Identifiability of Estimates of Two-Phase Reservoir Properties in History Matching

1984 ◽  
Vol 24 (06) ◽  
pp. 697-706 ◽  
Author(s):  
A.T. Watson ◽  
G.R. Gavalas ◽  
J.H. Seinfeld
Keyword(s):  
Pressure Drop ◽  
History Matching ◽  
Single Phase ◽  
Pressure Data ◽  
Relative Permeabilities ◽  
Unknown Parameters ◽  
Reservoir Properties ◽  
Matching Problem ◽  
Two Phase ◽  
Reservoir History Matching

Abstract Since the number of parameters to be estimated in a reservoir history match is potentially quite large, it is important to determine which parameters can be estimated with reasonable accuracy from the available data. This aspect can be called determining the identifiability of the parameters. The identifiability of porosity and absolute parameters. The identifiability of porosity and absolute and relative permeabilities on the basis of flow and pressure data in a two-phase (oil/water) reservoir is pressure data in a two-phase (oil/water) reservoir is considered. The question posed is: How accurately can one expect to estimate spatially variable porosity and absolute permeability and relative permeabilities given typical permeability and relative permeabilities given typical production and pressure data" To gain insight into this production and pressure data" To gain insight into this question, analytical solutions for pressure and saturation in a one-dimensional (1D) waterflood are used. The following, conclusions are obtained.Only the average value of the porosity can be determined on the basis of water/oil flow measurements.The permeability distribution can be determined from pressure drop data with an accuracy depending on the pressure drop data with an accuracy depending on the mobility ratio.Exponents in a power function representation of the relative permeabilities can he determined from WOR data alone but not nearly so accurately as when pressure drop and flow data are used simultaneously. Introduction The utility of reservoir simulation in predicting reservoir behavior is limited by the accuracy with which reservoir properties can be estimated. Because of the high costs properties can be estimated. Because of the high costs associated with coring analysis, reservoir engineers must rely, on history matching as a means of estimating reservoir properties. In this process a history match is carried out by choosing the reservoir properties as those that result in simulated well pressure and flow data that match as closely as possible those measured during production. In general, reservoir properties at each gridblock in the simulator represent the unknown values to be determined. Although there are efficient methods for estimating such a large number of unknowns, it has long been recognized from the results of single phase history matching exercises that many different sets of parameter values may yield a nearly identical match of observed and predicted pressures. The conventional single phase predicted pressures. The conventional single phase history matching problem is in fact a mathematically illposed problem, which explains its nonunique behavior. Such a situation is, in short, the result of the large number of unknowns to be estimated on the basis of the available data and the lack of sensitivity of the simulator solutions to the parameters. Because of this lack of sensitivity, the need to reduce the number of unknown Parameters or to introduce some additional constraints, such as "smoothness" of the estimated parameters, has been recognized. A problem as important as that of choosing which minimization method to employ in history matching is that of choosing, on the basis of the available well data. which properties actually should be estimated. This selection properties actually should be estimated. This selection depends on the relationship of the unknown parameters to the simulated well data. Ideally one would want to knowwhich parameters can be determined uniquely if the measurements were exact, andgiven the expected level of error in the measurements, how accurately we can expect to be able to estimate the parameters. The first question, that of establishing uniqueness of the estimated parameters, is notoriously difficult to answer, and for a parameters, is notoriously difficult to answer, and for a problem as complicated as reservoir history matching, problem as complicated as reservoir history matching, there are virtually no general results available that allow one to establish uniqueness for permeability or porosity. Thus, it is not possible in general to base our choice of which parameters to estimate on rigorous mathematical uniqueness results. In lieu of an answer to Question 1, the selection of parameters to be estimated can be based on Question 2, parameters to be estimated can be based on Question 2, which is amenable to theoretical analysis. If the expected errors in estimation of any of the parameters, or any linear combination of the parameters, are extremely large, then that parameter or set of parameters can be judged as not identifiable. In such a case, steps may be taken to reduce the number of unknown parameters. In summary, the reservoir history matching problem is a difficult parameter estimation problem, and understanding the relationship between the unknown parameters and the measured data is essential to obtaining meaningful estimates of the reservoir properties. Quantitative studies regarding the accuracy of estimates for single-phase history matching problems have been reported by Shah et al. and Dogru et al. Shah et al,. investigated the optimal level of zonation for use with 1D single-phase (oil) situations. SPEJ P. 697

