Brooks-Corey Model Research Articles

A prediction method of oil recovery for hot water chemical flooding in heavy oil reservoirs: Semi-analytical stream tube model

Abstract Chemical agents (polymer and surfactant) assisted hot water flooding is an effective means of enhancing oil recovery in heterogeneous and heavy oil reservoirs. The rapid prediction of oil recovery through hot water chemical flooding is very important. The stream tube model is a fast analytical method for predicting water flooding recovery, but it is not suitable for hot water chemical flooding. In this paper, a permeability distribution model for multi-stream tubes is established based on permeability variation coefficient using normal distribution function for reservoir heterogeneity, Using GOMPERTZ growth function model to characterize the changes in stream tube temperature, polymer viscosity, and surfactant concentration with injection volume, Using Brooks-Corey model to describe the influence of interfacial tension and temperature on the relative permeability curve. Finally, an analytical stream tube model for hot water chemical flooding of heavy oil reservoirs was established. The time discrete method is used to solve the model, then the graphs of the relationship between oil recovery and water cut are obtained. Compared with numerical simulation methods, the prediction error of oil recovery is less than 2%, and the calculation time is reduced by 89%. This model has been successfully applied to X oilfield. Based on historical fitting, it is predicted that hot water chemical flooding can enhance oil recovery by 6.3% compared to water flooding. This paper provides a fast calculation method for predicting and evaluating the effectiveness of hot water chemical flooding in heavy oil reservoirs.

Assessment of Two SWCC models for predicting the curve fitting parameters of lateritic soil: bentonite mixtures

This study compares the capabilities of soil water characteristic curve (SWCC) models by Brooks-Corey (BC) and van Genuchten (vG) in estimating the curve fitting parameters for lateritic soil—bentonite mixtures. The SWCCs of soil treated with 0–10% bentonite and compacted with British standard light (BSL) energy at compaction states representing dry of optimum, optimum, and wet of optimum conditions were measured by sequential desorption using pressure plate extractor. The fitting parameters of the two equations were determined by a non-linear fitting program. The fitting capabilities of the models on the measured data were compared by statistical indices namely the root mean square error (RMSE), linearity (R2) and index of agreement (d). Results revealed that volumetric water content increased as bentonite content increased with specimens containing 10% bentonite recording the highest and therefore has greater capacity to retain water/contaminants, while the air entry value (AEV) for the various soil mixtures also increased with higher bentonite content. The study also found that the estimated volumetric water contents approximated the measured values at all suctions with a high degree of accuracy with RMSE values that ranged from 0.0035 to 0.0150 for vG model which are somewhat lower than the values for BC equation. Similarly, R2 for the vG equation (≥ 0.99) are, on average, slightly higher than those of the BC fits. However, the d values associated with the BC model which varied between 0.788 and 0.971 are higher than those of the vG (0.784–0.968). Overall, the study established that the vG model proved marginally superior in respect of goodness of fit.

Analytical solution for steady vertical flux through unsaturated soils based on van Genuchten-Mualem model

Various analytical solutions for steady-state vertical flux through unsaturated soils exist in the literature. These solutions often assume specific hydraulic conductivity forms such as exponential form, rational power form, or power-law form such as the Brooks-Corey model. The van Genuchten-Mualem model is widely used in practical applications of flow through unsaturated soils. It is among the most complicated hydraulic conductivity forms for which no analytical solutions are available. This paper deals with the development of an analytical solution for steady-state vertical flux using van Genuchten-Mualem model. The analytical solution is obtained using Maclaurin series expansions and the multinomial theorem. The general form expresses the depth as a function of pressure head. The analytical solution can be used to model both infiltration (positive flux rate) and evaporation (negative flux rate). The derived analytical solution is compared with numerical results and shows excellent agreement. It can also be used to identify the hydraulic parameters analytically, eliminating the need of a numerical inverse procedure. As far as the author’s knowledge, the proposed solution is the first analytical solution dealing with van Genuchten-Mualem model, which is directly derived from the flow equation without any predefined mathematical form of the pressure head.

