lower than a critical Reynolds number（CRN）,the flow could not be well described by Darcy＇s law（so-called non-Darcy flow）,when the DCC was higher than CRN,the flow obeyed Darcy＇s law.Finally,this paper establishes a transient mathematical model considering Darcy flow and non-Darcy flow in low-permeability reservoirs,and proposes a methodology to solve the model.The solution technique,which is based on the Boltzmann transformation,is well suited for solving the flow model of low-permeability reservoirs.Based on the typical curves analysis,it was found that the pressure and its derivative curves were determined by such parameters as non-Darcy flow index and the flow characteristics.The results can be used for well test analysis of low-permeability reservoirs.

Non-Darcy flow in oil accumulation (oil displacing water) and relative permeability and oil saturation characteristics of low-permeability sandstones

Hydrocarbon resources in low-permeability sandstones are very abundant and are extensively distributed. Low-permeability reservoirs show several unique characteristics, including lack of a definite trap boundary or caprock, limited buoyancy effect, complex oil-gas-water distribution, without obvious oil-gas-water interfaces, and relatively low oil （gas） saturation. Based on the simulation experiments of oil accumulation in low-permeability sandstone （oil displacing water）, we study the migration and accumulation characteristics of non-Darcy oil flow, and discuss the values and influencing factors of relative permeability which is a key parameter characterizing oil migration and accumulation in low-permeability sandstone. The results indicate that： 1） Oil migration （oil displacing water） in low- permeability sandstone shows non-Darcy percolation characteristics, and there is a threshold pressure gradient during oil migration and accumulation, which has a good negative correlation with permeability and apparent fluidity; 2） With decrease of permeability and apparent fluidity and increase of fluid viscosity, the percolation curve is closer to the pressure gradient axis and the threshold pressure gradient increases. When the apparent fluidity is more than 1.0, the percolation curve shows modified Darcy flow characteristics, while when the apparent fluidity up＂ non-Darcy percolation curve; 3） Oil-water is less than 1.0, the percolation curve is a ＂concave- two-phase relative permeability is affected by core permeability, fluid viscosity, apparent fluidity, and injection drive force; 4） The oil saturation of low- permeability sandstone reservoirs is mostly within 35%-60%, and the oil saturation also has a good positive correlation with the permeability and apparent fluidity.

A fractal approach to low velocity non-Darcy flow in a low permeability porous medium

In this paper, the mechanism for fluid flow at low velocity in a porous medium is analyzed based on plastic flow of oil in a reservoir and the fractal approach. The analytical expressions for flow rate and velocity of non-Newtonian fluid flow in the low permeability porous medium are derived, and the threshold pressure gradient （TPG） is also obtained. It is notable that the TPG （J） and permeability （K） of the porous medium analytically exhibit the scaling behavior J ~ K-D＇r/（l＋Or）, where DT is the fractal dimension for tortuous capillaries. The fractal characteristics of tortuosity for capillaries should be considered in analysis of non-Darcy flow in a low permeability porous medium. The model predictions of TPG show good agreement with those obtained by the available expression and experimental data. The proposed model may be conducible to a better understanding of the mechanism for nonlinear flow in the low permeability porous medium.

Experimental Study of Seepage Properties of Non-Darcy Flow in Granular Coal Gangues

By using the steady-state seepage method, a patent seepage device together with the MTS815.02 Rock Mechanics Test System is used to test the seepage properties of non-Darcy flow in a granular gangue with five different grain sizes during the compaction. The experimental results show that the seepage properties are not only related to the stress or displacement level, but also to the grain size, the pore structure of the granular gangue, and the current porosity The permeability and the non-Darcy flow coefficient can be fitted respectively by the cubic polynomials and the power functions of the porosity, Formally, the flow in granular gangue satisfies the Forchheimer＇s binomial flow, but under the great axial and confining pressure and owing to the grain＇s crushing, the flow in granular gangues is different from that in rock-fills which are naturallv oiled un. As a result, the non-Darer flow coefficient may be negative.

Non-Darcy flows in layered porous media(LPMs)with contrasting pore space structures

Compared to single layer porous media,fluid flow through layered porous media(LPMs)with contrasting pore space structures is more complex.This study constructed three-dimensional(3-D)pore-scale LPMs with different grain size ratios of 1.20,1.47,and 1.76.The flow behavior in the constructed LPMs and single layer porous media was numerically investigated.A total of 178 numerical experimental data were collected in LPMs and single layer porous media.In all cases,two different flow regimes(i.e.,Darcy and Non-Darcy)were observed.The influence of the interface of layers on Non-Darcy flow behavior in LPMs was analyzed based pore-scale flow data.It was found that the available correlations based on single layer porous media fail to predict the flow behavior in LPMs,especially for LPM with large grain size ratio.The effective permeability,which incorporated the influence of the interface is more accurate than the Kozeny-Carman equation for estimating the Darcy permeability of LPMs.The inertial pressure loss in LPMs,which determines the onset of the Non-Darcy flow,was underestimated when using a power law expression of mean grain size.The constant B,an empirical value in the classical Ergun equation,typically equals 1.75.The inertial pressure loss in LPMs can be significantly different from it in single lager porous media.For Non-Darcy flow in LPMs,it is necessary to consider a modified larger constant B to improve the accuracy of the Ergun empirical equation.