Estimation of the Location of the Boundary of a Petroleum Reservoir

1975 ◽  
Vol 15 (01) ◽  
pp. 19-38 ◽  
Author(s):  
Wen H. Chen ◽  
John H. Seinfeld
Keyword(s):  
History Matching ◽  
Single Phase ◽  
Steepest Descent ◽  
Petroleum Reservoir ◽  
Pressure Data ◽  
Reservoir Properties ◽  
Matching Problems ◽  
Reservoir Property ◽  
Boundary Location

Abstract This paper considers the problem of estimating the shape of a petroleum reservoir on the basis of pressure data from wells within the boundaries of pressure data from wells within the boundaries of the reservoir. It is assumed that the reservoir properties, such as permeability and porosity, are properties, such as permeability and porosity, are known but that the location of the boundary is unknown. Thus, this paper addresses a new class of history-matching problems in which the boundary position is the reservoir property to be estimated. position is the reservoir property to be estimated. The problem is formulated as an optimal-control problem (the location of the boundary being the problem (the location of the boundary being the control variable). Two iterative methods are derived for the determination of the boundary location that minimizes a functional, depending on the deviation between observed and predicted pressures at the wells. The steepest-descent pressures at the wells. The steepest-descent algorithm is illustrated in two sample problems:the estimation of the radius of a bounded circular reservoir with a centrally located well, andthe estimation of the shape of a two-dimensional, single-phase reservoir with a constant-pressure outer boundary. Introduction A problem of substantial economic importance is the determination of the size and shape of a reservoir. Seismic data serve to define early the probable area occupied by the reservoir; however, probable area occupied by the reservoir; however, a means of using initial well-pressure data to determine further the volume and shape of the reservoir would be valuable. On the basis of representing the pressure behavior in a single-phase bounded reservoir in terms of an eigenfunction expansion, Gavalas and Seinfeld have shown how the total pore volume of an arbitrarily shaped reservoir can be estimated from late transient pressure data at the completed wells. We consider pressure data at the completed wells. We consider here the related problem of the estimation of the shape (or the location of the boundary) of a reservoir from pressure data at an arbitrary number of wells. For reasons of economy, the time allowable for closing wells is limited. It is important, therefore, that any method developed for estimating the shape of a reservoir be applicable, in principle, from the time at which the wells are completed until the current time. Thus, the problem we consider here may be viewed as one in the general realm of history matching, but also one in which the boundary location is the property to be estimated rather than the reserved physical properties. The formulation in the present study assumes that everything is known about the reservoir except its boundary. In actual practice, the reverse is generally true. (By the time sufficient information is available regarding the spatial distribution of permeability and porosity, the boundaries may be fairly well known.) Nevertheless, relatively early in the life of a reservoir, when initial drillstem tests have served to identify an approximate distribution of properties, it may be of some importance to attempt to estimate the reservoir shape. Since knowledge of reservoir properties such as permeability and porosity is at properties such as permeability and porosity is at best a result of initial estimates from well testing, core data, etc., the assumption that these properties are known will, of course, lead only to an approximate reservoir boundary. As the physical properties are identified more accurately, the reservoir boundary can be more accurately estimated. It is the object of this paper to formulate in a general manner and develop and initially test computational algorithms for the class of history-matching problems in which the boundary is the unknown property.There are virtually no prior available results on the estimation of the location of the boundary of a region over which the dependent variable(s) is governed by partial differential equations. The method developed here, based on the variation of a functional on a variable region, is applicable to a system governed by a set of nonlinear partial differential equations with general boundary conditions. The derivation of necessary conditions for optimality and the development of two computational gradient algorithms for determination of the optimal boundary are presented in the Appendix. To illustrate the steepest-descent algorithm we present two computational examples using simulated reservoir data. SPEJ P. 19