Prediction of relative permeability from capillary pressure based on the fractal capillary bundle model

The accurate prediction of relative permeability is one of the most important issues in the study of multiphase flow in porous media and is involved in many fields, such as geothermal energy development and petroleum engineering. Traditional models such as the Purcell model and Burdine model for predicting relative permeability from capillary pressure usually assume that the wetting phase (WP) flows in small pores and that the nonwetting phase (NWP) flows in large pores, which is a limiting assumption, such that these models cannot describe the process of WP being displaced by NWP. In this study, we propose a generalized theoretical model for calculating relative permeability based on the fluid distribution characteristics and the fractal capillary bundle model. The relative location α of the interface between NWP and WP is introduced to characterize the transport of the two-phase interface in porous media. The developed model was validated with relative permeability experimental data of gas–water/oil and compared with the Li model and the Brooks–Corey model. The comparison results show that the new model provides an improved match to experimental data than the Li model and the Brooks–Corey model. The sensitivity analysis results of the model show that tortuosity and pore size distribution play a key role in predicting relative permeability from the capillary pressure curve. The relative permeability of wetting phase decreases significantly with increasing fractal dimension of tortuosity and fractal dimension of pore size distribution. The effect of parameter α, which reflects fluid transport characteristics, on relative permeability of wetting phase is more pronounced in porous media with complex structures and low/ultralow permeability.

Characterization of multiphase flow in shaly caprock for geologic CO2 storage

Geologic carbon dioxide (CO2) storage is a promising strategy for reducing its atmospheric levels, but concerns regarding potential leakage through faults or high-permeability inclusions in caprock highlight the importance of understanding its multiphase flow mechanism. In this study, a comprehensive experimental approach is introduced to characterize the pore structure of heterogeneous and fractured caprock and its relationship to water/CO2 relative permeability. It is observed that intact caprock with higher clay content exhibits a higher decrease in water relative permeability in the drainage: the exponent values in the Brooks-Corey model are 3.89 for sandy Eau Claire Shale, 5.04 for the clayey shale, and 6.77 for intact Opalinus Clay - consistent with observations for porous and tight rock. CO2 relative permeability also exhibits an exponential increase with CO2 saturation, which is more pronounced for sandy and fractured caprock due to the higher endpoint CO2 relative permeability and CO2 saturation. Moreover, a method to estimate relative permeability based on the water-CO2 capillary pressure response derived from the pore structure analysis is adopted and its results are compared to the direct measurements. This approach provides a better fit for intact clay-rich caprock but tends to underestimate the relative permeability as a function of fluid saturation for heterogeneous and fractured caprock. The pore structure-based estimation allows for pre-assessing the multiphase flow through caprock, although the direct measurements are highly recommended, especially for fractured materials.

Analytical solutions for steady vertical flux through unsaturated soils with generalized hydraulic properties

In this paper, analytical solutions for one-dimensional nonlinear steady-state vertical flux through unsaturated soils are presented. The analytical solutions are applicable for homogeneous soils for which the hydraulic conductivity function is a generalized form of the well-known Brooks-Corey and rational models. The adopted generalized soil–water retention model involves two additional parameters which reflect the curvature of hydraulic functions near saturation. The main advantage of the used generalized hydraulic conductivity function is that it fits better experimental data. The soil domain is a finite-depth medium overlying a fixed groundwater level. The derived analytical solutions are obtained using the Maclaurin series expansion technique. Analytical solutions for both infiltration and evaporation are developed. For evaporation from a fixed groundwater level, the derived solutions can be used to predict the existence of a vaporization plane below the soil surface. For this case, analytical expression of the position of drying front separating liquid and gas regions is derived. The analytical results are compared to numerical ones for various infiltration and evaporation examples and excellent agreements are obtained. The generalized model is also used to fit experimental data obtained for two soils. The results show that the use of the standard Brooks-Corey model may lead to an overestimation of the position of the drying front.

