Mixing Cell Method Research Articles

Determination of Minimum Miscible Pressure and Phase Behavior of CO2–Shale Oil in Nanopores

CO2 injection into shale oil reservoirs has gained widespread attention as it not only enhances oil recovery but also facilitates the geological storage of CO2. Shale oil reservoirs are characterized by numerous nanopores, and the resident fluid exhibits unique phase behavior due to nanoconfinement effects, resulting in the failure of the traditional theoretical model and microfluidic experiment in depicting the minimum miscible pressure (MMP) of CO2–shale oil in nanopores. This study proposes a method for calculating the MMP of CO2–shale oil in nanopores. It combines the three‐phase equilibrium calculation method of gas phase–liquid phase–adsorbed phase with the multiple mixing cell (MMC) method. The method considers capillary pressure and corrects the critical properties influenced by the nano‐limited domain effect. Using this approach, the phase behavior of CO2–shale oil in nanopores and the MMP are investigated. It is found that as the pore radius decreases, the area of the two‐phase region surrounded by the phase envelope of multicomponent fluids gradually shrinks. The smaller the pore radius, the more significant the shrinkage of the two‐phase region. The entire phase envelope and the critical point shift to the left and down on the coordinates, and this phenomenon is more pronounced for pore radii less than 20 nm. The presence of CO2 increases the system’s critical condensation pressure and decreases the critical condensation temperature. The MMP of the 6‐component system of CO2–shale oil for a pore radius of 5 nm in this paper is 63.7 bar at 101°C.

Using water sources extent during inundation as a reliable predictor for vegetation zonation in a natural wetland floodplain

Distinctive zones of inundation water during floods were shown to originate from different sources in some major floodplains around the world. Recent research showed that the zonation of water in rivers and floodplains is related to vegetation patterns. In spite of this, water source zones were not used for vegetation modeling due to difficulties in their delineation. In this study, we used simulation results of a fully-coupled groundwater-surface water integrated hydrological model (IHM) HydroGeoSphere and the Hydraulic Mixing-Cell method to provide standard hydrological predictors (e.g. water depth, inundation length, groundwater depth, exchange flux) and the extent of inundation zones having a certain water source (discharged groundwater, river, rainfall, and snowmelt). These variables were used to train a vegetation model for the lower Biebrza floodplain (about 290 km2) using vegetation maps from 1960, 1980, and 2000. We used a one-at-a-time (OAT) approach, where each map was validated based on a model trained on the remaining two maps to obtain realistic error estimates. We also used a fractional approach in which a fraction of each map was used for training and validation. The single model from the fractional approach was used to assess the importance of predictors and to predict vegetation for the 20th century and for the 21st century using IHM simulation forced by the Twentieth Century Reanalysis data and EURO-CORDEX RCP 2.6, 4.5, and 8.5 model ensembles. The model which used both water sources extent and standard predictors performed the best overall and was sensitive to the future trends. The extent of river water within the inundation area was by far the most important vegetation predictor. The models that neither used the water sources extent predictors nor the exchange flux were not able to predict the trends of areas covered by certain vegetation types under future climate. The advantage of the water sources extent predictors was their ability to represent the spatial effect of local hydrological phenomena. This was not possible with the standard predictors, because they show only the source of the phenomena (e.g. groundwater discharge zone), but do not indicate the actual area affected by its physical and chemical properties, which is more relevant for vegetation development. Our results highlight the relevance of using water extent predictors due to their ability to explain spatiotemporal ecological processes, such as vegetation development. We suggest to use water extent predictors in modelling for developing more accurate decision support for wetland floodplains.

Nanometer-Scale CO2-Shale Oil Minimum Miscibility Pressure Calculations Based on Modified PR-EOS

CO2 flooding is recognized as an efficient method for enhancing shale oil recovery, while CO2-oil MMP (minimum miscibility pressure) in the micro-nanoscale is a crucial parameter. This paper presents a method for calculating the MMPs of pure hydrocarbons (C4H10, C6H14, C8H18, and C10H22) and CO2 systems in nanopores (3 nm to 10 nm) with temperature ranging from 290.15 K to 373.15 K. Firstly, we modify the Peng-Robinson equation of state (PR-EOS) by considering the influence of confinement effect and capillary pressure in nanopores. Secondly, the flash calculation algorithm is employed to determine whether the oil and gas phases in nanopores have reached an equilibrium state according to the equality of the fugacity of the two phases. Thirdly, we calculate the interfacial tension (IFT) between the two phases using the Macleod-Sugden equation. When the extrapolated IFT is zero, we treat the corresponding pressure as the MMP of the CO2-oil system in nanopores. Simulation results indicate that the calculated MMP using this method has a relative error of about 0.62% compared to the MMP calculated using the multiple mixing cell (MMC) method, indicating high reliability for MMP prediction. Moreover, the measured MMP at the nanoscale is generally smaller than that in the bulk phase due to the influence of the confinement effect. The MMP is positively correlated with the reservoir temperature, the carbon atom number in alkanes, and the nanopore radius.

