Academic literature on the topic 'Darcy flow model'
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Journal articles on the topic "Darcy flow model"
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Wang, Yuan, Yu-long Niu, and Qiang Feng. "Study on the REV Size of Fractured Rock in the Non-Darcy Flow Based on the Dual-Porosity Model." Geofluids 2018 (2018): 1–22. http://dx.doi.org/10.1155/2018/7535927.
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For the problem of whether the representative elementary volume (REV) obtained in the Darcy flow is also applicable to the case of the non-Darcy flow, the study on the REV size within the non-Darcy flow is proposed tentatively. The concept of the REV in the non-Darcy flow is based on the definition of the REV. According to the determination of the REV in the Darcy flow, the intrinsic permeability k and non-Darcy coefficient β are used simultaneously for the determination of the REV in the non-Darcy flow. The pore pressure cohesive element (PPCE) is developed with the subroutine in ABAQUS. Then the simulation method of the Darcy and non-Darcy flow in the fractured rock mass is built using the PPCE. The proposed plan is examined through the comparison with existing research results. It is validated that this technic is efficient and accurate in simulating the Darcy and non-Darcy flow in the fractured rock mass. Combined with fracture networks generated by Monte Carlo Simulation technique, the PPCE is applied to the study on the REV size. Both conditions of the Darcy and non-Darcy flow are simulated for comparison. The simulation results of this model show that the REV of the non-Darcy flow is inconsistent with the REV of the Darcy flow, and the REV of the non-Darcy flow is more significant than the REV of the Darcy flow. The intrinsic permeability k tensors obtained in the Darcy flow and the non-Darcy flow are basically the same.
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Di Nucci, Carmine, and Daniele Celli. "From Darcy Equation to Darcy Paradox." Fluids 7, no. 4 (March 22, 2022): 120. http://dx.doi.org/10.3390/fluids7040120.
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This theoretical paper focuses on the single-phase fluid flow through a granular porous medium. The emphasis is on the Darcy flow regime (without free boundary) of a linear viscous fluid in a saturated, deformable, homogeneous porous medium. The approach is developed at the Darcy scale (also referred to as macroscale or phenomenological scale). Within this framework, some discrete aspects of the flow model are highlighted, the governing equations are revisited, the thermodynamic state functions are reconsidered, and the Darcy paradox is presented. The Darcy paradox is illustrated for the isoshoric-isothermal flow of a viscous fluid in the liquid state, in a homogenous porous medium. After some remarks about the intrinsic assumption of this kind of flow, the governing equations are reduced to a well-known parabolic equation. According to this equation, infinitesimal pressure disturbances diffuse at an infinite speed. To remove this paradox, a mathematical model, based on the elementary scales method, is employed.
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Yang, Bin, Tianhong Yang, Zenghe Xu, Honglei Liu, Wenhao Shi, and Xin Yang. "Numerical simulation of the free surface and water inflow of a slope, considering the nonlinear flow properties of gravel layers: a case study." Royal Society Open Science 5, no. 2 (February 2018): 172109. http://dx.doi.org/10.1098/rsos.172109.
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Groundwater is an important factor of slope stability, and 90% of slope failures are related to the influence of groundwater. In the past, free surface calculations and the prediction of water inflow were based on Darcy's law. However, Darcy's law for steady fluid flow is a special case of non-Darcy flow, and many types of non-Darcy flows occur in practical engineering applications. In this paper, based on the experimental results of laboratory water seepage tests, the seepage state of each soil layer in the open-pit slope of the Yanshan Iron Mine, China, were determined, and the seepage parameters were obtained. The seepage behaviour in the silt layer, fine sand layer, silty clay layer and gravelly clay layer followed the traditional Darcy law, while the gravel layers showed clear nonlinear characteristics. The permeability increases exponentially and the non-Darcy coefficient decreases exponentially with an increase in porosity, and the relation among the permeability, the porosity and the non-Darcy coefficient is investigated. A coupled mathematical model is established for two flow fields, on the basis of Darcy flow in the low-permeability layers and Forchheimer flow in the high-permeability layers. In addition, the effect of the seepage in the slope on the transition from Darcy flow to Forchheimer flow was considered. Then, a numerical simulation was conducted by using finite-element software (FELAC 2.2). The results indicate that the free surface calculated by the Darcy–Forchheimer model is in good agreement with the in situ measurements; however, there is an evident deviation of the simulation results from the measured data when the Darcy model is used. Through a parameter sensitivity analysis of the gravel layers, it can be found that the height of the overflow point and the water inflow calculated by the Darcy–Forchheimer model are consistently less than those of the Darcy model, and the discrepancy between these two models increases as the permeability increases. The necessity of adopting the Darcy–Forchheimer model was explained. The Darcy–Forchheimer model would be applicable in slope engineering applications with highly permeable rock.
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Lai, Bitao, Jennifer L. Miskimins, and Yu-Shu Wu. "Non-Darcy Porous-Media Flow According to the Barree and Conway Model: Laboratory and Numerical-Modeling Studies." SPE Journal 17, no. 01 (October 19, 2011): 70–79. http://dx.doi.org/10.2118/122611-pa.
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Summary This paper presents the results of our new experimental studies conducted for high flow rates through proppant packs, which show that the Barree and Conway (2004) flow model is capable of overcoming limitations of the Forchheimer non-Darcy equation at very high flow rates. To quantify the non-Darcy flow behavior using the Barree and Conway model, a numerical model is developed to simulate non-Darcy flow. In addition, an analytical solution is presented for steady-state linear non-Darcy flow and is used to verify the numerical-simulation results. The numerical model incorporates the Barree and Conway model into a general-purpose reservoir simulator for modeling multidimensional, single-phase non-Darcy flow in porous and fractured media and supplements the laboratory findings. The numerical model is then used to perform sensitivity analysis of the Barree and Conway flow model's parameters and to investigate transient behavior of non-Darcy flow at an injection well.
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Sefidgar, Mostafa, M. Soltani, Kaamran Raahemifar, and Hossein Bazmara. "Effect of Fluid Friction on Interstitial Fluid Flow Coupled with Blood Flow through Solid Tumor Microvascular Network." Computational and Mathematical Methods in Medicine 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/673426.