Water-Gas Imbibition Relative Permeability: Literature Review, Direct versus Indirect Methods and Experimental Recommendations

When an active aquifer encroaches into a gas bearing reservoir or when an oil rim sweeps gas during late depletion of the gas cap, gas displacement by liquid is important for estimating the gas recovery. In the water displacing gas condition, the viscosity ratio is extremely favorable, resulting in a sharp waterfront in the reservoir matrix: it results that changing the relative permeability Kr shape has negligible effect, while endpoints water relative permeability Krw Max and residual gas saturation Sgr are much more important to understand gas flow performance for estimation of gas recovery with active aquifer or productivity decline after water breakthrough. Three main methods are used to determine water/gas relative permeability curves: imbibition unsteady-state, imbibition steady-state or indirect approaches such as co-current spontaneous imbibition if transient data are available. One of the other popular indirect methods is called Brooks-Corey approach: by measuring the drainage Pc curve using centrifuge or porous plate methods, it is possible to calculate a pore size distribution index c. This coefficient is used in a Brooks-Corey model to determine the drainage Kr curve. It is also required to measure and determine the relationship between the residual gas saturation Sgr and the initial gas saturation Sgi relationship. Finally, it is accepted that there is no hysteresis on the water relative permeability Krw curve, as water is always the wetting phase in the gas/water couple. As non-wetting phase, gas exhibits strong hysteresis between drainage and imbibition curves: it is therefore necessary to apply a correction on the drainage Krg curve to build the imbibition one using correcting models. The aim of this paper is to compare gas/water relative permeability of clastic rocks using direct waterflooding information and indirect approach using Brooks-Corey model. It is shown that using the indirect approach leads to results like those experimentally obtained. Also, additional numerical simulations are proposed to discuss the relevance of measuring the entire water-gas imbibition relative permeability curve using the steady-state approach.

Parameter identification of unsaturated seepage model of core rockfill dams using principal component analysis and multi-objective optimization

The seepage behavior of core walls is an important aspect of safety monitoring of core rockfill dams. The permeability coefficient of unsaturated soil of the core wall is a nonlinear function not a constant. Since there are many seepage monitoring points in the core wall, parameter identification of unsaturated seepage model of soil core wall based on the data of these points is a high-dimensional multi-objective optimization problem. In this paper, a parameter identification method of unsaturated seepage in the core wall is proposed, in which an unsaturated seepage model, principal component analysis (PCA), and elitist non-dominated sorting genetic algorithm (NSGA-II) are combined. The seepage monitoring sequence of measuring points in saturated and unsaturated regions of the core wall is synthesized to a few comprehensive variables by PCA respectively. The NSGA- II is adopted to optimize the objective functions established by these comprehensive variables to obtain the Pareto-optimal solutions. The efficiency, accuracy, and robustness to the error of the proposed method have been confirmed in a fictitious dam and an actual dam. The predicted seepage pressure with the Brooks-Corey (BC) model is in best agreement with the measured data, and the simulated seepage field is more in line with the actual situation.

A multiphase flow approach to gas permeability of composite facesheets during Co-cure of honeycomb core sandwich structures

Co-curing composites facesheets to produce sandwich structures involves heating a prepreg-core assembly in a vacuum bag and compacting it by an external pressure until the prepreg cures and bonds to the core. During this process, vacuum pressure can cause atmospheric air in the honeycomb core to bubble through uncured prepreg facesheets and impact material quality, such as excess resin bleed and void growth in either the facesheet or the bondline. In this paper, we propose a multiphase flow model to predict core gas pressure throughout the process. Fully impregnated prepreg facesheets only become permeable to gas as liquid resin desaturates pores in the fabric weave. To solve for the core gas pressure, resin and gas transport equations are solved simultaneously. The permeability of each phase is a function of saturation–defined by the Brooks-Corey model to be a power law function of the capillary pressure. Deviating from a single phase (gas only) model, using dimensionless analysis we identify four material properties which control the outcome: (i) an intrinsic permeability, K i, which only depends on the fabric architecture and is invariant to all process variables (time, temperature, pressure), (ii) the ratio of fluid viscosities, μ w/ μ g, where resin viscosity varies strongly with temperature and degree of cure, (iii) a bubbling (capillary) pressure, P b, which defines the minimum pressure needed to penetrate the largest pore, and (iv) a pore size distribution parameter, λ, which defines the percentage of pores which may desaturate given the evolving capillary pressure. Changes in observed gas permeability due to time and temperature are explained by the evolving desaturation of facesheet resin and the known relationship to resin viscosity. By applying this model, a reduced set to material characterization is required and complex process cycles can quickly be interrogated. With our experiments we demonstrate that one single value for K i and one single value for λ can be used to predict the gas pressure decay (time dependent) at different temperatures, confirming that the intrinsic permeability is constant for the fiber network and changes in the relative permeability to air are due to the decrease in viscosity only. The limitations of this model are presented and discussed.