Spatiotemporal variations in water sources and mixing spots in a riparian zone

Abstract. Riparian zones are known to modulate water quality in stream corridors. They can act as buffers for groundwater-borne solutes before they enter the stream at harmful, high concentrations or facilitate solute turnover and attenuation in zones where stream water (SW) and groundwater (GW) mix. This natural attenuation capacity is strongly controlled by the dynamic exchange of water and solutes between the stream and the adjoining aquifer, creating potential for mixing-dependent reactions to take place. Here, we couple a previously calibrated transient and fully integrated 3D surface–subsurface numerical flow model with a hydraulic mixing cell (HMC) method to map the source composition of water along a net losing reach (900 m) of the fourth-order Selke stream and track its spatiotemporal evolution. This allows us to define zones in the aquifer with more balanced fractions of the different water sources per aquifer volume (called mixing hot spots), which have a high potential to facilitate mixing-dependent reactions and, in turn, enhance solute turnover. We further evaluated the HMC results against hydrochemical monitoring data. Our results show that, on average, about 50 % of the water in the alluvial aquifer consists of infiltrating SW. Within about 200 m around the stream, the aquifer is almost entirely made up of infiltrated SW with practically no significant amounts of other water sources mixed in. On average, about 9 % of the model domain could be characterized as mixing hot spots, which were mainly located at the fringe of the geochemical hyporheic zone rather than below or in the immediate vicinity of the streambed. This percentage could rise to values nearly 1.5 times higher following large discharge events. Moreover, event intensity (magnitude of peak flow) was found to be more important for the increase in mixing than event duration. Our modeling results further suggest that discharge events more significantly increase mixing potential at greater distances from the stream. In contrast near and below the stream, the rapid increase in SW influx shifts the ratio between the water fractions to SW, reducing the potential for mixing and the associated reactions. With this easy-to-transfer framework, we seek to show the applicability of the HMC method as a complementary approach for the identification of mixing hot spots in stream corridors, while showing the spatiotemporal controls of the SW–GW mixing process and the implications for riparian biogeochemistry and mixing-dependent turnover processes.

How Is Ultrasonic-Assisted CO2 EOR to Unlock Oils from Unconventional Reservoirs?

CO2 enhanced oil recovery (EOR) has proven its capability to explore unconventional tight oil reservoirs and the potential for geological carbon storage. Meanwhile, the extremely low permeability pores increase the difficulty of CO2 EOR and geological storage processing in the actual field. This paper initiates the ultrasonic-assisted approach to facilitate oil–gas miscibility development and finally contributes to excavating more tight oils. Firstly, the physical properties of crude oil with and without ultrasonic treatments were experimentally analyzed through gas chromatography (GC), Fourier-transform infrared spectroscopy (FTIR) and viscometer. Secondly, the oil–gas minimum miscibility pressures (MMPs) were measured from the slim-tube test and the miscibility developments with and without ultrasonic treatments were interpreted from the mixing-cell method. Thirdly, the nuclear-magnetic resonance (NMR) assisted coreflood tests were conducted to physically model the recovery process in porous media and directly obtain the recovery factor. Basically, the ultrasonic treatment (40 KHz and 200 W for 8 h) was found to substantially change the oil properties, with viscosity (at 60 °C) reduced from 4.1 to 2.8 mPa·s, contents of resin and asphaltene decreased from 27.94% and 6.03% to 14.2% and 3.79%, respectively. The FTIR spectrum showed that the unsaturated C-H bond, C-O bond and C≡C bond in macromolecules were broken from the ultrasonic, which caused the macromolecules (e.g., resin and asphaltenes) to be decomposed into smaller carbon-number molecules. Accordingly, the MMP was determined to be reduced from 15.8 to 14.9 MPa from the slim-tube test and the oil recovery factor increased by an additional 11.7%. This study reveals the mechanisms of ultrasonic-assisted CO2 miscible EOR in producing tight oils.