One-dimensional non-Darcy flow in a semi-infinite porous media:a multiphase implicit Stefan problem with phases divided by hydraulic gradients

One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.

Research advances in non-Darcy flow in low permeability media

More and more experimental results show that Darcy’s law is not fully applicable in low permeability media,and non-Darcy flow has been identified.In this paper we reviewed the research of non-Darcy flow experiments in low-permeability media in recent decades,discuss the existence of non-Darcy flow,and summarize its constitutive equations.The reasons for the threshold gradient were also discussed and summarized for the criterion of the critical point of non-Darcy flow.On this basis,the future development of non-Darcy flow experiments in the rock and clay media were discussed,in order to provide a certain reference for subsequent research on seepage laws in low permeability media.

Investigation of Barree-Conway non-Darcy flow effects on coalbed methane production

Coalbed gas non-Darcy flow has been observed in high permeable fracture systems,and some mathematical and numerical models have been proposed to study the effects of non-Darcy flow using Forchheimer non-Darcy model.However,experimental results show that the assumption of a constant Forchheimer factor may cause some limitations in using Forchheimer model to describe non-Darcy flow in porous media.In order to investigate the effects of non-Darcy flow on coalbed methane production,this work presents a more general coalbed gas non-Darcy flow model according to Barree-Conway equation,which could describe the entire range of relationships between flow velocity and pressure gradient from low to high flow velocity.An expanded mixed finite element method is introduced to solve the coalbed gas non-Darcy flow model,in which the gas pressure and velocity can be approximated simultaneously.Error estimate results indicate that pressure and velocity could achieve first-order convergence rate.Non-Darcy simulation results indicate that the non-Darcy effect is significant in the zone near the wellbore,and with the distance from the wellbore increasing,the non-Darcy effect becomes weak gradually.From simulation results,we have also found that the non-Darcy effect is more significant at a lower bottom-hole pressure,and the gas production from non-Darcy flow is lower than the production from Darcy flow under the same permeable condition.

Non-Darcy flow seepage characteristics of saturated broken rocks under compression with lateral constraint

Using an MTS816.03 test system and self-designed seepage apparatus, seepage tests of saturated broken rocks were conducted, and the influence of lithology, axial stress, grain size distribution and loading rate on seepage characteristics was analyzed. The results show that： （1） Under the same axial stress （12 MPa）, the permeability of different lithologic samples increases in the order： gangue 〈 mudstone 〈 sandstone 〈 limestone. The permeability of gangue is 3 magnitudes lower than that of limestone. The absolute value of the non-Darcy coefficient β increases in the order： limestone 〈 sandstone 〈 mudstone 〈 gangue. The non-Darcy coefficient β of limestone, which is positive, is 5 magnitudes lower than that of gangue. （2） With increasing axial stress, the permeability of saturated broken sandstone decreases, and the absolute value of the non-Darcy coefficient β increases. After the axial stress exceeds 12 MPa, the curves of permeability and non-Darcy coefficient β all tend to be stable. （3） With increasing Talbol power exponent, the permeability increases, and the absolute value of the non-Darcy coefficient β decreases. （4） With increasing loading, the permeability increases, and the absolute value of the non-Darcy coefficient β decreases. When the loading rate is 0.5 kN/s, the non-Darcy coefficient β is positive.

Efect of double dispersion on non-Darcy mixed convective flow over vertical surface embedded in porous medium

A numerical study of a non-Darcy mixed convective heat and mass transfer flow over a vertical surface embedded in a dispersion, melting, and thermal radiation is porous medium under the effects of double investigated. The set of governing boundary layer equations and the boundary conditions is transformed into a set of coupled nonlinear ordinary differential equations with the relevant boundary conditions. The transformed equations are solved numerically by using the Chebyshev pseudospectral method. Comparisons of the present results with the existing results in the literature are made, and good agreement is found. Numerical results for the velocity, temperature, concentration profiles, and local Nusselt and Sherwood numbers are discussed for various values of physical parameters.