An approximate solution to one-dimensional upward infiltration in soils for a rapid estimation of soil hydraulic properties

Soil hydraulic properties quantify soil pore structure and are essential input parameters for simulating soil hydrological processes in the vadose zone. Although methods have been proposed to estimate soil hydraulic properties, efficient and applicable ones are still scarce. In this study, we derived an approximate solution for a one-dimensional upward infiltration process with initially uniform soil water content distribution. Founded on our solution, a nonlinearly constrained optimization method was developed to simultaneously estimate the parameters Ks and n, hd of the Brooks-Corey model by measuring the upward infiltration process in a 5-cm high soil column. Numerical experiments simulated with Hydrus-1D were performed on six theoretical soils to evaluate the accuracy of the new method. The results indicate that the approximate solution can accurately simulate the upward infiltration process at the low initial soil water content range, with a relative error (Re) of 2.85% to 5.47% for soil water content distribution in space and time. For the cumulative infiltration curve, the Re ranges from 0.2% to 1.01%. The estimated parameters Ks and n, hd of the Brooks-Corey model are close to the actual parameter values, with a Re of 0.59% to 4.68% for n, 0.00% to 7.91% for hd, and 0.50% to 8.83% for Ks, respectively. It is concluded that the new method provides a simple and valuable tool for the accurate estimation of soil hydraulic properties.

Experimental Investigation of the Different Polyacrylamide Dosages on Soil Water Movement under Brackish Water Infiltration.

The use of soil conditioners in conjunction with brackish water irrigation is critical for the efficient development and use of brackish water as well as the enhancement of the structure of saline soil and stimulating crop growth. This study investigated the effects of different polyacrylamide (PAM) dosages (0, 0.02%, 0.04%, and 0.06%) on the water flow properties of sandy loam during brackish water infiltration using one-dimensional vertical and horizontal soil column infiltration experiments. The results showed that: (1) PAM could lower the soil infiltration rate and increase soil water retention performance under brackish water infiltration conditions. (2) PAM had a significant effect on the parameters of the Philip and Kostiakov infiltration models. The soil sorption rate S and the empirical coefficient λ were the smallest, and the empirical index β was the largest when the PAM dosage was 0.04%. (3) PAM dosage displayed a quadratic polynomial connection with the soil saturated water content and the saturated hydraulic conductivity. The soil saturated water content was highest when the PAM dosage was 0.04%, the intake suction hd of the Brooks-Corey model increased by 15.30%, and the soil water holding capacity was greatly improved. (4) Soil treated with PAM could absorb more water under the same soil water suction, whereas the soil unsaturated hydraulic conductivity and its growth rate decreased. The soil saturated diffusion rate Ds, as well as the soil water diffusion threshold, rose. Finally, the 0.04% PAM dosage could improve soil hydrodynamic characteristics under brackish water infiltration, which is beneficial for the efficient utilization of brackish water.

Effect of Shelterbelt Construction on Soil Water Characteristic Curves in an Extreme Arid Shifting Desert