A New Predictive Method for CO2-Oil Minimum Miscibility Pressure

Gas injection processes are among the effective methods for enhanced oil recovery. Miscible and/or near miscible gas injection processes are among the most widely used enhanced oil recovery techniques. The successful design and implementation of a miscible gas injection project are dependent upon the accurate determination of minimum miscibility pressure (MMP), the pressure above which the displacement process becomes multiple-contact miscible. This paper presents a method to get the characteristic curve of multiple-contact. The curve can illustrate the character in the miscible and/or near miscible gas injection processes. Based on the curve, we suggest a new model to make an accurate prediction for CO2-oil MMP. Unlike the method of characteristic (MOC) theory and the mixing-cell method, which have to find the key tie lines, our method removes the need to locate the key tie lines that in many cases is hard to find a unique set. Moreover, unlike the traditional correlation, our method considers the influence of multiple-contact. The new model combines the multiple-contact process with the main factors (reservoir temperature, oil composition) affecting CO2-oil MMP. This makes it is more practical than the MOC and mixing-cell method, and more accurate than traditional correlation. The method proposed in this paper is used to predict CO2-oil MMP of 5 samples of crude oil in China. The samples come from different oil fields, and the injected gas is pure CO2. The prediction results show that, compared with the slim-tube experiment method, the prediction error of this method for CO2-oil MMP is within 2%.

A new empirical correlation of minimum miscibility pressure for produced gas reinjection

Minimum miscible pressure is a key parameter to screen and design miscible gas injection projects. The aim of this paper is to establish a correlation with only a few input parameters to easily and accurately predict minimum miscible pressure for the reinjection of produced gas with high acidic components. First, the critical parameters of equation of state for each component of the crude oil were obtained through fitting pressure-volume-temperature (PVT) experimental results. Based on the analytically calculated minimum miscible pressures from mixing-cell method, an empirical correlation for predicting minimum miscible pressure in the displacement of crude oil by produced gas was regressed. Finally, the correlation’s accuracy was tested by comparing the minimum miscible pressures predicted from the new proposed correlation to other previous correlations and 20 experimental slim-tube minimum miscible pressures of 12 oil samples. The results indicate that the analytically calculated minimum miscible pressures from the mixing-cell method have a relative error of 0.5% compared to the slim-tube experiment results, which supports its reliability. Furthermore, the new proposed correlation is observed to be superior in terms of the average relative error being only 6.4% for all the 75 analytically calculated minimum miscible pressures and 20 experimental slim-tube minimum miscible pressures, which is lower than the average relative error obtained from other previous correlations.

Minimum Miscibility Pressure Using the Multiple Mixing-cell Combined with the PC-SAFT Equation of State

The minimum miscibility pressure (MMP) is one of the crucial and substantial parameters in the gas injection projects for enhanced oil recovery (EOR). This parameter indicates the minimum pressure at constant temperature and composition conditions for reaching 100 percent of the oil production recovery which leads to a huge economic income. Therefore, an inaccurate prediction of the MMP may lead to increase the costs of oil production. Among the various methods for prediction of the MMP: slim-tube experiment, slim-tube simulation techniques and their simplified cell-to-cell could be considered as the most accurate methods. Although the experimental methods for calculation the MMP take into account the real crude oil, but they are always expensive; therefore, thermodynamic simulation is a better way for the estimation of this parameter. Indeed, the analytical methods are fast and some of them like multiple mixing-cell method used in this study lead to the reliable results and often show good agreements with the experimental methods. A combination of multiple mixing-cell model proposed by Ahmadi and Johns and pressure-temperature (P-T) flash calculations including the PC-SAFT equation of state (EoS), and the cubic equations of state of Peng-Robinson (PR) and Esmaeilzadeh-Roshanfekr (ER) were used in this work for the prediction of MMP. The results overally prove that the PC-SAFT EoS leads to more accurate results in comparison with the ER and PR equations of state.