Investigation of low water recovery based on gas-water two-phase low-velocity Non-Darcy flow model for hydraulically fractured horizontal wells in shale

Various mechanisms are employed to interpret the low water recovery during the shale-gas production period,such as extra-trapped water in the fracture network,water imbibition due to osmotic pressure and capillary pressure.These lead to the difficulty of water flow,which could be described by lowvelocity non-Darcy's law known as threshold pressure gradient(TPG).In this paper we firstly employ the low-velocity non-Darcy's law to describe the water flow and use Darcy flow accounting for slip flow and free molecular flow mechanisms to model gas flow in the shale formation.The sensitive study using numerical simulation shows that the proposed flow model could model the low fracturing liquid recovery and that large pseudo TPG leads to lower fracturing liquid recovery.Thus,the proposed model would give new insight to model the low water recovery in shale formations.

Gas Condensate Two Phase Flow Performance in Porous Media Considering Capillary Number and Non-Darcy Effects

Retrograde condensation frequently occurs during the development of gas condensate reservoirs. The loss of productivity is often observed due to the reduced relative permeability to gas as condensate accumulates near the well bore region. How to describe the condensate blockage effect exactly has been a continuous research topic. However, up to now, the present methods usually over-estimate or underestimate the productivity reduction due to an incorrect understanding of the mechanism of flow in porous medium, which inevitably results in an inaccurate prediction of production performance. It has been found in recent numerous theoretical and experimental studies that capillary number and non-Darcy flow have significant influence on relative permeability in regions near the well bore. The two effects impose opposite impacts on production performance, thus leading to gas condensate flow showing characteristics different from general understanding. It is significant for prediction of performance in gas condensate wells to understand the two effects exactly. The aim of the paper is to describe and analyze the flow dynamics in porous media accurately during the production of gas condensate reservoirs. Based on the description of three-zone flow mechanism, capillary number and non-Darcy effect are incorporated in the analysis of relative permeability, making it possible to describe the effect of condensate blockage. The effect of capillary number and inertial flow on gas and condensate relative permeability is analyzed in detail. Novel Inflow Performance Relation (IPR) models considering high velocity effects are formulated and the contrast analysis of different IPR models is conducted. The result shows that the proposed method can help predict the production performance and productivity more accurately than conventional methods.

4D-Flow MRI在肥厚型心肌病左室流出道血流评估中的价值探索

目的 探索四维血流(four-dimensional flow,4D-Flow)磁共振成像(magnetic resonance imaging,MRI)技术在左心室腔内应用的可行性。材料与方法 本研究为前瞻性、横断面研究,纳入2022年8月至2023年1月于我院接受心脏MRI检查的21例肥厚型心肌病患者,采用3.0 T MRI扫描仪进行二维血流(tow-dimensional flow,2D-Flow)及4D-Flow成像,收集患者一周内进行的超声心动图检查结果。采用组内相关系数(inter-class correlation coefficient,ICC)、变异系数(coefficients of variation,COV)及Bland-Altman分析比较2D-Flow、4D-Flow评估左室流出道峰值流速的可重复性及一致性,并通过Pearson相关性分析探究二者与超声心动图测量结果的关系。结果 2D-Flow和4D-Flow观察者内/观察者间的ICC分别为0.999/0.999和0.995/0.992,COV分别为0.5%/0.5%和2.4%/2.6%。4D-Flow与超声心动图的测量结果呈中度相关,相关系数r值为0.574(P=0.006),但一致性较差,ICC为0.375(P=0.013)。2D-Flow与4D-Flow和超声心动图间无显著的一致性及相关性。结论 4D-Flow技术能够可视化心腔内血流模式,对左室流出道峰值流速的测量具有高度可重复性,且与超声心动图的测量结果具有显著的一致性。

A numerical study of non-Darcy flow in EGS heat reservoirs during heat extraction

Underground non-Darcy fluid flow has been observed and investigated for decades in the petroleum industry. It is deduced by analogy that the fluid flow in enhanced geothermal system (EGS) heat reservoirs may also be in the non-Darcy regime under some conditions. In this paper, a transient 3D model was presented, taking into consideration the non-Darcy fluid flow in EGS heat reservoirs, to simulate the EGS long-term heat extraction process. Then, the non-Darcy flow behavior in water- and supercritical CO2 (SCCO2)-based EGSs was simulated and discussed. It is found that non-Darcy effects decrease the mass flow rate of the fluid injected and reduce the heat extraction rate of EGS as a flow resistance in addition to the Darcy resistance which is imposed to the seepage flow in EGS heat reservoirs. Compared with the water-EGS, the SCCO2-EGS are more prone to experiencing much stronger non-Darcy flow due to the much larger mobility of the SCCO2. The non-Darcy flow in SCCO2- EGSs may thus greatly reduce their heat extraction performance. Further, a criterion was analyzed and proposed to judge the onset of the non-Darcy flow in EGS heat reservoirs. The fluid flow rate and the initial thermal state of the reservoir were taken and the characteristic Forchheimer number of an EGS was calculated. If the calculated Forchheimer number is larger than 0.2, the fluid flow in EGS heat reservoirs experiences non-negligible non-Darcy flow characteristic.