To explore the impact of artificial shelterbelt construction with saline irrigation on the soil water characteristic curve (SWCC) of shifting sandy soil in extreme arid desert areas, three treatments including under the shelterbelt (US), bare land in the shelterbelt (BL) and shifting sandy land (CK) in the hinterland of the Taklimakan Desert were selected. The age of the shelterbelt is 16, and the vegetation cover is mainly Calligonum mongolicum. The soils from different depths of 0–30 cm were taken keeping in view the objective of the study. The SWCCs were determined by the centrifugal method and fitting was performed using various models such as the Gardner (G) model, Brooks–Corey (BC) model and Van Genuchten (VG) model. Then, the most suitable SWCC model was selected. The results showed that electrical conductivity (EC) and organic matter content of BL and US decreased with the increasing soil depth, while the EC and organic matter content of CK increased with the soil depth. The changes in soil bulk density, EC and organic matter of 0–5 cm soil were mostly significant (p < 0.05) for different treatments, and the differences in SWCCs were also significant among different treatments. Moreover, the construction of an artificial shelterbelt improved soil water-holding capacity and had the most significant impacts on the surface soil. The increase in soil water-holding capacity decreased with increasing soil depth, and the available soil water existed in the form of readily available water. The BC model and VG model were found to be better than the G model in fitting results, and the BC model had the best fitting result on CK, while the VG Model had the best fitting result on BL with higher organic matter and salt contents. Comparing the fitting results of the three models, we concluded that although the fitting accuracy of the VG model tended to decrease with increasing organic matter and salinity, the VG model had the highest fitting accuracy when comparing with BC and G models for the BL treatment with high organic matter and salinity. Therefore, the influence of organic matter and salinity should be considered when establishing soil water transfer function.

The Control of Plant and Soil Hydraulics on the Interannual Variability of Plant Carbon Uptake Over the Central US

AbstractThe interannual variability (IAV) of gross primary productivity (GPP) reflects the sensitivity of GPP to climate variations and contributes substantially to the variations and long‐term trend of the atmospheric CO2 growth rate. Analyses of three observation‐based GPP products indicate that their IAVs are consistently correlated to terrestrial water storage anomaly over the central US, where episodic droughts occur. A land surface model explicitly representing plant hydraulics and groundwater capillary rise with an adequate soil hydraulics well captures the observed GPP IAV. Our sensitivity experiments indicate that, without representations of plant hydraulics and groundwater capillary rise or using an alternative soil hydraulics, the land model substantially overestimates the GPP IAV and the GPP sensitivity to water in the central US. This study strongly suggests the use of the van Genuchten water retention model to replace the most commonly used Brooks–Corey model, which generally produces too strong matric suction of soil water especially in dry conditions, in land surface modeling. This study highlights the importance of plant and soil hydraulics and surface–groundwater interactions to Earth system modeling for projections of future climates that may experience more intense and frequent droughts.

An Investigation of Fractional One-Dimensional Groundwater Recharge by Spreading Using an Efficient Analytical Technique

In the present work, the q-homotopy analysis transform method (q-HATM) was used to generate an analytical solution for the moisture content distribution in a one-dimensional vertical groundwater recharge problem. Three scenarios for the Brooks–Corey model are studied based on linear and nonlinear diffusivity and conductivity functions. The governing nonlinear fractional partial differential equations are solved effectively by the combination of a hybrid analytical technique, which is the combination of the q-homotopy analysis method and the Laplace transform method. Figures and tables are used to discuss the outcomes for fractional values of the time derivative. Mathematica software is used to plot the figures. The examples used in this paper demonstrate the accuracy and competence of the considered algorithm. The acquired results demonstrate the efficiency and reliability of the projected scheme and are also suitable to carry out the highly nonlinear complex problems in a real-world scenario.

Part 2: Oil displacement by foam injection in Hele-Shaw cell with a pattern of impermeable regions

Foam flooding is an enhanced oil recovery (EOR) technique that is used to control gas mobility, particularly in heterogeneous and fractured reservoirs. This paper aims to numerically investigate the effect of foam flooding on oil displacement efficiency at the fracture scale. Also, an empirical model for relative permeability, which remains difficult and troublesome to measure, is proposed and implemented in the numerical model. The experimental data used in this study include five injection methods that are water, gas, tertiary gas, foam, and tertiary foam. Fluid injection was performed through 2D-printed models of an actual rock cross-section, which was obtained from high-resolution micro-computed tomography. In the numerical model used, pressure and saturation equations of injected fluids were solved using the COMSOL software.Based on the findings, both experimental data and numerical simulation results of the five injection methods were in good agreement, which shows the robustness of the proposed relative permeability model with a single fitting parameter. The oil recovery factor predicted using the proposed model outperformed that of the Buckley-Leverett and Brooks-Corey models. Also, the relative permeability model was further tested against coreflooding experimental data from the literature. This study provides more insight into numerical modeling of foam flooding at the fracture scale. Moreover, we believe that the proposed relative permeability model is robust and capable of capturing the evolution of relative permeability with saturation along the injection process, which encourages further investigation and improvement by the subsurface community. Further field-scale studies, including sensitivity and optimization, can be performed based on this work.