Prediction of minimum miscibility pressure of pure CO2, carbon dioxide gas mixtures and polymer-supercritical CO2 in oil using modified quadrupole Cubic Plus Association Equation of State (mqCPA EoS)

In this work, the modified quadrupole Cubic Plus Association Equation of State (mqCPA EoS) is used to predict the minimum miscibility pressure (MMP) of pure carbon dioxide, carbon dioxide gas mixtures and polymer-supercritical carbon dioxide (scCO2) systems with the different oils. Carbon dioxide is accounted as a quadruple molecule and by applying Ahmadi and Johns's algorithm, which is a multiple mixing cell (MMC) method, minimum miscibility pressures (MMP) for some scCO2-oil systems are predicted. Moreover, a specific parameterization work was performed so that the new binary interaction parameters, kij, were optimized for the hydrocarbon-CO2 pairs using mqCPA with the quadropolar term. For comparison of the present equation of state (EoS) with the common EoSs, the predictive Peng-Robinson 78 (PPR78) and Soave-Redlich-Kwong (SRK) are applied along with the binary interaction parameter correlations of the Pedesen and Xu et al. kij (Industrial & Engineering Chemistry Research, 54 (2015) 2816–2824). The obtained MMP values by mqCPA EoS are compared to the experiments and those obtained using the PPR78 and SRK EoSs. It is shown that the calculated results by mqCPA EoS demonstrate a good agreement with the experimental values, especially for injected of pure CO2 and leads to the better results in comparison with PPR78 and SRK EoSs. To predict MMP of the polymer-scCO2 systems, three types of poly(lactic acid) (PLA), three types of poly(1-O-(vinyloxy) ethyl-2,3,4,6-tetra-O-acetyl-β-d-glucopyranoside) (P(AcGlcVE)) and seven types of poly(vinyl acetate-alt-dibutyl maleate) (PVAc-alt-PDBM) are used. To our best knowledge, this is the first effort to predict the MMP of these polymer-scCO2-oil systems.

Modified Multiple Mixing Cell Method for Determining Minimum Miscibility Pressure

We briefly review methods for determining minimum miscibility pressure (MMP). We propose a modified multiple mixing cell method for determining the minimum miscibility pressure, based on the variation in the tie-lines at different pressures. First using the conventional multiple mixing cell method, we calculated the tie-line length, studied the tie-line pattern, and found the pressure-dependent critical points. Then we proposed some steps to improve the existing algorithm. The results obtained using the modified method are compared with the results of other methods. We have established that our algorithm can solve the problem of the complexity of searching for key tie lines, as is done in the calculation by the conventional mixing cell method, and thus our algorithm ensures that we find an accurate value of the MMP.

Technology Focus: Reservoir Simulation (July 2016)

Overview Reservoir-simulation-model inputs are numerous, and uncertainty is pervasive— before, during, and after development. On top of that, there is always pressure to deliver quality results as quickly as possible. This gives rise to a simple question, one that has yet to find a simple answer: How refined is refined enough and how coarse is too coarse? I run the risk of oversimplification here, but it seems we are faced with a classic dichotomy, one that is exacerbated by the pull of advances in high-performance computing that permit ever-greater model refinement, while, simultaneously, we have the push of (possibly stochastic) sampling of uncertainty, thereby encouraging the development of simplified (or surrogate) models that can run hundreds, even thousands, of times. The question is one of striking the right balance between two apparently contradictory approaches to simulation. The adage “horses for courses” is not particularly helpful in itself, even though it is probably appropriate. Does one have two distinct models with roughly commensurate scaling, or can we build a single all-purpose model with different scales within it that is both fast and accurate with multiscale grid? Dimensional scale represents just one aspect of the term “multiscale,” which I have mistakenly taken to mean just the juxtaposition of, essentially, geometrical scale within a single model (such as that found in coupling a simulation grid and the wellbore). The large ratio associated with domain size, and the resolution of the geological data, is usually managed by upscaling. However, so-called multiscale methods represent a new avenue of research, one that may provide a bridge between the aforementioned push and pull of refinement resulting from the needs of different decision makers. Multiscale, which has been the subject of ongoing study over the past decade, knits together geometrical quantities (dimensional scale) with tailored computational schemas (numerical scale). This multifaceted multiscale concept may offer a means to construct an accurate coarser-scaled model, one honoring the attributes of the fine-scale heterogeneous geological data from both numerical and spatial standpoints. This method class computes local basis functions for the solution variables, to construct a smaller (coarse) system for computing an approximate solution on the original simulation grid. While it is too early to say whether this broader notion of multiscale (numerical and geometrical) will provide a single, unifying, model for engineers, it is possible that this, or some other such method, may strike that elusive balance between refinement (accuracy) and surrogacy (speed). For those interested in reading up on this topic, the peer-reviewed SPE papers SPE 119183 and SPE 163649 provide more detail and clarify the status of some ongoing research. JPT Recommended additional reading at OnePetro: www.onepetro.org. SPE 169063 Application of Multiple- Mixing-Cell Method To Improve Speed and Robustness of Compositional Simulation by Mohsen Rezaveisi, The University of Texas at Austin, et al. SPE 177634 Multiscale Geomechanics: How Much Model Complexity Is Enough? by Gerco Hoedeman, Baker Hughes SPE 174905 Experimental Design or Monte Carlo Simulation? Strategies for Building Robust Surrogate Models by Jared Schuetter, Battelle Memorial Institute, et al. SPE 169357 Reduced-Order Modeling in Reservoir Simulation Using the Bilinear Approximation Techniques by Mohammadreza Ghasemi, Texas A&M University, et al.

