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ECMOR XIII - 13th European Conference on the Mathematics of Oil Recovery
- Conference date: 10 Sep 2012 - 13 Sep 2012
- Location: Biarritz, France
- ISBN: 978-90-73834-30-9
- Published: 10 September 2012
101 - 114 of 114 results
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A Solution of the Completion Problem of Symmetric Positive Definite Matrices
By J. LeguijtA solution of the completion problem of symmetric positive definite matrices. Symmetric positive definite matrices play an important role in statistics. Typical examples are the covariance and correlation matrices which capture the second order statistics between random variables. In modelling studies these matrices are usually estimated from available data. Unfortunately, this data is not always sufficient to estimate all coefficients of the matrix. For statistical modelling, a complete matrix is needed and for this reason sensible estimates are needed for undefined coefficients. The entropy is a well accepted measure for the information content of a probability density function (pdf). The values given to the undefined coefficients can be chosen such that the information content of the corresponding multi-normal distribution is minimal. In this way a pdf is obtained that honours the available data and does not impose unnecessary extra constraints. A constraint cross entropy minimisation problem has to be solved to compute the values of the missing coefficients. The cross entropy optimization problem can be easily derived from first principles and results in an equation with Lagrange multipliers. The system of equations for the Lagrange multipliers is solved with a Newton iteration. A generalisation is possible such that a complete covariance matrix can be computed from second order statistics between linear combinations of random variables. The method has for instance been used to construct a prior pdf for reservoir models or to compute a cross variogram between two correlated lateral continuous random variables.
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Comparison of High-order Numerical Schemes for Flows in Heterogeneous and Anisotropic Porous Media
Authors V. Baron, Y. Coudière and P. SochalaWe consider unstationnary flows in porous media through Richards equation. Backward Differentiation Formulas are used to discretize the unstationnary term and we propose to compare DG (Discontinuous Galerkin) and DDFV (Discrete Duality Finite Volume) schemes for the discretization of the diffusive term. On the one hand, the flexibility of DG methods and their ample theoretical foundation make them a reliable choice for a number of computational problems. Two DG methods are used for Richards equation; we consider here the Symmetric Interior Penalty Galerkin (SIPG) method because it preserves the natural symmetry of the discrete diffusion operator, and the Local Discontinuous Galerkin (LDG) method which ensures convergence with a positive penalty parameter. On the other hand, the e fficiency of DDFV methods has also been proved to approximate the diffusive fl ux; besides, as finite voume methods, they ensure good preservation of physical properties and offer superconvergence in the L2-norm on a regular basis. Accuracy and robustness of theses schemes are tested and compared on relevant test cases, specially in heterogeneous and anisotropic medium.
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Method of Interpretation Interference Tests of Horizontal Wells in Anisotropic Reservoir
Authors D.V. Posvyanskii, L.A. Gaidukov and A.O. PotapovInterference between wells provide an important information about reservoir characteristics such as permeability, lateral anisotropy (azimuth and ratio), vertical anisotropy. There are some well-known analytical methods for interpretation of interference tests, however they are valid for vertical wells and can be applied to interpretation of horizontal wells test only under the assumption of large distance between wells. Analysis of horizontal well interference tests is a difficult problem which is based on mathematical model of the reservoir. The mathematical model demands solution of the diffusivity equation which describes the single-phase fluid flow in porous media. This equation can be solved numerically, but it can be time-consuming process that restricts the application of this techniques for inverse problems of interference test analysis. In [1] it was proposed method for solution the diffusivity equation used Green’s function technique together with Ewald’s summation algorithm. The last one is a special case of the Poisson summation formula and it allows to solve the diffusivity equation efficiently. It was obtained analytical expression for time-dependent pressure behavior in a reservoir penetrated by horizontal well. The purpose of this work is to provide tools to evaluate reservoir parameters from horizontal observation well pressure response influencing pressure transient created by horizontal and vertical wells in reservoirs. Within the method presented in [1], the analytical expression for observation well pressure time response is obtained. This expression is obtained at the suggestion of the bottom hole pressure is a constant along well trajectory, but it can be time dependent variable. Any assumptions about sufficiently large distance between wells were not used. We consider inference tests between several horizontal wells and a vertical well in anisotropic reservoir which was realized as a research work in a Western Siberian greenfield. Estimation of reservoir characteristics is achieved by solving inverse problem, where the mathematical model of reservoir generates the pressure response to the actual one closely. A detailed sedimentology study was used to choose the initial value of lateral anisotropy azimuth. It is shown this value is in a good agreement with the same one defined from hydrodynamic computation. The series of multidirectional interference tests allow to verify the obtained analytical results. The results of analytical interpretation were validated by matching with numerical results, performed in 3D simulators. [1] E.S.Makarova, S.V. Milutin, D.V.Posvyanskii, V.S.Posvyanskii ECMOR XII P007 2010
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Reservoir Spatial Heterogeneity Modeling Through Pattern-Based Stochastic Simulations
Authors H. Mustapha, R. Dimitrakopoulos and S. ChatterjeeIn this paper a pattern-based stochastic simulation method is presented to model 2D and 3D heterogeneous reservoirs. The method uses a spatial template to extract information in the form of patterns from a training image; i.e., a prior representation of the reservoir. The patterns are grouped into a pattern database and are classified to reduce the computational time. The classification algorithm proposed consists of reducing the dimension of spatial patterns and performing a class centers selection in the lower dimensional space. Conditional and unconditional simulations of reservoir channels are presented. The method is compared with the Filtersim method. Results show that the proposed method is better at producing the multipoint configurations and the main characteristics of the reference images including continuity of channels, and it is very stable with respect to the number of classes and spatial templates used in the simulations.