Using Time-Lapse Resistivity Imaging Methods to Quantitatively Evaluate the Potential of Groundwater Reservoirs

In this study, we attempt to establish an alternative method for estimating the groundwater levels and the specific yields of an unconfined aquifer for the evaluation of potential groundwater reservoirs. We first converted the inverted resistivity into the normalized water content. Then, we inverted the parameters of the Brooks-Corey model from the vertical profiles of the water content by assuming that the suction head was in proportion to the elevation regarding a predefined base level. Lastly, we estimated the groundwater level, the theoretical specific yield, and the specific yield capacity from the Brooks-Corey parameters at every survey site in the study area. The contour maps of the time-lapse groundwater levels show that the groundwater flows downstream, with a higher hydraulic gradient near the river channel than in the area away from the main channel. We conclude that the estimated maximum specific yield capacities are consistent with that derived from the pumping tests in the nearby observation well. Additionally, the specific yield capacities are only three quarters to two thirds of the theoretical specific yields derived from the difference between the residual and saturated water contents in the Brooks-Corey model. We conclude that the distribution pattern of the specific yields had been subjected to the distribution of natural river sediments in the Minzu Basin, since the modern channel was artificially modified. Although we had to make some simple assumptions for the estimations, the results show that the surface resistivity surveys provide reasonable estimations of the hydraulic parameters for a preliminary assessment in an area with few available wells.

Approximate analytical solutions for one-dimensional soil water movement equation with arbitrary initial conditions

Simple analytical solutions of one-dimensional horizontal and vertical soil water movement are of great importance for the soil hydraulic parameters estimation and precision irrigation. We present the approximate analytical solutions for the problem of horizontal absorption and vertical infiltration in soil under arbitrary initial conditions and constant boundary conditions based on the principle of least action and the variational principle. The algebraic functions of soil water content (SWC) profile, cumulative infiltration, infiltration rate and infiltration time about the wetting front distance/depth (WFD) and the parameters of Brooks-Corey model are obtained. Compared with the numerical results simulated by HYDRUS-1D, the algebraic results calculated by approximate analytical solutions have high precision to predict the horizontal and vertical soil water infiltration based on the known WFD. Contrary to the linear relationship in horizontal absorption with uniform initial conditions, the relationships between cumulative infiltration and WFD, infiltration rate and reciprocal of WFD, square of WFD and infiltration time are nonlinear for the arbitrary heterogeneous initial conditions. However, the initial conditions have little effect on the relationship between infiltration rate and WFD when the initial SWCs are lower than 0.6 θs. Based on the nonlinear relationships of infiltration rate and WFD in vertical infiltration, the WFD and time of the stable infiltration are obtained.

Investigating NMR-based absolute and relative permeability models of sandstone using digital rock techniques

Absolute permeability determines fluid production capacity, and relative permeability determines the produced fluid types, and their accurate calculations are critical for evaluating hydrocarbon-bearing formations. In oil/gas fields, the nuclear magnetic resonance (NMR) logging tool has been widely used due to its unique advantage of directly detecting the rock pores. However, this tool cannot measure absolute permeability or relative permeability directly. There exist many approaches for estimating NMR-based permeabilities, but most of them lack the basic understandings of fluid flow characteristics in micro-pores. To investigate NMR-based absolute and relative permeability models of sandstone , we suggest three types of pores, i.e., non-flowing bound water pores , slow-flowing free water pores, and fast-flowing free water pores, and incorporate them into the permeability calculations. The digital rock technique was utilized to study fluid flow mechanisms. Lattice Boltzmann method (LBM) was applied on two constructed three-dimensional digital rocks to calculate fluid flow velocity field distributions and absolute permeabilities. From the fluid flow simulation results, it was found that bound water exists not only in small pores but also in medium-sized pores and big pores. By comparing the permeability values of digital rocks with various porosities, we demonstrated that fast-flowing free water pores have the highest contribution to absolute permeability. Five relative permeability calculation models were investigated, where we found that the best link between the T 2 spectrum and relative permeability curve was the pore size distribution index. The obtained results indicated that the traditional Coates model combined with the spectral bulk volume irreducible method and the Brooks-Corey model are high-precision models for calculating the NMR-based absolute and relative permeabilities in sandstone reservoirs. • Three fluid flow velocity-based types of pores were investigated. • Fluid flow characteristics and their impacts on permeability were analyzed. • The three-dimensional spatial distribution of bound water was studied. • The accuracies of five relative permeability calculation models were compared.