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A solid tumor is investigated as porous media for fluid flow simulation. Most of the studies use Darcy model for porous media. In Darcy model, the fluid friction is neglected and a few simplified assumptions are implemented. In this study, the effect of these assumptions is studied by considering Brinkman model. A multiscale mathematical method which calculates fluid flow to a solid tumor is used in this study to investigate how neglecting fluid friction affects the solid tumor simulation. The mathematical method involves processes such as blood flow through vessels and solute and fluid diffusion, convective transport in extracellular matrix, and extravasation from blood vessels. The sprouting angiogenesis model is used for generating capillary network and then fluid flow governing equations are implemented to calculate blood flow through the tumor-induced capillary network. Finally, the two models of porous media are used for modeling fluid flow in normal and tumor tissues in three different shapes of tumors. Simulations of interstitial fluid transport in a solid tumor demonstrate that the simplifications used in Darcy model affect the interstitial velocity and Brinkman model predicts a lower value for interstitial velocity than the values that Darcy model predicts.
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Fu, Xiang, Xiang Fang Li, Shi Qing Cheng, Liang Huang, and Xiang Rong Nie. "Pressure Behavior of a Coupling Model with Variable Permeability Effect." Applied Mechanics and Materials 152-154 (January 2012): 364–68. http://dx.doi.org/10.4028/www.scientific.net/amm.152-154.364.
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It is indicated that the fluid flow does not abide by the Darcy law in certain low permeability reservoirs. The results of pressure date analysis in these low permeability reservoirs would be inaccurate when generalized using Darcy flow models. To improve analysis results, based on the concept of variable permeability effect, the dual non-Darcy coupling flow model is established in this paper. The iterative algorithm is adopted to solve the differential set of equations. This work also describes the pressure behaviors by pressure response curves.
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Fu, Xiang, Xiang Fang Li, Shi Qing Cheng, Liang Huang, and Xiang Rong Nie. "Pressure Behavior of a Coupling Model with Variable Permeability Effect." Applied Mechanics and Materials 152-154 (January 2012): 689–93. http://dx.doi.org/10.4028/www.scientific.net/amm.152-154.689.
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It is indicated that the fluid flow does not abide by the Darcy law in certain low permeability reservoirs. The results of pressure date analysis in these low permeability reservoirs would be inaccurate when generalized using Darcy flow models. To improve analysis results, based on the concept of variable permeability effect, the dual non-Darcy coupling flow model is established in this paper. The iterative algorithm is adopted to solve the differential set of equations. This work also describes the pressure behaviors by pressure response curves.
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Aryanti, N., Y. Bindar, and I. G. Wenten. "Two Dimentional Numerical Models Of Hollow Fiber Membrane Contactor." REAKTOR 6, no. 2 (June 19, 2017): 77. http://dx.doi.org/10.14710/reaktor.6.2.77-84.
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Membrane contactor is separation processing unit using membrane as a contacting device. The major advantage of membrane contactor relies on its high contact area compared to conventional scrubber. One of the important applications of membrane contactor is to reduce emission of acid gases. In this work, modeling of membrane contactor is conductedto describe concentration distribution along fiber length used to predict effective fiber length by solving mass conservation equation. Solving of mass conservation equation required information of fluid flow distribution obtained by solving continuity and momentum equation simultaneously. The finite volume method is used to obtain the solution. Modeling of fluid flow was carried out by adding Darcy`s and Brinkman-Darcy flow models into Navier-Stokes equation. The momentum and continuity equation are solved for two-dimentional cylindrical coordinate. The result of velocity profile at axial direction were validated with Pangrle et.al. (1992) experimental data. The comparison shows that consideration using Brinkman-Darcy flow model give agood agreement with experimental data in which maximal axial velocity achieved is 0,047 m/s for this model and 0,05 m/s for experimental data.the concentration profile at radial direction using Darcy and Brickman-Darcy flow models have also been investigated. Furthermore, concentration profile at axial direction using the both two flow models indicate a decrease of concentration along fiber length. The comparison between models and experimental data by Subhakti and Azmier (1997) agree very closely to the Brinkman- Darcy flow model. The prediction of effective fiber length was conducted based on minimum economical flux oe\f membrane contactor. The calculation gives the effective fiber length obtained is 0.19 m at gas concentration, gas flow rate, and sorbent concentration of 0.02 mol/L, 0.8 m/s and 0.256 M respectively.Keywords : modeling, membrane contactor, Darcy, Brinkman-Darcy
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Hdhiri, Najib, and Brahim Ben Beya. "Numerical study of laminar mixed convection flow in a lid-driven square cavity filled with porous media." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 4 (April 3, 2018): 857–77. http://dx.doi.org/10.1108/hff-04-2016-0146.
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Purpose The purpose of this study is to produce a numerical model capable of predicting the mixed convection flows in a rectangular cavity filled with a porous medium and to analyze the effects of several parameters on convective flow in porous media in a differentially heated enclosure. Design/methodology/approach The authors used the finite volume method. Findings The authors predicted and analyzed the effects of Richardson number, Darcy number, porosity values and Prandtl number in heat transfer and fluid flow. On other hand, the porosity and Richardson number values lead to reducing the heat transfer rate of mixed convection flow in a porous medium. Originality/value A comparison between Darcy–Brinkman–Forchheimer model and Darcy–Brinkman model is discussed and analyzed. The authors finally conclude that the Darcy–Brinkman model overestimates the heat transfer rate.
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Yan, Liang Dong, Zhi Juan Gao, and Feng Gang Dai. "Effective use Model of Low Permeability Oil Reservoir." Advanced Materials Research 753-755 (August 2013): 53–57. http://dx.doi.org/10.4028/www.scientific.net/amr.753-755.53.
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The throat of low permeability oil reservoir is narrow and small, the reservoir fluid flow resistance is big, and with the start-up pressure gradient, compare with medium and high permeability reservoir fluid flow, the characteristics are obviously different in performance for non-darcy flow at low speed. This kind of oil field reservoir started in the process of mining scope is small, the degree of use and the development effect is low. To solve these problems, this paper established considering start-up pressure gradient of the new unstable seepage flow mathematical model of non-darcy radial flow which the analytical solution and the productivity equation is deduced, established the effective radius of the use of low permeability reservoirs, and systemicly researched the calculation method of area well pattern of different types of non-darcy seepage.