History Matching in Two-Phase Petroleum Reservoirs: Incompressible Flow

1977 ◽  
Vol 17 (06) ◽  
pp. 398-406 ◽  
Author(s):  
Bruno van den Bosch ◽  
John H. Seinfeld
Keyword(s):  
Relative Permeability ◽  
Incompressible Flow ◽  
History Matching ◽  
Single Phase ◽  
Absolute Permeability ◽  
Petroleum Reservoir ◽  
Reservoir Properties ◽  
Two Phase ◽  
Pressure Behavior ◽  
Three Phase

Abstract The estimation of porosity, absolute permeability, and relative permeability-saturation relations in a two-phase petroleum reservoir is considered The data available for estimation are assumed to be the oil flow rates and the pressures at the wells. A situation in which the reservoir may be represented by incompressible flow of oil and water also is considered. A hypothetical, circular reservoir with a centrally located producing well is studied in detail. In principle, the porosity can be estimated on the basis of saturation behavior, absolute permeability on The basis of pressure behavior, and permeability on The basis of pressure behavior, and coefficients in the relative permeability-saturation relations on the basis of both saturation and pressure behavior. The ability to achieve good pressure behavior. The ability to achieve good estimates was found to depend on the nature of the flow in a given situation. Introduction The estimation of petroleum reservoir properties on the basis of data obtained during production, so-called history matching, has received considerable attention. By and large, the development of theories for history matching and their application have been confined to reservoirs that can be modeled as containing a single phase. (Wasserman et al. considered the estimation of absolute permeability and porosity in a three-phase reservoir permeability and porosity in a three-phase reservoir by the use of pseudo single-phase model.) Since in the single-phase case only a single partial-differential equation is needed to describe partial-differential equation is needed to describe the reservoir, identification techniques can be tested most conveniently on such a system. The customary parameters to be estimated are the rock porosity (or the storage coefficient) and the porosity (or the storage coefficient) and the directional permeabilities (or the transmissibilities), which are not uniform throughout the reservoir but a function of location. The history matching of single-phase reservoirs through the estimation of these functional properties now appears to be understood quite well. Numerical algorithms have been thoroughly studied and tested. The most difficult aspect is the ill-conditioned nature of the problem arising from the large number of unknowns problem arising from the large number of unknowns relative to the available data. A recent study has elucidated the basic structure of single-phase history-matching problems and has shown how the degree of ill-conditioning may be assessed quantitatively. Reservoirs generally contain more than one fluid phase, however, and consequently are described by phase, however, and consequently are described by mathematical models accounting for the multiphase nature of the system. The porosity and absolute permeabilities still must be estimated as in the permeabilities still must be estimated as in the single-phase case. In addition, it may be necessary to estimate the relative permeability-saturation relationships. Ordinarily, relative permeability vs saturation curves are determined through experiments on core samples. Because it may be difficult to reproduce actual reservoir flow conditions in a laboratory core sample, it is desirable to consider the direct estimation of relative permeability-saturation relationships on the basis of permeability-saturation relationships on the basis of reservoir data that ordinarily would be available during the course of production. This paper represents an initial investigation of the complex identification problem in two-phase reservoirs. The major objective problem in two-phase reservoirs. The major objective of this study is to investigate the feasibility of parameter estimation in two-phase reservoirs in parameter estimation in two-phase reservoirs in which the reservoir is described by a two-phase incompressible flow model. In the next section we present basic equations governing two-phase (oil-water) reservoirs. We first define the general history-matching problem for these reservoirs and then consider a hypothetical reservoir, circularly symmetric with a central producing well in which the flow may be taken as producing well in which the flow may be taken as incompressible. The radial flow reservoir represents a situation in which oil is produced from a water drive. We wanted to estimate reservoir properties based on data obtained at the well. Considering the flow as incompressible enables us to draw a direct comparison to the classic incompressible linear-flow case for which the problem of estimating relative permeabilities is well established. Thus, we permeabilities is well established. Thus, we seek to understand fully the incompressible flow case as a prelude to the general problem of history matching in two-phase compressible flow reservoirs. SPEJ P. 398

Hydrodynamic and Thermal Performance of a Vapor-Venting Microchannel Copper Heat Exchanger