One-dimensional large-strain consolidation of soft clay with non-Darcian flow and nonlinear compression and permeability of soil

Geometrical nonlinearity of the soft soil and the deviation of water flow in the soft clay from Darcy's law have been well recognized in practice. However, the theory of consolidation, which can account for both the geometrical nonlinearity and the non-Darcian flow, has not been reported so far. In this contribution, a model for the consolidation of soft clay which can allow for these two factors simultaneously is proposed. Utilizing the finite difference method, the numerical model for this problem is developed. With the numerical model, the effects of the geometrical nonlinearity and the non-Darcian flow on the consolidation of the soft soil are investigated. The results show that when the self-weight stress is calculated by the same method, the rate of the non-Darcian consolidation for the large-strain case is larger than that for the small-strain case, but the difference between them is limited. However, the difference between the consolidation rates caused by the non-Darcian and Darcian flows is significant. Therefore, when the geometrical nonlinearity of the soft clay is considered in calculating the consolidation settlement, due to the complexity of the large-strain assumption, the small-strain assumption can be used to replace it if the self-weight stress for the small-strain assumption is calculated by considering its sedimentation. However, due to the aforementioned large difference between the consolidation rates with consideration of the non-Darcian flow in soft clay or not, it is better to consider the non-Darcian flow law for both the small and large stain assumptions.

针对无线控制环境的OpenFlow流表申请优化机制

在无线控制环境下,受限于控制通道的带宽和时延,OpenFlow交换机通过Packet-In消息申请流表的过程耗时较长,这个时段内交换机可能收到同一条流的大量后续报文,并分别产生与首包报文完全相同的Packet-In消息发往控制器,明显浪费网络资源。同时,针对每条流独立发送一个Packet-In报文的方式容易导致控制通道负载以小包为主、包头开销占比过大、网络资源利用率不高的问题。因此,针对无线控制条件下报文粒度Packet-In消息触发和离散化Packet-In消息发送机制存在的问题,提出了一种基于流粒度实现Packet-In消息触发和Packet-In消息聚合发送的方案,并通过仿真试验,验证了方案的效果。

Non-Darcian Flow Toward a Finite-Diameter Vertical Well in a Confined Aquifer

Non-Darcian radial flow toward a finite-diameter, fully penetrating well in a confined aquifer was analyzed on the basis of the Izbash equation with consideration of the wellbore storage effect. We derived semi-analytical solutions of drawdown by using the Boltzmann transform, and obtained approximate analytical solutions of the drawdown at early and late times. MATLAB programs were developed to facilitate computation of the semi-analytical solutions. The turbulence factor v which was directly related to the pumping rate appeared to have negligible influence upon the wellbore well function at early times, but imposed significant influence at intermediate and late times. However, the turbulence factor v imposed non-negligible influence upon the aquifer well function during the entire pumping period, provided that the observation point was not sufficiently close to the wellbore. Sensitivity analysis indicated that the power index n in the Izbash equation had less influence on the type curves at the face of the pumping wellbore, but had much greater influence upon the well function in the aquifer. As the n values increased, the drawdown in the aquifer decreased at early times and increased at late times. The Boltzmann transformation could only be used in an approximate sense for radial non-Darcian flow problems. This approximation would provide accurate solutions at early times, and introduce small but consistent discrepancies at intermediate and late times for the wellbore well function.

基于 Moldflow 的汽车中控台框架翘曲变形分析及优化

以某汽车中控台框架为研究对象,测量试模样品发现其翘曲变形量超过了装配要求。通过Moldflow软件模拟了该塑件实际的注塑过程,翘曲变形量的模拟值与实测平均值的最大误差为5.98%,发现该塑件翘曲变形的主要因素为冷却不均和收缩不均。本文在原物料中添加质量分数为25%的玻璃纤维以及优化工艺参数后,翘曲变形量的模拟值与初始方案相比降低了86.22%。试模验证表明,优化后的翘曲变形量模拟值与实测平均值的最大误差为4.35%,证明了Moldflow软件模拟分析的准确性。试模后各检测点的最大翘曲变形量降到了1.6 mm以下,较优化之前降低了80%以上,为类似大型复杂注塑件的翘曲变形分析及优化提供了思路。