Dissertations / Theses on the topic "Darcy flow model"
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Fahs, Amin. "Modeling of naturel convection in porous media : development of semi-analytical and spectral numerical solutions of heat transfer problem in special domains." Thesis, Strasbourg, 2021. https://publication-theses.unistra.fr/restreint/theses_doctorat/2021/Fahs_Amin_2021_ED269.pdf.
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Le problème de la cavité poreuse carrée est largement utilisé comme cas de référence courant pour les problèmes de Convection Naturelle (CN) en milieux poreux. Il peut être utilisé pour plusieurs applications numériques, théoriques et pratiques. Par ailleurs, toutes les solutions de haute précision existantes dans la littérature scientifique sont développées dans des conditions de régime permanent. Cependant, il est bien connu que les processus de CN dans les milieux poreux se produisent naturellement dans un régime dépendant du temps, car les conditions aux limites peuvent être variables dans le temps. Pour surmonter cette difficulté, la solution en régime permanent est souvent simulée comme une solution transitoire qui évolue jusqu'à atteindre l'état d'équilibre. Ces régimes dépendant du temps sont très efficaces pour détecter les effets des variations de paramètres sur le processus physique de CN, en particulier pour les sujets d'intérêt de cette thèse: la variation du niveau d'inclinaison du domaine et la prise en compte des variations de température de la paroi chaude dans le temps. À cet effet, trois objectifs sont identifiés dans cette thèse: 1. Développer une solution de convection naturelle en fonction du temps dans des milieux poreux en utilisant le Modèle Darcy en deux modes: transitoire et instable. 2. Étudier le comportement en fonction du temps de la convection naturelle dans des milieux poreux ayant le niveau d'inclinaison du domaine comme paramètre variable dans deux modes: transitoire et instable. 3. Développer une solution de convection naturelle en fonction du temps dans des milieux poreux en utilisant le Modèle Darcy-Lapwood-Brinkman en deux modes: transitoire et instable. Pour ce faire, du fait de la grande précision dans les domaines simplement connectés, une méthode spectrale de résidus pondérés de type Galerkin est choisie pour développer une solution au problème de CN dans une cavité carrée poreuse. L’application de la procédure de Fourier-Galerkin (FG), deux configurations traitant des régimes instables sont considérées où chaque solution est dérivée pour une large gamme des nombres de Rayleigh (Ra) avec d'autres conditions spéciales. Ce travail de thèse est subdivisé en cinq chapitres. Dans le premier chapitre, nous avons présenté un aperçu physique du processus de convection naturelle en milieux poreux. Dans le deuxième chapitre, le développement mathématique des équations, la méthode de résolution et la procédure de résolution sont décrits en détails. Dans le chapitre trois, la première étude de cas de cette thèse, la solution dépendante du temps de la convection naturelle dans une cavité carrée remplie de milieux poreux saturé utilisant le modèle de Darcy est développé. Dans le chapitre quatre, le problème de variation temporelle de Darcy-Lapwood- Brinkman de CN dans une enceinte poreuse saturée carrée est étudié. Dans le chapitre cinq, les solutions dépendant du temps sont développées pour le problème de convection naturelle utilisant la loi de Darcy dans une cavité poreuse inclinée et considéré comme une étude complète sur les effets de l'inclinaison du domaine sur le processus physique du problème de convection libre. Pour tous les cas, les régimes transitoires et instables sont considérésThe problem of the porous square cavity is extensively used as a common benchmark case for Natural convection (NC) problem in porous media. It can be used for several numerical, theoretical, and practical purposes. All the existing high accurate solutions are developed under steady-state conditions. However, it is well known that the processes of NC in porous media occurs naturally in a time-dependent procedure, as boundary conditions can be variable in time. Also, the convergence of the steady-state solution is known to be difficult. To overcome this difficulty, the steady-state solution is often simulated as a transient solution that evolves until reaching the steady-state condition. These time-dependent modes are very efficient to detect the effects of the parameter variations on the physical process of NC, especially for the subject of interest in this thesis: the domain inclination level and hot wall temperature variation in time. For this purpose, three goals are identified in this Thesis: 1. Developing a time-dependent solution of natural convection in porous media using the Darcy model in two modes: Transient and unsteady. 2. Investigating the time-dependent behavior of natural convection in porous media having the domain inclination level as a variable parameter in two modes: Transient and unsteady. 3. Developing a time-dependent solution of natural convection in porous media using the Darcy-Lapwood-Brinkman model in two modes: Transient and unsteady. To do so, according to the high accuracy in the simply connected domains, one of the Galerkin spectral weighted residual method is chosen to develop a space-time dependent solution for NC problem in a square porous cavity. Applying the Fourier-Galerkin (FG) procedure, two configurations dealing with transient and unsteady regimes are considered where each solution is derived for a wide range of Rayleigh numbers with other special conditions. This work of thesis is explained in details as five chapters.The NC physical process with the time-dependent variations is described in the transient mode to reach the steady-state solution and for the unsteady mode during a one period using periodic sinusoidal boundary conditions on the cavity hot wall. Finally, the work of this thesis is described in details in five chapters; while the sixth and last chapter is devoted to the summary and conclusion.The results in this thesis work provide a set of high-accurate data that are published in three papers to be used for testing numerical codes of heat transfer in time-dependent configurations
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Hyde, Eoin Ronan. "Multi-scale parameterisation of static and dynamic continuum porous perfusion models using discrete anatomical data." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:4c7df64f-b134-4b5c-8502-e34fb2c937c9.