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Importance and Applicability of a Generalized Shape Factor by Modeling Dual Porosity Reservoirs
By M.T. AmiryShape factor is regarded as a fundamental property of dual porosity reservoirs. Nevertheless, there are some confusion in the literature and simulation practice about its form and applicability. This paper discusses the different definitions showing that the most general is the one given by Heinemann and Mittermeir (2012). The paper tries to present a proper understanding of the nature of the shape factor and demonstrates a detailed mathematical approach to measure all the required parameters to calculate the shape factor for isolated matrix blocks accurately, as well as a simpler practical approach to determine the shape factor by outcrop investigations which can be routinely performed as it regards the practical usability of the shape factor as essential. A fine-scale single matrix block simulation model is used to demonstrate the difference in matrix-fracture interflow using the most widely-used shape factor after Kazemi et al (1992) which ignores permeability anisotropy versus a case that considers the anisotropy tensor. It is demonstrated that isotropic permeability assumption can lead to significantly wrong results for cases of high anisotropy and therefore, the generalized Heinemann and Mittermeir (2012) shape factor which considers all the parameters that affect the flow behavior, is recommended to be used routinely.
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Mathematical Modelling of Acid Treatment of the Bottom Hole Zone for Carbonate Reservoirs
Authors R.R. Tukhvatullina and V.S. PosvyanskiiAcid treatment of the bottom hole formation zone is successfully applied in oil industry to increase well production rates. The value of permeability of the bottom hole zone strongly depends on well production. Acid impacts on porous media and it increases the permeability in a neighbourhood of well. The simulation of the effect of acid injection on permeability evolution is an important task, which demands accounting of various physical phenomena in bottom hole zone. Solution is based on the numerical consideration of the mathematical model of chemical reaction in carbonate reservoir. The main aim of this study is to estimate influence of the Carbon-dioxide gas, which is one of the chemical reaction products, on the permeability of bottom hole zone and well skin-factor. Two-dimensional model of two-phase flow of acid aqueous solution and gas is considered.This phenomena has effect on acid filtration in porous media and it takes into account in numerical simulation. It is shown that in some important cases neglecting of gas phase leads to significant errors in estimation of parameters of bottom hole zone.Well skin-factor after acid treatment is calculated as a function of acid volume and injection time.
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Elliptic Functions in Modeling of Oil Recovery
Authors V. Astafiev, A. Kasatkin and P. RotersThe objective of this paper is to investigate the inflow performance of multiple vertical wells producing from and injecting into a closed reservoir of constant thickness under pseudosteady-state conditions. For this case we represent, like the method of imaginary sources, a closed reservoir as an element of unbounded doubly periodic array of wells and use the elliptic Weierstrass zeta- and sigma-functions to describe this inflow performance. This approach allow us: • to find the pressure distribution and the field of fluid velocities in the reservoir; • to calculate the productivity index (PI) and the Dietz’s shape factor for any shape of reservoirs, not only, like Dietz (1965), for rectangular and triangular ones; • to analyze the influence of a reservoir shape on the Dietz’s shape factor; • to establish, like Valko, Doublet and Blasingame (2000) for a rectangular reservoir, the influence matrix (IM) and the multiwell productivity matrix (MPM) for any shape of reservoirs; • to introduce the multiwell productivity index (MPI) as a norm of MPM and to find the optimal placement of producing wells in a closed reservoir, based on the maximum MPI condition; • to introduce, like Kaviani and Valko (2010) MPI-based method for a rectangular reservoir, the connectivity matrix (CM) for any shape of reservoirs and to evaluate on the base of the CM-approach the interwell connectivity of injector/producer wells in waterflooding of any shape of reservoirs.