Analytical Solutions for Gas Displacements With Bifurcating Phase Behavior

Summary Minimum miscibility pressure (MMP) is one of the most important parameters in the design of a successful gasflooding process. The most-reliable methods to calculate the MMP are based on slimtube experiments, 1D slimtube simulations, mixing-cell calculations, and the analytical methods known as the method of characteristics (MOC). The calculation of MMP by use of MOC is the fastest method because it relies solely on finding the key tie lines in the displacement path. The MOC method for MMP estimation in its current form assumes that the composition path is a series of shocks from one key tie line to the next. For some oils, however, these key tie lines do not control miscibility, and the MMP calculated by use of the key-tie line approach can be significantly in error. The error can be as high as 5,000 psia for heavier oils or CO2 displacements at low temperature in which three-phase hydrocarbon regions can exist (L1–L2–V). At higher pressures, the two- or three-phase region can split (or bifurcate) into two separate two-phase regions (L1–L2 and L1–V regions). Thus, for the MMP calculation from MOC to be correct, we must calculate the entire composition path for this complex phase behavior, instead of relying on the shock assumption from one key tie line to the next. In this paper, the MOC-composition route is developed completely for the bifurcating phase-behavior displacement for pure CO2 injection by use of a simplified pseudoternary system that is analogous to the complex phase behavior observed for several real displacements with CO2. We develop the MOC analytical solutions by honoring all constraints required for a unique solution—velocity, mass balance, entropy, and solution continuity. The results show that a combination of shocks and rarefaction waves exists along the nontie-line path, unlike previous MOC solutions reported to date. We show that by considering the entire composition path, not just the key tie lines, the calculated MMP agrees with the mixing-cell method. We also show that, in this complex ternary displacement, the displacement mechanism has features of a both condensing and vaporizing (C/V) drive, which was thought to be possible only for gasfloods with four or more components. For pure CO2 injection, the solution also becomes discontinuous for oils that lie on the tie line envelope curve. Finally, we show that shock paths within the two-phase region are generally curved in composition space and that there is no MMP for some oil compositions considered in the displacements by CO2. Recovery can be large even though the MMP is not reached.

Multiple-Mixing-Cell Method for MMP Calculations

SummaryThe minimum miscibility pressure (MMP) is a key parameter governing the displacement efficiency of gasfloods. There are several methods to determine the MMP, but the most accurate methods are slim-tube experiments, analytical methods, and numerical-simulation/cell-to-cell methods. Slim-tube experiments are important to perform because they use actual crude oil, but they are costly and time consuming. Analytical methods that use the method of characteristics (MOC) are very fast and help to understand the structure of gasfloods. MOC, however, relies on finding the unique and correct set of key tie lines in the displacements, which can be difficult. Slim-tube simulation methods and their simplified cell-to-cell derivatives require tedious fluid and rock inputs, and their MMP estimates can be clouded by dispersion.This paper presents a simple and accurate multiple-mixing-cell method for MMP calculations that corrects for dispersion, and is faster and less cumbersome than 1D simulation methods. Unlike previous mixing-cell methods, our cell-to-cell mixing model uses a variable number of cells, and is independent of gas/oil ratio, volume of the cells, excess oil volumes, and the amount of gas injected. The new method only relies on robust P/T flash calculations using any cubic equation-of-state (EOS). The calculations begin with only two cells and perform additional cell-to-cell contacts between resulting equilibrium-phase compositions based on equilibrium gas moving ahead of the equilibrium liquid phase. We show for a variety of oil and gas compositions that all key tie lines can be found to the desired accuracy, and that they are nearly identical to those found using analytical MOC methods. Our approach, however, is more accurate and robust than those from MOC because we do not make approximations regarding shocks along nontie-line paths, and the unique set of key tie lines converges automatically.The MMP using our mixing-cell method can be calculated in minutes using an Excel spreadsheet and is estimated from a novel bisection method of the minimum tie-line lengths observed in the cells at four or five pressures. Our multiple-mixing-cell method can calculate either the MMP or the minimum miscibility for enrichment (MME) independent of the number of components in the gas or oil. Our approach further supports the notion that the MMP is independent of fractional flow because we obtain the same key tie lines independent of how much fluid is moved from one cell to another.