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The aim of this thesis is to replace the intractable problem of using discrete flow models within large vascular networks with a suitably parameterised and tractable continuum perfusion model. Through this work, we directly address the hypothesis that discrete vascular data can be incorporated within continuum perfusion models via spatially-averaged parameterisation techniques. Chapter 1 reviews biological perfusion from both clinical and computational modelling perspectives, with a particular focus on myocardial perfusion. In Chapter 2, a synthetic 3D vascular network was constructed, which was controllable in terms of its size and properties. A multi-compartment static Darcy perfusion model of this discrete system was parameterised via a number of techniques. Permeabilities were derived using: (i) porosity-scaled isotropic (ϕI); (ii) Huyghe and Van Campen (HvC); and (iii) projected-PCA parameterisation methods. It was found that HvC permeabilities and pressure-coupling fields derived from the discrete data produced the best comparison to the spatially-averaged Poiseuille pressure. In Chapter 3, the construction and analysis of high-resolution anatomical arterial vascular models was undertaken. In Chapter 4, various anatomically-derived vascular networks were used to parameterise our perfusion model, including a microCT-derived rat capillary network, a single arterial subtree, and canine and porcine whole-organ arterial models. Allowing for general-connectivity (as opposed to strictly-hierarchical connectivity) yielded a significant improvement on the continuum model pressure. For the whole-organ model however, it was found that the best results were obtained by using porosity-scaled isotropic permeabilities and anatomically-derived pressure-coupling fields. It was also discovered that naturally occurring small length but relatively large radius vessels were not suitable for the HvC method. In Chapter 5, the suitability of derived parameters for use within a dynamic perfusion model was examined. It was found that the parameters derived from the original static network were adequate for application throughout the cardiac cycle. Chapter 6 presents a concluding discussion, highlighting limitations and future directions to be investigated.
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Thibaud, Laurent. "Contribution à l'étude de la convection naturelle à l'intérieur d'un cylindre vertical poreux soumis à une densité de flux thermique parietal constante : application aux silos à grains." Poitiers, 1988. http://www.theses.fr/1988POIT2299.
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Brihi, Sarra. "Mathematical analysis and numerical approximation of flow models in porous media." Thesis, Normandie, 2018. http://www.theses.fr/2018NORMC263/document.
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Cette thèse est consacrée à l'étude des équations du Darcy Brinkman Forchheimer (DBF) avec des conditions aux limites non standards. Nous montrons d'abord l'existence de différents type de solutions (faible, forte) correspondant au problème DBF stationnaire dans un domaine simplement connexe avec des conditions portants sur la composante normale du champ de vitesse et la composante tangentielle du tourbillon. Ensuite, nous considérons le système Brinkman Forchheimer (BF) avec des conditions sur la pression dans un domaine non simplement connexe. Nous prouvons que ce problème est bien posé ainsi que l'existence de la solution forte. Nous établissons la régularité de la solution dans les espaces L^p pour p >= 2.L'étude et l'approximation du problème DBF non stationnaire est basée sur une approche pseudo-compressibilité. Une estimation d'erreur d'ordre deux est établie dans le cas o\`u les conditions aux limites sont de types Dirichlet ou Navier.Enfin, une méthode d'éléments finis Galerkin Discontinue est proposée et la convergence établie concernant le problème DBF linéarisé et le système DBF non linéaire avec des conditions aux limites non standardThis thesis is devoted to Darcy Brinkman Forchheimer (DBF) equations with a non standard boundary conditions. We prove first the existence of different type of solutions (weak, strong) of the stationary DBF problem in a simply connected domain with boundary conditions on the normal component of the velocity field and the tangential component of the vorticity. Next, we consider Brinkman Forchheimer (BF) system with boundary conditions on the pressure in a non simply connected domain. We prove the well-posedness and the existence of a strong solution of this problem. We establish the regularity of the solution in the L^p spaces, for p >= 2.The approximation of the non stationary DBF problem is based on the pseudo-compressibility approach. The second order's error estimate is established in the case where the boundary conditions are of type Dirichlet or Navier. Finally, the finite elements Galerkin Discontinuous method is proposed and the convergence is settled concerning the linearized DBF problem and the non linear DBF system with a non standard boundary conditions
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Terblanche, Luther. "The prediction of flow through two-dimensional porous media." Thesis, Stellenbosch : University of Stellenbosch, 2006. http://hdl.handle.net/10019.1/1722.
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Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2006.When considering flow through porous media, different flow regimes may be identified. At very small Reynolds numbers the relation between the pressure gradient and the velocity of the fluid is linear. This flow regime ...
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Hennicker, Julian. "Discrétisation gradient de modèles d’écoulements à dimensions hybrides dans les milieux poreux fracturés." Thesis, Université Côte d'Azur (ComUE), 2017. http://www.theses.fr/2017AZUR4057/document.
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Cette thèse porte sur la modélisation et la discrétisation d’écoulements Darcy dans les milieux poreux fracturés. Nous suivons l’approche des modèles, dits à dimensions hybrides, qui représentent les réseaux de fractures comme des surfaces de codimension 1 immergées dans la matrice. Les modèles considérés prennent en compte les interactions entre matrice et fractures et permettent de traiter des fractures agissant comme conduites ou comme barrières, ce que nécessite de prendre en compte les sauts de pression aux interfaces matrice-fracture. Dans le cas des écoulements diphasiques, nous proposons des modèles, qui prennent en compte les sauts de saturations aux interfaces matrice-fracture, dû à la capillarité. L’analyse numérique est menée dans le cadre général de la méthode de discrétisations gradients, qui est étendue aux modèles considérés. Deux familles de schémas numériques, le schéma Vertex Approximate Gradient et le schéma Volumes Finis Hybrides sont adaptées aux modèles à dimensions hybrides. On prouve via des résultats de densité que ce sont des schémas gradients, pour lesquels la convergence est établie. En diphasique, l’existence d’une solution est obtenue en passant. Plusieurs cas tests sont présentés. En monophasique, on observe la convergence sur des différents types de mailles pour une famille de solutions dans un milieux fracturé hétérogène et anisotrope. En diphasique, nous présentons une série de cas tests afin de comparer les modèles à dimensions hybrides au modèle de référence, dans lequel les fractures ont la même dimension que la matrice. A part quantifier le gain en performance de calcul, ces tests montrent la qualité des différents modèles réduitsThis thesis investigates the modelling of Darcy flow through fractured porous media and its discretization on general polyhedral meshes. We follow the approach of hybrid dimensional models, invoking a complex network of planar fractures. The models account for matrix-fracture interactions and fractures acting either as drains or as barriers, i.e. we have to deal with pressure discontinuities at matrix-fracture interfaces. In the case of two phase flow, we present two models, which permit to treat gravity dominated flow as well as discontinuous capillary pressure at the material interfaces. The numerical analysis is performed in the general framework of the Gradient Discretisation Method, which is extended to the models under consideration. Two families of schemes namely the Vertex Approximate Gradient scheme (VAG) and the Hybrid Finite Volume scheme (HFV) are detailed and shown to fit in the gradient scheme framework, which yields, in particular, convergence. For single phase flow, we obtain convergence of order 1 via density results. For two phase flow, the existence of a solution is obtained as a byproduct of the convergence analysis. Several test cases are presented. For single phase flow, we study the convergence on different types of meshes for a family of solutions. For two phase flow, we compare the hybrid-dimensional models to the reference equidimensional model, in which fractures have the same dimension as the matrix. This does not only provide quantitative evidence about computational gain, but also leads to deep insight about the quality of the proposed reduced models
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Nchabeleng, Mathibele Willy. "Hydraulic fracture with Darcy and non-Darcy flow in a porous medium." Thesis, 2017. http://hdl.handle.net/10539/22740.