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Simulation of the Wave Action Effect on the Two-phase flow Process and the Reservoir Recovery
Authors A.N. Cheremisin and N.A. CheremisinThere exist a number of researches, which prove that the frequencies ranging from one to several thousands hertz as applied to oil reservoirs enhance the well stream and reduce watercut. Well-elaborated are the physical basis and various cases of basic technologies and solutions for commercial applications of vibroacoustic bottom hole treatment to restore the permeability and the increase the inflow in a well, but the question is still open regarding the mechanism to affect the remote flushed-out zone. This work addresses the wave EOR method. We have proven the relation of the natural frequency spectrum of the capillary trapped oil with the pore space in the reservoir, its current oil saturation and the external pressure gradient. The model was developed for the two-phase flow in the acoustic field based on the resonance effect on capillary trapped residual oil saturation. The model was backtested on site in Samotlor and other oil fields. Some model cases resulted in the methodic basis for efficient wave action application for various types of reservoirs and different development conditions. We provide evidence that oil recovery is enhanced locally if the oil reservoir is exposed to acoustic frequencies of 300 to 3000 Hz.
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Computational Method for Relative Permeability Curves based on the Bernoulli Generalized Porous Media Equations
Authors S. Stepanov, N. Cheremisin, S. Sokolov, A. Altunin and A. ShabarovObtaining the relperm curves as a result of multiphase flow calculation immediately in the porous media can be of current interest. Some researches have been carried out to date where relperms are derived by solving the Navier–Stokes equations for communicating pore channels. Those approaches have two significant drawbacks: (1) need to reconstruct the pore space based on the core CT scanning, and (2) intensive work for solving the Navier–Stokes equation system which requires supercomputers. The results of our work help to compute relperm curves using normal computers. We were able to do this due to several factors: (1) simplified pore space reconstruction based on capillary curve data, (2) using the flow-cluster notion as an element that averagely describe the pore space geometry, and (3) using the Bernoulli generalized two-phase flow equation. This approach helped to compute the relperm curves within plausible timing and compare them with actual test curves. We demonstrate that a material factor to affect the relperm curve shape is the pressure loss due to interfacial forces. We propose the empirical functions and parameters to simulate those effects. The computed relperm curves show a pretty good match with the actual test results.
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Rapoport -Leas Model for Two-phase Flow in Anisotropic Porous Media
Authors M.N. Kravchenko, N.M. Dmitriev and M.N. DmitrievAs real hydrocarbon collectors have anisotropic properties the problem of generalization of classical theory describing two-phase flow in porous media for is very actual now. In classical Rapoport -Leas model [1] it used concept determination of absolute and phase permeability. For anisotropic media the tensor model for anisotropic porous media was created [2-3]. The tensor connection between speed of filtration and pressure gradient was established. Tensors phase and relative phase permeability are written out and examined and also laboratory method of determination of absolute permeability tensor has been developed . However problem of description two-phase flow in anisotropic layer had been solved only as generalization of the theory of Bakli-Leverett in which it is necessary that pressure in both phases was equal. In model pressure in various phases is different. Therefore for generalization of the Rapoport -Leas theory for anisotropic media it is necessary to take into consideration anisotropy for definition of the capillary pressure which equal to a difference of pressure in phases. In this report generalization of the mathematical model of Rapoport -Leas for anisotropic porous media is given. The formula for definition of the capillary pressure, setting communication between pressure in phases, is represented by scalar function from vector argument. For definition that scalar function the tensor of capillary pressure and a inverse tensor of characteristic linear dimension are introduced. It is shown that the Leverett function of saturation entered for isotropic porous media ( can be generalized by the fourth rank tensor. For generic the common theory of Rapoport -Leas for anisotropic media case new function (type of Leverett function ) and new relative phase permeability functions were created.