Study of the Saffman–Taylor Instability in an Oil Reservoir Formation in Two Dimensions

The paper is devoted to the simulation of oil displacement by water and the formation of the Saffman–Taylor instability. The problem is solved in two-dimensional formulation. A circular domain with one injection well and eight production wells located along the contour around the injection well is considered as geometry. To study the patterns of oil displacement by water, hydrostatic pressure, oil and water seepage rates, and oil saturation are calculated. The graphical analysis considers mainly the oil saturation field. The hydrostatic pressure field is calculated by solving the steady-state piezoconductivity equation; the field of the oil-water seepage rate is calculated using the linear Darcy filtration law; and the oil saturation field is calculated from the solution of the advection transport equation. The two-phase nature of the flow considered in the problem lies in the fact that the oil and water phases have their own relative phase permeabilities calculated using the Brooks–Corey model. The equations are solved numerically using the finite volume method. An irregular triangular grid is used to discretize the computational domain. As a result of the simulation, it is established that the type of the Saffman–Taylor instability, due to its randomness, strongly depends on the computational grid. After the production wells are flooded, the displacement front stabilizes. The instability increases as the ratio of dynamic viscosities of oil and water increases.

A comparative study of soil-water characteristic curves for compacted lateritic soil – bacillus coagulans mixtures

A comparative study of soil-water characteristic curves (SWCCs) for compacted lateritic soil ̶ Bacillus coagulans (B. coagulans) mixtures for municipal solid waste (MSW) application was studied. Soil treatment was performed at approximately about one-third pore volume of the microbes (i.e., B. coagulans) for suspension densities of 0, 1.5×108, 6.0×108, 1.2×109, 1.8×109 and 2.4×109cells/ml, correspondingly. Soil specimens were prepared at optimum moisture content (OMC) of British Standard light (BSL) compaction energy. Cementation reagent was applied on the compacted soil and permitted to penetrate until partial saturation was achieved. A set-up of pressure plate extractor was employed to measure the volumetric water content, θ (VWC) in the laboratory for varying matric suctions with a minimum of 10 kPa up to a maximum of 1,500 kPa. The unsaturated hydraulic conductivity (UHC) and VWC were assessed using Brooks - Corey (BC) and Fredlund - Xing (FX) models. Largely, BC and FX models overrated the VWC. Also, the VWC decreased with higher matric suction for the two models considered and the laboratory measured values. The UHC predicted for matric suctions of 500 and 1,500 kPa initially decreased for B. coagulans suspension density up to 1.2×109 cells/ml for BC and FX models, with the exception of a few cases, but thereafter increased with increase in microbial density. For FX model at 1,500 kPa, UHC values of 2.42×10–9, 2.02×10–9, 9.31×10–10, 8.09×10–10 , 1.29×10–9 and 2.27×10–9m/s were recorded at 0, 1.5×108, 6.0×108, 1.2×109, 1.8×109 and 2.4×109cells/ml, respectively. In the case of BC model, values of 2.26×10–17, 1.41×10–14, 2.2×10–14, 4.6×10–19 , 3.25×10–17 and 2.45×10–14m/s were recorded at 0, 1.5×108, 6.0×108, 1.2×109, 1.8×109 and 2.4×109cells/ml, respectively. Thus, the FX model met the design maximum hydraulic conductivity value of 1 x 10–9 m/s requirement for MSW system when lateritic soil was treated with B. coagulans suspension density of 1.2×109 cells/ml, while the BC model satisfied the requirement for all the microbial densities considered and it is recommended for modelling of UHC of lateritic soil admixed with B. coagulans for MSW containment application.