The permeability and location horizontal and vertical of the low- or high-permeability inclusions were set to be variable. Their effects on free convection were quantitatively assessed using three assessment indicators, including the total solute mass TM , Sherwood number Sh , and solute center of gravity COG.
The expected results will advance our understanding of unstable density-drive flow and solute transport processes under the effect of structured heterogeneity. The Elder problem is a classic example of free convection phenomena [ 33 , 34 ]. The Elder problem is chosen here for two reasons: 1 it is a classic example for free convection in which the flow is purely driven by density gradients and 2 the Elder problem, especially under the homogeneous condition, has been extensively investigated and discussed in both experimental and numerical simulation researches.
The Elder problem used in this study Figure 1 is modified from the one used in [ 21 ], by transforming the fixed-concentration boundary at the bottom into a no-flux boundary while keeping the domain geometry and boundary conditions unchanged. This transformation makes the solute transport process more realistic under natural conditions.
Initially, the domain is filled with freshwater with a hydrostatic head distribution. The domain boundaries are impermeable except that constant heads are prescribed at the top left and right corners as two outlets for fluid and solute.
A fixed concentration boundary condition [ 20 , 23 ] is imposed at the source zone. As abundant solutes diffused from the source zone accumulate at the top boundary, flow is generated by density gradients in the model domain. The solute transport model involving advection, molecular diffusion, and mechanical dispersion is given as where ML -3 is dissolved concentration; L 2 T -1 is the hydrodynamic dispersion coefficient tensor; and LT -1 is the pore water velocity vector.
An empirical linear relationship is present between salt concentration and density e. Stratification formed by a denser fluid overlying a lighter fluid develops high a density gradient and causes gravity-driven instability. The instability of the Elder problem can be quantified by the dimensionless Rayleigh number [ 15 ]: where is the dimensionless density contrast coefficient; L is the height of the model domain; LT -2 is the gravitational acceleration; ML -1 T -1 is the dynamic viscosity; L 2 T -1 is the molecular diffusion coefficient; and L 2 is the intrinsic permeability.
The value of Ra measures the relation between buoyancy-driven transport and diffusion. The values of the parameters in the equation can be found in Table 1. The critical Ra is for Horton-Rogers-Lapwood problem with infinitely extended horizontal porous layer and constant temperature at upper and lower boundaries, but is zero for the classic Elder problem [ 23 , 39 ] since the fixed concentration boundary at both upper and lower boundaries decides the inevitable concentration gradient and thus free convection.
The calculated Rayleigh number of our modified Elder problem approaches , which is far greater than the critical value Ra cr , ensuring the occurrence of free convection. Previous studies have proved that free convection processes in the Elder problem are largely influenced by grid discretization [ 19 , 40 ].
As suggested by [ 41 ], horizontal and vertical are adopted in this study to avoid the influences of grid discretization, resulting in a total of rectangular cells. As the bottom boundary condition of the modified Elder problem is transformed to be a no-flux boundary, the system cannot reach its steady state until the entire domain is filled with saltwater with a concentration equal to that of the source zone.
The permeability contrast is set to be 0. Due to the symmetry of the model domain, only the left half of the domain was subjected to the simulations Figure 2. Their locations, labeled by their horizontal and vertical positions Figure 2 , were selected to assess the behavior of the free convection in response to the changing location of the low high -permeability inclusion.
Note that the leftmost two inclusions are partially overlapped. Preliminary simulations showed that the low high -permeability inclusions located close to the top boundary affect the free convection mostly.
Therefore, emphasis was more put on the low high -permeability inclusions situated in regions from 1, 1 to 1, 4 Figure 2. For cases of , , and , four positions of inclusions are considered i. Table 2 lists the cases with low high -permeability inclusions. By considering a homogeneous case, a total of 45 cases are simulated. TM is a parameter to quantify time-variable solute mass normalized present in the model domain, calculated as [ 21 ] where is the normalized concentration ranging within [0, 1] at.
The value of TM varies from 0 when the domain has no solute mass to 1 when the domain is fully filled with saltwater. Sh is defined as the ratio of actual mass transfer due to free convection during the transient state to the rate of mass transfer due to diffusion, calculated as [ 14 ] where is the solute mass flux across the top boundary, and are the width and length of the source zone, and is the maximum concentration difference between freshwater and saltwater.
A stable system with only diffusive transport has. Conversely, an unstable system with convection and diffusion, such as the classic Elder problem, has. COG describes the horizontal and vertical transient gravity center of solute plume present in the domain.
It is one of the most important stochastic characteristics and can be calculated through spatial moments: where the summation is executed for all cells of the model grid; is the normalized concentration of cell ; and are coordinates of the cell; and and are integer power exponents that define the moment order.
Subsequently, the gravity center is expressed by. COG can indicate the movement of the salt plumes in the heterogeneous porous media in both vertical and horizontal direction. Comparing the concentration distributions for the case of low-permeability inclusion , Figure 3 a , the homogeneous case , Figure 3 b , and the case of high-permeability inclusion , Figure 3 c , it can be observed that after three years five separate fingers were developed in the case of high-permeability region with one main finger generated underneath the high-permeability region in the middle, while only four fingers were generated in the case of low-permeability region and the homogeneous case.
This indicates that a higher instability existed in the system of high-permeability region than the other two cases. The high-permeability region aggravates the imbalance by accelerating its transport in the local region, resulting in the deeper and wider penetration of the salt plumes.
Conversely, the low-permeability region slows down the process of descending and spreading, such that the solute distributes more uniformly. Conclusion can be made that, in terms of instability, a higher permeability region in the domain has a pronounced impact on free convection, which would result in more solute mass entering into the domain at the same loading time.
