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%.

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.

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.

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.

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.

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.

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.

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.

Nonlinear convection in a non-Darcy porous medium

In this paper, the natural convection in a non-Darcy porous medium is studied using a temperature-concentration-dependent density relation. The effect of the two parameters responsible for the nonlinear convection is analyzed for different values of the inertial parameter, dispersion parameters, Rayleigh number, Lewis number, Soret number, and Dufour number. In the aiding buoyancy, the tangential velocity increases steeply with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter when the inertial effect is zero. However, when the inertial effect is non-zero, the effect of the nonlinear temperature parameter and the nonlinear concentration parameter on the tangential velocity is marginal. The concentration distribution varies appreciably and spreads in different ranges for different values of the double dispersion parameters, the inertial effect parameter, and also for the parameters which control the nonlinear temperature and the nonlinear concentration. Heat and mass transfer varies extensively with an increase in the nonlinear temperature parameter and the nonlinear concentration parameter depending on Dacry and non-Darcy porous media. The variation in heat and mass transfer when all the effects, i.e., the inertial effect, double dispersion ef- fects, and Soret and Dufour effects, are simultaneously zero and non-zero. The combined effects of the nonlinear temperature parameter, the nonlinear concentration parameter and buoyancy are analyzed. The effect of the nonlinear temperature parameter and the nonlinear concentration parameter and also the cross diffusion effects on heat and mass transfer are observed to be more in Darcy porous media compared with those in non- Darcy porous media. In the opposing buoyancy, the effect of the temperature parameter is to increase the heat and mass transfer rate, whereas that of the concentration parameter is to decrease.

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.

Numerical investigation of Dufour and Soret effects on unsteadyMHD natural convection flow past vertical plate embedded innon-Darcy porous medium

The Dufour and Soret effects on the unsteady twodimensional magnetonyaro dynamics （MHD） doublediffusive free convective flow of an electrically conducting fluid past a vertical plate embedded in a nonDarcy porous medium are investigated numeri cally. The governing nonlinear dimensionless equations are solved by an implicit finite difference scheme of the CrankNicolson type with a tridiagonal matrix manipulation. The effects of various parameters entering into the problem on the unsteady dimension less velocity, temperature, and concentration profiles are studied in detail. Furthermore, the time variation of the skin friction coefficient, the Nusselt number, and the Sherwood number is presented and analyzed. The results show that the unsteady velocity, tem perature, and concentration profiles are substantially influenced by the Dufour and Soret effects. When the Dufour number increases or the Soret number decreases, both the skin friction and the Sherwood number decrease, while the Nusselt number increases. It is found that, when the magnetic parameter increases, the velocity and the temperature decrease in the boundary layer.

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.

Hydromagnetic thin film flow of Casson fluid in non-Darcy porous medium with Joule dissipation and Navier＇s partial slip

In this paper, the effects of viscous and Ohmic heating and heat genera- tion/absorption on magnetohydrodynamic flow of an electrically conducting Casson thin film fluid over an unsteady horizontal stretching sheet in a non-Darcy porous medium are investigated. The fluid is assumed to slip along the boundary of the sheet. Similar- ity transformation is used to translate the governing partial differential equations into ordinary differential equations. A shooting technique in conjunction with the 4th order Runge-Kutta method is used to solve the transformed equations. Computations are car- ried out for velocity and temperature of the fluid thin film along with local skin friction coefficient and local Nusselt number for a range of values of pertinent flow parameters. It is observed that the Casson parameter has the ability to enhance free surface velocity and film thickness, whereas the Forchheimer parameter, which, is responsible for the inertial drag has an adverse effect on the fluid velocity inside the film. The velocity slip along the boundary tends to decrease the fluid velocity. This investigation has various applications in engineering and in practical problems such as very large scale integration （VLSI） of electronic chips and film coating.

Non-Darcy Mixed Convection between Differentially Heated Vertical Walls Filled with a Porous Material: Application of New Modified Adomian Decomposition Method

This paper presents non-Darcy mixed convective flow of an incompressible and viscous fluid in a differentially heated vertical channel filled with a porous material in the presence of a temperature dependent source/sink. The analytical solution of fourth order non-linear ordinary differential equation for temperature field, which is formed by eliminating velocity field from system of governing equations in non-dimensional form, is obtained by using new modified Adomian decomposition method (NMADM) in terms of various parameters. In order to illustrate the interactive influences of governing parameters on the temperature and velocity fields, a numerical study of the analytical solution is performed with respect to three categories of transport processes i) when forced convection is dominated, ii) when forced and natural convection are equal and iii) when natural convection is dominated. Analysis of all categories has revealed that the temperature and velocity profiles are increasing function of modified Darcy number while decreasing function of Forchheimer number.

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.

Laminar MHD natural convection of nanofluid containing gyrotactic microorganisms over vertical wavy surface saturated non-Darcian porous media

Magnetohydrodynamic （MHD） bioconvection of an incompressible electrically conducting nanofluid near a vertical wavy surface saturated porous medium containing both nanoparticle and gyrotactic microorganisms is investigated. The nanofluid is represented by a model that includes both Brownian motion and thermophoresis effects. A suitable set of non-dimensional variables are used to transform the governing boundary layer equations into a dimensionless form. The resulting nonlinear system is mapped to the vertical flat plate domain, and a non-similar solution is used to the obtained equations. The obtained non-similar system is then solved numerically using the fourth-order Runge-Kutta method. The influence of various physical parameters on the local Nusselt number, the local Sherwood number, the local density number of the motile microorganisms, the dimensionless velocity, the dimensionless temperature, and the rescaled density of motile microorganisms is studied. It is found that the local Nusselt number, the local Sherwood number, and the local density number of the motile microorganisms decrease by increasing either the Grashof number or the magnetic field parameter.

Combined Effect of Magnetic Field and Thermal Dispersion on a Non-Darcy Mixed Convection

This paper is devoted to investigate the influences of thermal dispersion and magnetic field on a hot semi-infinite vertical porous plate embedded in a saturated Darcy-Forchheimer-Brinkman porous medium. The coefficient of thermal diffusivity has been assumed to be the sum of the molecular diffusivity and the dynamic diffusivity due to mechanical dispersion. The effects of transverse magnetic field parameter (Hartmann number Ha), Reynolds number Re (different velocities), Prandtl number Pr (different types of fluids) and dispersion parameter on the wall shear stress and the heat transfer rate are discussed.