Limitations of Current Method-of-Characteristics (MOC) Methods Using Shock-Jump Approximations To Predict MMPs for Complex Gas/Oil Displacements

Summary An accurate minimum miscibility pressure (MMP) is one of the key factors in miscible-gasflood design. There is a variety of experimental and analytical methods to determine the MMP, but the most-reliable methods are slimtube experiments, 1D slimtube simulations, mixing-cell models, and the fast key-tie-line approach using the method of characteristics (MOC). Direct comparisons of all these methods generally agree well, but there are cases in which they do not. No explanation has yet been given for these anomalies, although the MMP is critically important to recovery. The focus of this paper is to explain when current MOC results assuming that shocks exist from one key tie line to the next may not be reliable, and how to identify when this is the case. We demonstrate, using fluid characterizations from Middle East oils, that the MMPs using this MOC method can be more than 6,500 psia greater than those calculated using a recently developed mixing-cell method. The observed differences in the MMP increase substantially as the API gravity of the oil decreases, likely the result of the onset of L1-L2-V behavior. We show that the key tie lines determined using this MOC method do not control miscibility for such cases. We explain the reasons for these differences using simplified pseudoternary models and show how to determine when an error exists. We also offer a way to correct the MMP predictions using the MOC for these complex gas/oil displacements without solving for the complete compositional path.

A hydraulic mixing-cell method to quantify the groundwater component of streamflow within spatially distributed fully integrated surface water–groundwater flow models

The complexity of available hydrological models continues to increase, with fully integrated surface water–groundwater flow and transport models now available. Nevertheless, an accurate quantification of streamflow generation mechanisms within these models is not yet possible. For example, such models do not report the groundwater component of streamflow at a particular point along the stream. Instead, the groundwater component of streamflow is approximated either from tracer transport simulations or by the sum of exchange fluxes between the surface and the subsurface along the river. In this study, a hydraulic mixing-cell (HMC) method is developed and tested that allows to accurately determine the groundwater component of streamflow by using only the flow solution from fully integrated surface water–groundwater flow models. By using the HMC method, the groundwater component of streamflow can be extracted accurately at any point along a stream provided the subsurface/surface exchanges along the stream are calculated by the model. A key advantage of the HMC method is that only hydraulic information is used, thus the simulation of tracer transport is not required. Two numerical experiments are presented, the first to test the HMC method and the second to demonstrate that it quantifies the groundwater component of streamflow accurately.

A Practical Method for Minimum-Miscibility-Pressure Estimation of Contaminated CO2 Mixtures

Summary Minimum miscibility pressure (MMP) is a key parameter in the design of gasfloods. Injection-gas compositions often vary during the life of a gasflood owing to reinjection and mixing of fluids in situ. Understanding the impact of the gas compositional changes on the MMP is essential to optimal design of fieldwide pressure management and carbon dioxide (CO2) use. Determining the MMP by slimtube or other methods for each possible variation in the gas-mixture composition is impractical. This paper gives an easy and accurate way to determine impure CO2 MMPs for variable field solvent compositions on the basis of just a few MMPs. Alternatively, the approach could be used to estimate the enrichment level required to lower the MMP to a desired pressure. The MMP-estimation method relies on determining the MMP for pure CO2 injection, and also for a few impure binary MMPs at small CO2-contaminant levels. The number of MMPs needed for the method is equal to the number of components in the injection gas. We use the method of characteristics (MOC) and our newly developed mixing-cell method to estimate the required MMPs, although any reliable MMP analytical or experimental method can be used. We demonstrate how to calculate MMPs for several multicomponent oils displaced by CO2 contaminated by mixtures of N2, CH4, C2, C3, and H2S. The results show that the predicted MMPs for a west Texas crude displaced by contaminated-CO2 injection streams are nearly linear over the range from pure-CO2 injection to any mole fraction combination of the five contaminants. The accuracy of the predicted MMPs is within ±15 psia of that from calculations using mixing-cell simulations, slimtube simulations, and slimtube experiments where available. For another example oil displacement by impure CO2, however, the linear trend in MMPs with contamination mole fractions is accurate only for total contamination levels less than approximately 20% mole fraction, but this is still within a useful range for CO2-gasflood design and optimization. We also examine the sensitivity of local displacement efficiency to dispersion for binary gas mixtures using 1D simulation.