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A dissertation submitted to the Faculty of Science,University of the
Witwatersrand, in fulfilment of the requirements for the
degree of Master of Science.
December 2016.This research is concerned with the analysis of a two-dimensional Newtonian fluid-driven fracture in a permeable rock. The fluid flow in the fracture is laminar and the fracture is driven by the injection of a Newtonian fluid into it. Most of the problems in litera- ture involving fluid flow in permeable rock formation have been modeled with the use of Darcy's law. It is however known that Darcy's model breaks down for flows involv- ing high fluid velocity, such as the flow in a porous rock formation during hydraulic fracturing. The Forchheimer flow model is used to describe the non-Darcy fluid flow in the porous medium. The objective of this study is to investigate the problem of a fluid-driven fracture in a porous medium such that the flow in the porous medium is non-Darcy. Lubrication theory is applied to the system of partial di erential equations since the fracture that is considered is thin and its width slowly varies along its length. For this same reason, Perkins-Kern-Nordgren approximation is adopted. The theory of Lie group analysis of differential equations is used to solve the nonlinear coupled sys- tem of partial differential equations to obtain group invariant solutions for the fracture half-width, leak-o depth and length of the fracture. The strength of fluid leak-off at the fracture wall is classi ed into three forms, namely, weak, strong and moderate. A group invariant solution of the traveling wave form is obtained and an exact solution for the case in which there is weak fluid leak-off at the interface is found. A dimensionless parameter, F0, termed the Forchheimer number was obtained and investigated. Nu- merical results are obtained for both the case of Darcy and non-Darcy flow. Computer generated graphs are used to illustrate the analytical and numerical results.
MT2017
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Lehr, Heather Lyn. "Analysis of a Darcy-Stokes system modeling flow through vuggy porous media." Thesis, 2004. http://hdl.handle.net/2152/1234.
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"Optimized Reduced Models for Discrete Fracture Networks Used in Modeling Particle Flow and Transport." Tulane University, 2020.
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archives@tulane.eduDiscrete fracture networks (DFNs) can be modeled with polygonal representations that are useful for geophysical modeling of nuclear waste containment and hydrofrac- turing. Flow and transport calculations are possible, but computationally expensive, limiting the feasibility for model uncertainty quantification. Graphs are used to re- duce model complexity and computation time. We present the formulation of using a graph as a reduced model for DFNs and pose the inversion problem central to this research. We present a novel alternative to Darcy’s law on graphs using the well known Brinkman formulation on the continuum. We apply the Levenberg-Marquardt algorithm to optimize graphs, calibrating them to observed data through the inversion problem. We present the deficiencies in physically motivated graphs, and show how optimized graphs produce better results overall. Our solution finds lumped parameters representing the fracture properties, and is used to reduce the computational time required for particle transport calculations. Breakthrough curves are produced on our obtained solutions, which closely match the high fidelity model. We present examples of creating these reduced models for DFNs with 500 fractures to illustrate the methodology and optimization scheme used to obtain an improved result over a previous formulation.
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Jaime Lopez-Merizalde
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Thomas, Sunil George. "On some problems in the simulation of flow and transport through porous media." 2009. http://hdl.handle.net/2152/6575.
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The dynamic solution of multiphase flow through porous media is of
special interest to several fields of science and engineering, such as petroleum,
geology and geophysics, bio-medical, civil and environmental, chemical engineering
and many other disciplines. A natural application is the modeling of
the flow of two immiscible fluids (phases) in a reservoir. Others, that are broadly
based and considered in this work include the hydrodynamic dispersion (as in
reactive transport) of a solute or tracer chemical through a fluid phase. Reservoir
properties like permeability and porosity greatly influence the flow of these
phases. Often, these vary across several orders of magnitude and can be discontinuous
functions. Furthermore, they are generally not known to a desired level
of accuracy or detail and special inverse problems need to be solved in order
to obtain their estimates. Based on the physics dominating a given sub-region
of the porous medium, numerical solutions to such flow problems may require
different discretization schemes or different governing equations in adjacent regions.
The need to couple solutions to such schemes gives rise to challenging
domain decomposition problems. Finally, on an application level, present day
environment concerns have resulted in a widespread increase in CO₂capture and
storage experiments across the globe. This presents a huge modeling challenge
for the future. This research work is divided into sections that aim to study various
inter-connected problems that are of significance in sub-surface porous media
applications. The first section studies an application of mortar (as well as nonmortar,
i.e., enhanced velocity) mixed finite element methods (MMFEM and
EV-MFEM) to problems in porous media flow. The mortar spaces are first
used to develop a multiscale approach for parabolic problems in porous media
applications. The implementation of the mortar mixed method is presented for
two-phase immiscible flow and some a priori error estimates are then derived
for the case of slightly compressible single-phase Darcy flow. Following this,
the problem of modeling flow coupled to reactive transport is studied. Applications
of such problems include modeling bio-remediation of oil spills and other
subsurface hazardous wastes, angiogenesis in the transition of tumors from a
dormant to a malignant state, contaminant transport in groundwater flow and
acid injection around well bores to increase the permeability of the surrounding
rock. Several numerical results are presented that demonstrate the efficiency
of the method when compared to traditional approaches. The section following
this examines (non-mortar) enhanced velocity finite element methods for solving
multiphase flow coupled to species transport on non-matching multiblock grids.
The results from this section indicate that this is the recommended method of
choice for such problems.