2008 ◽  
Author(s):  
Milnes P. David ◽  
Amy Marconnet ◽  
Kenneth E. Goodson
Keyword(s):  
Heat Exchanger ◽  
Carbon Fiber ◽  
Pressure Drop ◽  
Thermal Resistance ◽  
Heat Exchangers ◽  
Single Phase ◽  
Two Phase ◽  
Microchannel Heat Exchanger ◽  
Large Pressure ◽  
Dry Out

Two-phase microfluidic cooling has the potential to achieve low thermal resistances with relatively small pumping power requirements compared to single-phase heat exchanger technology. Two-phase cooling systems face practical challenges however, due to the instabilities, large pressure drop, and dry-out potential associated with the vapor phase. Our past work demonstrated that a novel vapor-venting membrane attached to a silicon microchannel heat exchanger can reduce the pressure drop for two-phase convection. This work develops two different types of vapor-venting copper heat exchangers with integrated hydrophobic PTFE membranes and attached thermocouples to quantify the thermal resistance and pressure-drop improvement over a non-venting control. The first type of heat exchanger, consisting of a PTFE phase separation membrane and a 170 micron thick carbon-fiber support membrane, shows no improvement in the thermal resistance and pressure drop. The results suggest that condensation and leakage into the carbon-fiber membrane suppresses venting and results in poor device performance. The second type of heat exchanger, which evacuates any liquid water on the vapor side of the PTFE membrane using 200 ml/min of air, reduces the thermal resistance by almost 35% in the single-phase regime in comparison. This work shows that water management, mechanical and surface properties of the membrane as well as its attachment and support within the heat exchanger are all key elements of the design of vapor-venting heat exchangers.

Two-Phase Flow Behavior in Channels With Sudden Area Change Using Experimental and Computational Approach

2017 ◽  
Author(s):  
Ashish Kotwal ◽  
Che-Hao Yang ◽  
Clement Tang
Keyword(s):  
Experimental Data ◽  
Pressure Drop ◽  
Experimental Analysis ◽  
Single Phase ◽  
Computational Analysis ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Two Phase ◽  
Flow Rates ◽  
Area Change

The current study shows computational and experimental analysis of multiphase flows (gas-liquid two-phase flow) in channels with sudden area change. Four test sections used for sudden contraction and expansion of area in experiments and computational analysis. These are 0.5–0.375, 0.5–0.315, 0.5–0.19, 0.5–0.14, inversely true for expansion channels. Liquid Flow rates ranging from 0.005 kg/s to 0.03 kg/s employed, while gas flow rates ranging from 0.00049 kg/s to 0.029 kg/s implemented. First, single-phase flow consists of only water, and second two-phase Nitrogen-Water mixture flow analyzed experimentally and computationally. For Single-phase flow, two mathematical models used for comparison: the two transport equations k-epsilon turbulence model (K-Epsilon), and the five transport equations Reynolds stress turbulence interaction model (RSM). A Eulerian-Eulerian multiphase approach and the RSM mathematical model developed for two-phase gas-liquid flows based on current experimental data. As area changes, the pressure drop observed, which is directly proportional to the Reynolds number. The computational analysis can show precise prediction and a good agreement with experimental data when area ratio and pressure differences are smaller for laminar and turbulent flows in circular geometries. During two-phase flows, the pressure drop generated shows reasonable dependence on void fraction parameter, regardless of numerical analysis and experimental analysis.

Friction Factor Equations Accuracy for Single and Two-Phase Flows

2020 ◽  
Author(s):  
Germano Scarabeli Custódio Assunção ◽  
Dykenlove Marcelin ◽  
João Carlos Von Hohendorff Filho ◽  
Denis José Schiozer ◽  
Marcelo Souza De Castro
Keyword(s):  
Pressure Drop ◽  
Friction Factor ◽  
Single Phase ◽  
Computational Cost ◽  
Two Phase Flows ◽  
Two Phase ◽  
Explicit Equation ◽  
Implicit Equation ◽  
Lower Accuracy ◽  
Statistical Metrics