Application of the Hybrid Finite Element Mixing Cell method to an abandoned coalfield in Belgium

The Hybrid Finite Element Mixing Cell (HFEMC) method is a flexible modelling technique particularly suited to mining problems. The principle of this method is to subdivide the modelled zone into several sub-domains and to select a specific equation, ranging from the simple linear reservoir equation to the groundwater flow in porous media equation, to model groundwater flow in each sub-domain. The model can be run in transient conditions, which makes it a useful tool for managing mine closure post-issues such as groundwater rebound and water inrushes. The application of the HFEMC method to an abandoned underground coal mine near the city of Liege (Belgium) is presented. The case study zone has been discretised taking advantage of the flexibility of the method. Then, the model has been calibrated in transient conditions based on both hydraulic head and water discharge rate observation and an uncertainty analysis has been performed. Finally, the calibrated model has been used to run several scenarios in order to assess the impacts of possible future phenomena on the hydraulic heads and the water discharge rates. Among others, the simulation of an intense rainfall event shows a quick and strong increase in hydraulic heads in some zones coupled with an increase in associated water discharge rates. This could lead to stability problems in local hill slopes. These predictions will help managing and predicting mine water problems in this complex mining system.

4-Point FDF of Muskingum method based on the complete St Venant equations

Using the Froude number, a nonlinear convection–diffusion equation was derived from the St Venant equations of continuity and momentum. It was made applicable to discrete space using a mixing-cell method, resulting in a first-order nonlinear ordinary differential equation. A 4-point finite difference scheme was used to solve the first-order nonlinear ordinary differential equation to achieve a nonlinear algebraic form that is very similar in structure to the traditional Muskingum form thus its naming as the ‘4-point FDF of Muskingum Method (4-point FDF M-M)’. In this method, the space interval Δ x is a characteristic channel length and a variable that is dependent on flow discharge, water depth, and flow velocity. An iterative search method was applied to simultaneously obtain the flow discharge and the optimal space interval Δ x. The method developed was tested with both synthetic numerical examples and observed events and the results were compared with those of the Lambda scheme and the method of characteristics. The outflow hydrographs produced by this new 4-point FDF M-M were of comparable accuracy. The parameters used in the new method are based on the physical attributes of the channel and thus do not need calibration as required for the Muskingum method.

Nonlinear Convection-Diffusion Equation with Mixing-Cell Method for Channel Flood Routing

Using a reference Froude number, a nonlinear convection-diffusion equation was derived from the Saint-Venant equations of continuity and momentum, and was solved by a mixing-cell method. The method involves discretizing the nonlinear diffusion equation in space and transforming it to a first-order nonlinear ordinary differential equation where the optimal space interval is obtained to be the same as the characteristic reach length. The nonlinear ordinary differential equation was solved by the fourth-order Runge-Kutta method. The method was tested with numerical examples, and compared with the looped-rating Muskingum-Cunge model and a lambda scheme. The outflow hydrographs produced by this method were of comparable accuracy.

A semianalytical solution of the Saint‐Venant equations for channel flood routing

A semianalytical solution for channel hydraulic routing has been derived from the Saint‐Venant equations. The Saint‐Venant equations were converted into a nonlinear diffusion equation by introducing the Froude number. Subsequently, the mixing cell method is used to discretize a nonlinear diffusion equation in space, transforming it into a first‐order nonlinear ordinary differential equation where the optimal space interval is obtained to be the same as the characteristic reach length. The nonlinear ordinary differential equation was solved by integration with respect to time to achieve a nonlinear implicit algebraic equation. The resulting implicit equation is practical and easy to program on a computer. This method is based on the results of a numerical simulation using the Lambda scheme which indicates that the Froude number is only dependent on the channel roughness and bottom slope. The method was tested with numerical examples and compared with the Lambda scheme and observed data. The hydrographs produced by this method were of comparable accuracy.