Next, a mortar finite element method is formulated and implemented
that extends the scope of the classical mortar mixed finite element method
developed by Arbogast et al [12] for elliptic problems and Girault et al [62] for
coupling different numerical discretization schemes. Some significant areas of
application include the coupling of pore-scale network models with the classical
continuum models for steady single-phase Darcy flow as well as the coupling
of different numerical methods such as discontinuous Galerkin and mixed finite
element methods in different sub-domains for the case of single phase flow [21,
109]. These hold promise for applications where a high level of detail and
accuracy is desired in one part of the domain (often associated with very small
length scales as in pore-scale network models) and a much lower level of detail at other parts of the domain (at much larger length scales). Examples include
modeling of the flow around well bores or through faulted reservoirs.
The next section presents a parallel stochastic approximation method
[68, 76] applied to inverse modeling and gives several promising results that
address the problem of uncertainty associated with the parameters governing
multiphase flow partial differential equations. For example, medium properties
such as absolute permeability and porosity greatly influence the flow behavior,
but are rarely known to even a reasonable level of accuracy and are very often
upscaled to large areas or volumes based on seismic measurements at discrete
points. The results in this section show that by using a few measurements of
the primary unknowns in multiphase flow such as fluid pressures and concentrations
as well as well-log data, one can define an objective function of the
medium properties to be determined, which is then minimized to determine the
properties using (as in this case) a stochastic analog of Newton’s method. The
last section is devoted to a significant and current application area. It presents a
parallel and efficient iteratively coupled implicit pressure, explicit concentration
formulation (IMPEC) [52–54] for non-isothermal compositional flow problems.
The goal is to perform predictive modeling simulations for CO₂sequestration
experiments.
While the sections presented in this work cover a broad range of topics
they are actually tied to each other and serve to achieve the unifying, ultimate
goal of developing a complete and robust reservoir simulator. The major results
of this work, particularly in the application of MMFEM and EV-MFEM
to multiphysics couplings of multiphase flow and transport as well as in the
modeling of EOS non-isothermal compositional flow applied to CO₂sequestration,
suggest that multiblock/multimodel methods applied in a robust parallel
computational framework is invaluable when attempting to solve problems as
described in Chapter 7. As an example, one may consider a closed loop control
system for managing oil production or CO₂sequestration experiments in huge
formations (the “instrumented oil field”). Most of the computationally costly activity occurs around a few wells. Thus one has to be able to seamlessly connect
the above components while running many forward simulations on parallel
clusters in a multiblock and multimodel setting where most domains employ an
isothermal single-phase flow model except a few around well bores that employ,
say, a non-isothermal compositional model. Simultaneously, cheap and efficient
stochastic methods as in Chapter 8, may be used to generate history matches of
well and/or sensor-measured solution data, to arrive at better estimates of the
medium properties on the fly. This is obviously beyond the scope of the current
work but represents the over-arching goal of this research.text
Books on the topic "Darcy flow model"
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Dudgeon, C. R. Non-Darcy flow of groundwater. Manly Vale, N.S.W: University of New South Wales, Water Research Laboratory, 1985.
Find full textBook chapters on the topic "Darcy flow model"
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Nazarenko, Nelli N., and Anna G. Knyazeva. "Transfer of a Biological Fluid Through a Porous Wall of a Capillary." In Springer Tracts in Mechanical Engineering, 503–20. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-60124-9_22.
Full textAbstract:
AbstractThe treatise proposes a model of biological fluid transfer in a dedicated macropore with microporous walls. The distribution of concentrations and velocity studies in the capillary wall for two flow regimes—convective and diffusive. The largest impact on the redistribution of concentration between the capillary volume and its porous wall is made by Darcy number and correlation of diffusion coefficients and concentration expansion. The velocity in the interface vicinity increases with rising pressure in the capillary volume or under decreasing porosity or without consideration of the concentration expansion.
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Chen, Junbin, Jia’en Lin, Liang Wu, and Xiaoming Wang. "Well Test Model and Analytical Method of Finite Conductivity Vertical Fracture Bilinear Flow of Low-Speed Non-darcy Flow." In Springer Series in Geomechanics and Geoengineering, 1825–37. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-10-7560-5_164.
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Pradeepa, T., and Ch RamReddy. "Micropolar Fluid Flow over a Frustum of Cone Subjected to Convective Boundary Condition: Darcy–Forchheimer Model." In Lecture Notes in Electrical Engineering, 129–46. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-1824-7_9.
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Madhava Reddy, Ch, Ch RamReddy, and D. Srinivasacharya. "Joule Heating and Thermophoresis Effects on Unsteady Natural Convection Flow of Doubly Stratified Fluid in a Porous Medium with Variable Fluxes: A Darcy–Brinkman Model." In Numerical Heat Transfer and Fluid Flow, 103–12. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-1903-7_13.
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Dine, Houssein Nasser El, Mazen Saad, and Raafat Talhouk. "A Finite Volume Scheme for Darcy-Brinkman’s Model of Two-Phase Flows in Porous Media." In Progress in Industrial Mathematics at ECMI 2016, 695–703. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-63082-3_104.
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Sheikholeslami, Mohsen. "Darcy Model for Nanofluid Flow in a Porous Media by Means of CVFEM." In Application of Control Volume Based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer, 441–82. Elsevier, 2019. http://dx.doi.org/10.1016/b978-0-12-814152-6.00013-8.
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Sheikholeslami, Mohsen. "Non-Darcy Model for Nanofluid Hydrothermal Treatment in a Porous Medium Using CVFEM." In Application of Control Volume Based Finite Element Method (CVFEM) for Nanofluid Flow and Heat Transfer, 483–546. Elsevier, 2019. http://dx.doi.org/10.1016/b978-0-12-814152-6.00014-x.
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Filipovic, Nenad, Milos Radovic, Dalibor D. Nikolic, Igor Saveljic, Zarko Milosevic, Themis P. Exarchos, Gualtiero Pelosi, Dimitrios I. Fotiadis, and Oberdan Parodi. "Computer Predictive Model for Plaque Formation and Progression in the Artery." In Coronary and Cardiothoracic Critical Care, 220–45. IGI Global, 2019. http://dx.doi.org/10.4018/978-1-5225-8185-7.ch012.