Abstract Estimate pressure drop throughout petroleum production and transport system has an important role to properly sizing the various parameters involved in those complex facilities. One of the most challenging variables used to calculate the pressure drop is the friction factor, also known as Darcy–Weisbach’s friction factor. In this context, Colebrook’ s equation is recognized by many engineers and scientists as the most accurate equation to estimate it. However, due to its computational cost, since it is an implicit equation, several explicit equations have been developed over the decades to accurately estimate friction factor in a straightforward way. This paper aims to investigate accuracy of 46 of those explicit equations and Colebrook implicit equation against 2397 experimental points from single-phase and two-phase flows, with Reynolds number between 3000 and 735000 and relative roughness between 0 and 1.40 × 10−3. Applying three different statistical metrics, we concluded that the best explicit equation, proposed by Achour et al. (2002), presented better accuracy to estimate friction factor than Colebrook’s equation. On the other hand, we also showed that equations developed by Wood (1966), Rao and Kumar (2007) and Brkić (2016) must be used in specifics conditions which were developed, otherwise can produce highly inaccurate results. The remaining equations presented good accuracy and can be applied, however, presented similar or lower accuracy than Colebrook’s equation.

scholarly journals Thermo-Fluidic Characteristics of Two-Phase Ice Slurry Flows Based on Comparative Numerical Methods

Processes ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 898 ◽  
Author(s):  
Shehnaz Akhtar ◽  
Haider Ali ◽  
Cheol Woo Park
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Pressure Drop ◽  
Flow Velocity ◽  
Mass Fraction ◽  
Single Phase ◽  
Ice Slurry ◽  
Two Phase ◽  
Ice Concentration ◽  
Eulerian Models

Ice slurry is a potential secondary refrigerant for commercial refrigeration systems because of its remarkable thermal properties. It is necessary to optimize the heat transfer process of ice slurry to reduce the energy consumption of the refrigeration system. Thus, this study investigates the heat transfer performance of single-phase (aqueous solution) and two-phase (ice slurry) refrigerants in a straight horizontal tube. The numerical simulations for ice slurry were performed with ice mass fraction ranging from 5% to 20%. The effects of flow velocity and ice concentration on the heat transfer coefficient were examined. The results showed that heat transfer coefficient of ice slurry is considerably higher than those of single-phase flow, particularly at high flow velocity and ice content, where increase in heat transfer with a factor of two was observed. The present results confirmed that ice slurry heat transfer ability is considerably affected by flow velocity and ice concentration in laminar range. Moreover, the second part of this paper reports on the credibility three distinct two-phase Eulerian–Eulerian models (volume of fluid (VOF), mixture, and Eulerian) for the experimental conditions reported in the literature. All two-phase models accurately predict the thermal field at low ice mass fraction but underestimate that at high ice mass fractions. Regardless of the thermal discrepancies, the Eulerian–Eulerian models provide quite reasonable estimation of pressure drop with reference to experimental data. The numerical predictions from the VOF model are more accordant with the experimental results and the maximum percentage error is limited to ~20% and ~13% for thermal and pressure drop predictions, respectively.

Correlations Between Pressure Drop and Two-Phase Flow Regimes in Microchannel Networks

2009 ◽  
Author(s):  
Christian Weinmu¨ller ◽  
Dimos Poulikakos
Keyword(s):  
Experimental Data ◽  
Pressure Drop ◽  
Flow Regime ◽  
Single Phase ◽  
Two Phase Flow ◽  
Flow Regimes ◽  
Phase Flow ◽  
Two Phase ◽  
Research Activities ◽  
Wide Range