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Alternative to Darcy’s Law for the Description of the Newtonian Fluid Flow in Porous Media
Authors V.V. Kadet and P.S. Chagirovt. The flow analysis at the micro level has yielded to law for macroscopic flow. It may be used as for general description of filtration at any values of pressure gradient and for establishment the critical values of pressure gradients that define the boundaries of Darcy's law applicability. The influence of parameters as porous media and fluid on the value of the threshold pressure gradients has been investigated. It has been proven that at high pressure gradients Darcy's law works well for low-porous reservoirs and it doesn’t work at all over the range of gradients for high-porous reservoirs. At low gradients the linear filtration law takes place only in high- porous rocks.
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Lithotype Clustering in Multidimentional Space
More LessThe focus of this work is lithotype clustering using Artificial Neural Network (ANN) for carbonate reservoirs. In contrast to terrigenous deposits, the carbonates are characterized by many parameters. It is quite a difficult problem to resolve carbonates into different lithotypes based on well data. In this work we used various data of real carbonate reservoir: well logs, core samples, petrophysical researches. To classify the carbonates several characteristics were picked out: class of carbonate (dolomites, limestone, etc.), biota, structure, recrystallization, leaching, incorporation, secondary mineral formation, type of collector, capacity. It is known that many types of data bear the curse of dimensionality. This can be mitigated by using Principal Component Analysis (PCA). PCA enables to determine the significant and uncorrelated variables. Then, the hypothesis of lithotypes discriminability on the multidimensional cross-plot should be confirmed. Finally, ANN for each characteristic was built. Laboratory results of litotype interpretation were used as training and testing data. Input neuron layer was formed by PCA output variables. Output neuron layer consisted of values of given characteristic. Lithotypes were determined by obtained characteristics and compared with laboratory results. Developed lithotype clusterization technique was successfully applied for real carbonate reservoirs
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Analysis and Evaluation of Heat and Mass Transfer Processes in Porous Media Based on Darcy-Stefan's Model
Authors D. Akhmed-Zaki, N. Danaev, S. Mukhambetzhanov and T. ImankulovOil and gas companies and research organizations developed and implemented in industry for new ways and processes of extraction and refining of oil and gas. For example, in today's world there is an enormous amount of research in the theory of filtration with a sufficient set of various mathematical models and different approaches to solve them, but unfortunately the reality of developments in the oil and gas fields, more complex processes of filtration options, taking into account the kinetics heat and mass transfer, etc., which of course directly affects the process flow operation of the facility and information systems requires a "fast" response (calculation) and forecasting. The latter implies the formation of an adequate IT systems, computer simulation and its early settlement for the shortest time that can not be achieved without the use of modern programming packages. Similary problems in using of surfactants. Injection of heat-carriers include mass and heat transfers in two different areas, combined transition interfaces through the moving front and accompanied by phase changes. The last two characteristics describe a general class of problems known as the Stefan and Verigin types of problem. We developed and studied a variety of mathematical models of filtration in a porous media with kinetic relations of heat and mass transfer. Created a computational algorithm of the task. In studying the process of pumping Surfactants in a productive oil reservoir must consider thermal effects that are more convenient to define a kinetic equation of heat and mass transfer characteristics for two different areas, combined transition interfaces through the moving front and accompanied by phase changes, which defines a single constructive approach solving a general class of problems like Stefan and Verigin or Darcy-Stefan. We offered mathematical model describing the mass transfer processes for nonisothermal filtration with conditions of formation boundaries of "transition" zones and computational parallel algorithm for 3D case based on Web oriented applications.
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New Method for Connection Transmissibility Calculation
By A.Y. YushkovThe paper reviews the problem of very low or zero productivity of simulated horizontal wells, when "connections" are placed in cells with zero vertical permeability (PERMZ). Such cells may imply, for instance, thin interlayering of permeable and impermeable interbeds inside one cell. The formulas are considered to calculate parameters describing the fluid flow to the "connection": the equivalent radius (R0j), transmissibility (khj), and the connection factor (Tj or CCFj). The conditional calculated examples show shortcomings of formulas and the reasons that lead to zero productivity of simulated wells. The author proposes refined formula to calculate Tj , R0j и khj, that account the well diameter going through the model cell. The formula imply that thinly-bedded vertically impermeable formation unit which has a plane-parallel filtration flow running across the area equal to well diameter multiplied by completion length. The applicability of the model for bigger cells is demonstrated by comparison of results obtained on the detailed grid. Bigger cells-based calculation results are close to those obtained on the detailed grid. The proposed formula can be used in dynamic simulators for more accurate simulation of fluid flowing to horizontal wells in case of bigger cells.
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