Full textAbstract:
In this chapter we described predictive model for plaque formation and progression in the coronary and carotid artery. A full three-dimensional model for plaque formation and progression, coupled with blood flow and LDL concentration is analysed. The Navier-Stokes equations together with the Darcy law for model blood filtration and Kedem-Katchalsky equations are implemented. Additionally, the system of three additional reaction-diffusion equations for simulation of the inflammatory process is coupled with full incremental iterative procedure. We developed hybrid genetic algorithm for fitting parameters of ODE model for oxidized LDL, macrophage, smooth muscle cell and foam cell concentration evolution in time. The animal carotid and coronary artery after 2 month of high fat diet are examined. We compared with CT our computer model of the plaque size for three groups of patients: De-novo, Old-lesions and Control patients. Detailed shear stress distributions for baseline and follow-up for these patients are given. There is a good matching for plaque size and location.
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Filipovic, Nenad, Milos Radovic, Dalibor D. Nikolic, Igor Saveljic, Zarko Milosevic, Themis P. Exarchos, Gualtiero Pelosi, Dimitrios I. Fotiadis, and Oberdan Parodi. "Computer Predictive Model for Plaque Formation and Progression in the Artery." In Handbook of Research on Trends in the Diagnosis and Treatment of Chronic Conditions, 279–300. IGI Global, 2016. http://dx.doi.org/10.4018/978-1-4666-8828-5.ch013.
Full textAbstract:
In this chapter we described predictive model for plaque formation and progression in the coronary and carotid artery. A full three-dimensional model for plaque formation and progression, coupled with blood flow and LDL concentration is analysed. The Navier-Stokes equations together with the Darcy law for model blood filtration and Kedem-Katchalsky equations are implemented. Additionally, the system of three additional reaction-diffusion equations for simulation of the inflammatory process is coupled with full incremental iterative procedure. We developed hybrid genetic algorithm for fitting parameters of ODE model for oxidized LDL, macrophage, smooth muscle cell and foam cell concentration evolution in time. The animal carotid and coronary artery after 2 month of high fat diet are examined. We compared with CT our computer model of the plaque size for three groups of patients: De-novo, Old-lesions and Control patients. Detailed shear stress distributions for baseline and follow-up for these patients are given. There is a good matching for plaque size and location.
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Marzougui, Souad, and Mourad Magherbi. "Irreversibility and Heat Transfer in Darcy-Forchheimer Magnetized Flow in a Porous Double Lid-Driven Cavity Filled With Copper-Water Nanofluid." In Advances in the Modelling of Thermodynamic Systems, 134–53. IGI Global, 2022. http://dx.doi.org/10.4018/978-1-7998-8801-7.ch008.
Full textAbstract:
The present work reports a numerical simulation of entropy generation and heat transfer in a lid-driven porous cavity filled with a nanofluid using Darcy-Forchheimer model. Given the large number of dimensionless parameters related to this problem, some of them are kept constant and therefore the other governing dimensionless number such as number, the Hartman number, and the nanoparticles volume fraction, 0£Ha£50, 2%£φ£8%, respectively. The effects of the nanoparticles volume fraction and Hartman number on the different irreversibilities are studied. Results show that the entropy generation is strongly affected by the increase of Hartmann number and the volume fraction. Results reveal that the irreversibility in the nanofluid decrease with the nanoparticle volume fraction for different Hartmann numbers.
Conference papers on the topic "Darcy flow model"
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Zhang, Andi. "Multiphase flow model of the transition between Darcy flow and Forchheimer flow." In World Environmental and Water Resources Congress 2013. Reston, VA: American Society of Civil Engineers, 2013. http://dx.doi.org/10.1061/9780784412947.050.
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Peng, Xiaolong, Zhilin Qi, Baosheng Liang, and Xueli Liu. "A New Darcy-Stokes Flow Model for Cavity-Fractured Reservoir." In Production and Operations Symposium. Society of Petroleum Engineers, 2007. http://dx.doi.org/10.2118/106751-ms.
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Xiong, Yi, Jinbiao Yu, Hongxia Sun, Jiangru Yuan, Zhaoqin Huang, and Yu-shu Wu. "A New Non-Darcy Flow Model for Low Velocity Multiphase Flow in Tight Reservoirs." In SPE Europec featured at 78th EAGE Conference and Exhibition. Society of Petroleum Engineers, 2016. http://dx.doi.org/10.2118/180072-ms.
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Saboorian-Jooybari, Hadi, and Peyman Pourafshary. "Non-Darcy Flow Effect in Fractured Tight Reservoirs: How Significant Is It at Low Flow Rates and Away from Wellbores?" In SPE Middle East Unconventional Resources Conference and Exhibition. SPE, 2015. http://dx.doi.org/10.2118/spe-172948-ms.
Full textAbstract:
Abstract Reservoir development is increasingly moving towards unconventional resources such as tight reservoirs due to the rapid decline in conventional reserves. Several researches have been done in the petroleum industry addressing significant influence of non-Darcy flow behavior on well deliverability and reservoir performance. In all of the previous works, it is generally believed that the non-Darcy flow in a reservoir, if occurs, becomes considerable within a few feet around the wellbores beyond which it is negligible. Furthermore, when mentioned non-Darcy effect in the literature, attentions are usually attracted to gas wells with high production rates (i.e. rates higher than 10 MMSCF / day). In this paper, correlations that are typically used for determination of non-Darcy coefficient are reviewed. Then, it is shown that the correlations are distinct and lead to considerably different values of the non-Darcy coefficient for the same rock sample. A simple guideline is also presented for choosing the most appropriate correlation for a reservoir. Main body of this paper is directed at accurate description of non-Darcy flow in fractured tight reservoirs. This study evaluates the validity of a widely accepted assumption, which considers non-Darcy effect significant only within a few feet around wellbore of gas wells producing at high rates. A synthetic simulation model is made using the data of a well from one of the Iranian fractured tight reservoirs. A full feature compositional simulator is used for the computations in this study. The distance out of the wellbore is subdivided into a number of regions, and then the necessity of including the non-Darcy component in each of the regions for better performance predictions is investigated. The effect of accounting the non-Darcy term in each of the regions on the simulation results such as production rate, final recovery and pressure behavior is studied. Furthermore, three distinct reservoir fluid types including dry gas, gas condensate and black oil are used with the objective well model to evaluate dependence of non-Darcy effect on the type of flowing fluid. The study results are used to provide guidelines about the necessity of global consideration of the non-Darcy term in simulation of different fluid systems even at low production rates. The results highlight that contribution of the non-Darcy component to flow can be significant even far away from wellbores, thus it must be considered globally in the bulk of reservoirs. Additionally, the study demonstrates that the role of non-Darcy component can be crucial even at low production rates regardless of reservoir fluid type. Therefore, in order to have accurate modeling, design and successful implementation of projects, simulations of fractured tight reservoirs must be performed with the global inclusion of the non-Darcy flow formulations regardless of the type of flowing stream and levels of production rate.