Microfluidics has experienced a significant increase in research activities in recent years with a wide range of applications emerging, such as micro heat exchangers, energy conversion devices, microreactors, lab-on-chip devices and micro total chemical analysis systems (μTAS). Efforts to enhance or extend the performance of single phase microfluidic devices are met by two-phase flow systems [1, 2]. Essential for the design and control of microfluidic systems is the understanding of the fluid/hydrodynamic behavior, especially pressure drop correlations. These are well established for single phase flow, however, analytical correlations for two-phase flow only reflect experimentally obtained values within an accuracy of ± 50% [3, 4]. The present study illustrates the effect of two-phase flow regimes on the pressure drop. Experimental measurement data is put into relation of calculated values based on established correlations of Lockhart-Martinelli with Chisholm modifications for macroscopic flows [5, 6] and Mishima-Hibiki modifications for microscale flows [7]. Further, the experimental pressure drop data is superimposed onto two-phase flow maps to identify apparent correlations of pressure drop abnormalities and flow regimes. The experiments were conducted in a square microchannel with a width of 200 μm. Optical access is guaranteed by an anodically bonded glass plate on a MEMS fabricated silicon chip. Superficial velocities range from 0.01 m/s to 1 m/s for the gas flow and from 0.0001 m/s to 1 m/s for the liquid flow with water as liquid feed and CO2 as gas. The analysis of the flow regimes was performed by imaging the distinct flow regimes by laser induced fluorescence microscopy, employing Rhodamine B as the photosensitive dye. The pressure drop was synchronically recorded with a 200 mbar, 2.5 bar and 25 bar differential pressure transmitter and the data was exported via a LabView based software environment, see Figure 1. Figure 2 illustrates the experimentally obtained pressure drop in comparison to the calculated values based on the Lockhard-Martinelli correlation with the Chisholm modification and the Mishima-Hibiki modification. For both cases the predications underestimate the two-phase pressure drop by more than 50%. Nevertheless, the regression of the experimental data has an offset of linear nature. Two-phase flow is assigned to flow regime maps of bubbly, wedging, slug or annular flow defined by superficial gas and liquid velocities. In Figure 3 the pressure drop is plotted as a surface over the corresponding flow regime map. Transition lines indicate a change of flow regimes enclosing an area of an anticline in the pressure data. In the direct comparison between the calculated and the measured values, the two surfaces show a distinct deviation. Especially, the anticline of the experimental data is not explained by the analytical correlations. Figure 4 depicts the findings of Figure 3 at a constant superficial velocity of 0.0232 m/s. The dominant influence of the flow regimes on the pressure drop becomes apparent, especially in the wedging flow regime. The evident deviation of two-phase flow correlations for the pressure drop is based on omitting the influence of the flow regimes. In conclusion, the study reveals a strong divergence of pressure drop measurements in microscale two-phase flow from established correlations of Lockhart-Martinelli and recognized modifications. In reference to [8, 9], an analytical model incorporating the flow regimes and, hence, predicting the precise pressure drop would be of great benefit for hydrodynamic considerations in microfluidics.

Flow Structures and Frictional Characteristics on Two-Phase Flow in Microchannels in PEM Fuel Cells

2005 ◽  
Author(s):  
Eon Soo Lee ◽  
Carlos H. Hidrovo ◽  
Julie E. Steinbrenner ◽  
Fu-Min Wang ◽  
Sebastien Vigneron ◽  
...  
Keyword(s):  
Pressure Drop ◽  
Gas Phase ◽  
Single Phase ◽  
Film Flow ◽  
Two Phase Flow ◽  
Water Injection ◽  
Injection Rate ◽  
Flow Structures ◽  
Phase Flow ◽  
Two Phase

This experimental paper presents a study of gas-liquid two phase flow in rectangular channels of 500μm × 45μm and 23.7mm long with different wall conditions of hydrophilic and hydrophobic surface, in order to investigate the flow structures and the corresponding friction factors of simulated microchannels of PEMFC. The main flow in the channel is air and liquid water is injected at a single or several discrete locations in one side wall of the channel. The flow structure of liquid water in hydrophilic wall conditioned channel starts from wavy flow, develops to stable stratified film flow, and then transits to unstable fluctuating film flow, as the pressure drop and the flow velocity of air increase from around 10 kPa to over 100 kPa. The flow structure in hydrophobic channel develops from the slug flow to slug-and-film flow with increasing pressure drop and flow velocity. The pressure drop for single phase flow is measured for a base line study, and the fRe product is in close agreement with the theoretical value (fRe = 85) of the conventional laminar flow of aspect ratio 1:11. At the low range of water injection rate, the gas phase fRe product of the two phase flow based on the whole channel area was not substantially affected by the water introduction. However, as the water injection rate increases up to 100 μL/min, the gas phase fRe product based on the whole channel area deviates highly from the single phase theoretical value. The gas phase fRe product with the actual gas phase area corrected by the liquid phase film thickness agrees with the single phase theoretical value.

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