Hence, emphasis for discussion will be put on the cases with higher permeability region afterwards. The concentration distributions in the domain with the high-permeability inclusion located at three different horizontal positions are plotted in Figure 4.
The salt plumes tend to preferentially develop underneath the high-permeability region, consistent with our aforementioned conclusion that a higher permeability region can result in a faster flow speed and a less uniform concentration distribution. After one year, an additional finger is generated when the high-permeability region is located within the source zone Figure 4 c.
By three years, except case 1, 1 with a less effective area, the main finger developed from the high-permeability region starts to reach the domain bottom, earlier than that in the homogeneous case.
Surprisingly, at approximately 15 years, all fingers have merged into a single dense plume. This may be because the area of the higher-permeability inclusion is relatively small in comparison to the total domain area. The development of the salt plumes under all cases experiences such a process, i.
However, diversity existed in the solute migration paths that depended on the position of the high-permeability region. Eventually, as the density gradient dissipates, diffusion becomes the primary transport mechanism, indicating that the system state was gradually transformed from instability to stability.
The concentration distributions for the domain with low-permeability regions located at three horizontal positions are different Figure 5 from that for the high-permeability regions.
By one year, it can be observed that downward solutes transport in all heterogeneous cases tends to be blocked by the low-permeability region, such that only one finger was developed in two of the cases Figures 5 a and 5 b. Three years after, the distributions show that part of salt plumes immigrate bypass the low-permeability regions rather than passing through them.
Fifty years after, the systems approached their steady state with the plumes symmetrically distributed. Yes, that water is black!
Photo: Matt Herod. However, there are still lots of misconceptions about how people envision groundwater. Many envision large underground lakes and rivers, and while those do exist, they represent an infinitesimally small percentage of all groundwater. Generally speaking groundwater exists in the pore spaces between grains of soil and rocks.
Imagine a water filled sponge. All of the holes in that sponge are water-filled. By squeezing that sponge we force the water out, similarly, by pumping an aquifer we force the water out of pore spaces. There are lots of terms in hydrogeology, most of which are very simple, but essential. Here are a few of the big ones and their meanings. Porosity is an intrinsic property of every material. It refers to the amount of empty space within a given material.
In a soil or rock the porosity empty space exists between the grains of minerals. However, in a material like a gravel, sand and clay mixture the porosity is much less as the smaller grains fill the spaces.
The amount of water a material can hold is directly related to the porosity since water will try and fill the empty spaces in a material. We measure porosity by the percentage of empty space that exists within a particular porous media. Figure 2. Porosity in two different media. The image on the left is analagous to gravel whereas on the right smaller particles are filling some of the pores and displacing water. Therefore, the water content of the material on the right is less. Source: Wikipedia.
Rock Mech. Benjamin, B. Porosity evolution of two Upper Carboniferous tight-gas-fluvial sandstone reservoirs: Impact of fractures and total cement volumes on reservoir quality.
Blessent, D. Coupling geological and numerical models to simulate groundwater flow and contaminant transport in fractured media. Chen, Y. Consistency analysis of Hoek-Brown and equivalent Mohr-coulomb parameters in calculating slope safety factor Bull. Davies, J. Gao, S.
Experimental research status and several novel understandings on gas percolation mechanism in low-permeability sandstone gas reservoirs. Gas Industry 30, 52— Google Scholar. Guglielmi, Y. Mesoscale characterization of coupled hydromechanical behavior of a fractured-porous slope in response to free water-surface movement. Hakan, A.
Percolation model for dilatancy-induced permeability of the excavation damaged zone in rock salt. Jing, L. Modeling of fluid flow and solid deformation for fractured rocks with discontinuous deformation analysis DDA method. Lei, D. Effect of cyclic freezing-thawing on the shear mechanical characteristics of nonpersistent joints.
Li, H. Imbibition model of tight sandstone based on distribution characteristics of roar. Li, J. Time-dependent dilatancy for brittle rocks. Lin, H.
Effect of non-persistent joints distribution on shear behavior. An empirical statistical constitutive relationship for rock joint shearing considering scale effect. Determination of the stress field and crack initiation angle of an open flaw tip under uniaxial compression. Liu, W. Stability study and optimization design of small-spacing two-well SSTW salt caverns for natural gas storages. Energy Storage. Marcel, N. Experimental investigation on anisotropy in dilatancy, failure and creep of Opalinus Clay.
Earth 32, — Mitchell, T. Experimental measurements of permeability evolution during triaxial compression of initially intact crystalline rocks and implications for fluid flow in fault zones. Oda, M. Damage growth and permeability change in triaxial compression tests of Inada granite. Ord, A. Mechanical Controls on dilatant shear zones. Wang, H.
Analysis of the stability of water-filled rock slope with fracture networks seepage and discrete element coupling. Wang, J. Fluid permeability of sedimentary rocks in a complete stress-strain process. Worthington, P. A diagnostic approach to quantifying the stress sensitivity of permeability. Wu, Y. Experimental study on relation between seepage and stress of sandstone in CT scale. Xiang, Z. Technologies for the benefit development of low-permeability tight sandstone gas reservoirs in the Yan'an Gas Field, Ordos Basin.
Natural Gas Industry B 6, — Yang, F. The permeability of fractured tight sandstone reservoirs is evaluated by stoneley wave. Oil Gas Technol. You, L. The amount and quality of our groundwater supply is affected by the following:. The porosity of the soil is the percent of the soil that is air space.
Porosity ultimately affects the amount of water a particular rock type can hold and depends on a couple of different factors. The ability of the ground water to pass through the pore spaces in the rock is described as the rock's permeability. Permeable layers of rock that store and transport water are called aquifers. While porosity and permeability usually go hand-in-hand, though some porous rocks are not permeable and some impermeable rocks are porous.
0コメント