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Belhaj, Hadi, Shabbir Mustafiz, Fuxi Ma, M. Satish, and M. R. Islam. "Modeling Horizontal Well Oil Production Using Modified Brinkman’s Model." In ASME 2005 International Mechanical Engineering Congress and Exposition. ASMEDC, 2005. http://dx.doi.org/10.1115/imece2005-81726.
Full textAbstract:
Horizontal well oil production has been numerically studied by using the Modified Brinkman’s Model. This model has been used along with the Darcy-Weisbach pipe flow equation in modeling of coupled porous medium/pipe flow. The results include seepage flow rate along the horizontal well, velocity distribution, pressure drop, and production pressure drop between the two ends of the horizontal well. They have been compared with those from Darcy model. It is found that when the fluid’s viscosity is low, there is a big difference between the results from the two models. However, when the fluid’s viscosity is high, the difference tends to vanish. In addition, two striking findings have been observed: (a) the curves for the distribution of the seepage flow rate along the pipeline are more flat than that from Darcy model. However, a higher viscosity makes the curve more uneven. This reverses the trend from Darcy model. (b) The velocity in the pipe is more uniform by MBM than that by Darcy model. The curves of V ~x become more uniform in the pipe when the fluid has a lower viscosity. This again reveres the trend from the Darcy model.
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Zeng, Fanhua, and Gang Zhao. "Semi-Analytical Model for Reservoirs with Forchheimer's Non-Darcy Flow." In SPE Gas Technology Symposium. Society of Petroleum Engineers, 2006. http://dx.doi.org/10.2118/100540-ms.
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Balhoff, Matthew Thomas, and Mary Fanett Wheeler. "A Predictive Pore-Scale Model for Non-Darcy Flow in Anisotropic Media." In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, 2007. http://dx.doi.org/10.2118/110838-ms.
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Civan, Faruk. "Phenomenological Filtration Model for Highly Compressible Filter Cakes Involving Non-Darcy Flow." In SPE Mid-Continent Operations Symposium. Society of Petroleum Engineers, 1999. http://dx.doi.org/10.2118/52147-ms.
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Li, Dachang, Corneliu-Liviu Ionescu, Ivbade Thaddeus Ehighebolo, Byron Haynes Jr., Ainur Zhazbayeva, Bakyt Yergaliyeva, and Luigi Francia. "Modeling and Simulation of Non-Darcy or Turbulent Flow for Oil Wells." In SPE Annual Caspian Technical Conference. SPE, 2022. http://dx.doi.org/10.2118/212067-ms.
Full textAbstract:
Abstract Modeling and simulation of non-Darcy or turbulent flow are well documented in the literature and available in commercial reservoir simulators (E300, Intersect) only for gas wells rather than oil wells. There is a need to model non-Darcy or turbulent flow in reservoir simulation for oil wells in the carbonate reservoirs with highly connected and densely distributed fractures and karst. This paper proposes a new non-Darcy or turbulent flow modeling and simulation method for oil wells. Unlike the industry's existing methods for non-Darcy or turbulent flow that focus on the non-Darcy coefficient only, this paper presents a new method that models the ratio between non-Darcy and Darcy flows such that a unified model for a field or a region can be created, which significantly simplifies the non-Darcy or turbulent flow modeling process for multiple wells, especially for future wells. The ratio-based method is simple and comprehensive. It can be easily calibrated with MRT (multiple-rate test) data and implemented into in-house or commercial reservoir simulators using a simulator supported scripting language, e.g., Python etc. Kashagan is the world's largest oil reservoir discovered in the last 30 years that contains highly connected and densely distributed fractures and karst in its rim. The oil production rate for a well in the rim can be higher than several tens KSTB/D if it is not constrained by the facility. The current MRT data in all tested wells clearly show non-Darcy flow phenomenon and confirm that modeling non-Darcy flow is necessary to the field. Kashagan had experienced difficulties to match BHP (bottom hole pressure) and large errors in the blind test due to the OPEC's production curtailment and high-rate tests. Build-up pressure curves were miss-matched and HM (history match) of the crossflows (10 KSTB/D with less than 10 psi) in the bottomhole of a PLT (production logging tool) well during shut-in was challenging. Since modeling non-Darcy flow for oil wells in the commercial simulators, e.g., E300 and Intersect, is unavailable, the simulation team in NCOC has created a new method for the needs of non-Darcy modeling and simulation. The applications of the new method have resulted in the excellent results and solved the issues of history matching BHP, high/low-rate tests, build-up pressure trends, and bottomhole crossflows.
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Shokir, E., and A. Showman. "A New Model for Estimating the Non Darcy Flow Coefficient Using Genetic Programming." In 82nd EAGE Annual Conference & Exhibition. European Association of Geoscientists & Engineers, 2020. http://dx.doi.org/10.3997/2214-4609.202012167.
Full textReports on the topic "Darcy flow model"
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Lohne, Arild, Arne Stavland, Siv Marie Åsen, Olav Aursjø, and Aksel Hiorth. Recommended polymer workflow: Interpretation and parameter identification. University of Stavanger, November 2021. http://dx.doi.org/10.31265/usps.202.
Full textAbstract:
Injecting a polymer solution into a porous medium significantly increases the modeling complexity, compared to model a polymer bulk solution. Even if the polymer solution is injected at a constant rate into the porous medium, the polymers experience different flow regimes in each pore and pore throat. The main challenge is to assign a macroscopic porous media “viscosity” to the fluid which can be used in Darcy law to get the correct relationship between the injection rate and pressure drop. One can achieve this by simply tabulating experimental results (e.g., injection rate vs pressure drop). The challenge with the tabulated approach is that it requires a huge experimental database to tabulate all kind of possible situations that might occur in a reservoir (e.g., changing temperature, salinity, flooding history, permeability, porosity, wettability etc.). The approach presented in this report is to model the mechanisms and describe them in terms of mathematical models. The mathematical model contains a limited number of parameters that needs to be determined experimentally. Once these parameters are determined, there is in principle no need to perform additional experiments.