Well Testing in Heterogeneous Media: A General Method to Calculate the Permeability Weighting Function

1996 ◽  
Author(s):  
R.K. Romeu ◽  
A.Q. Lara ◽  
B. Noetinger ◽  
G. Renard
Keyword(s):  
Heterogeneous Media ◽  
Weighting Function ◽  
Well Testing ◽  
General Method

General Method to Calculate the Permeability Weighting Function for Well Testing in Heterogeneous Media

1999 ◽  
Vol 2 (03) ◽  
pp. 281-287 ◽  
Author(s):  
R.K. Romeu ◽  
A.Q. Lara ◽  
Benoit Nœtinger ◽  
Ge´rard Renard
Keyword(s):  
Boundary Conditions ◽  
Analytical Solutions ◽  
Heterogeneous Media ◽  
Weighting Function ◽  
Approximate Solutions ◽  
Pressure Solution ◽  
Well Testing ◽  
First Order ◽  
Approximate Analytical Solutions ◽  
General Method

Summary The perturbation method provides approximate solutions of the well pressure for arbitrarily heterogeneous media. Although theoretically limited to small permeability variations, this approach has proved to be very useful, providing qualitative understanding and valuable quantitative results for many applications. The well pressure solution using this method is expressed by an integral equation where the permeability variations are weighted by a kernel, the permeability weighting function. As discussed in previous papers, deriving such permeability weighting functions appears to be a complicated calculation, available only for special cases. In this article we present simple and general method to calculate the permeability weighting function. In the Laplace domain, the permeability weighting function is easily related to the pressure solution of the background problem. Since Laplace pressure solutions are known for many situations (various boundary conditions, stratified and composite media etc.), the associated permeability weighting function can be derived immediately. Among other examples, we calculate and discuss the well pressure solution for a horizontal well that is producing from a heterogeneous reservoir. Introduction The trend for reservoir characterization has stimulated the study of well testing in more complex heterogeneous media. Well testing in heterogeneous media has been studied by three approaches: exact analytical solutions, numerical simulations and approximate analytical solutions. Exact analytical solutions exist for a restricted class of problems that involve some simple symmetry: layered reservoirs, single linear discontinuities, radial composite systems etc.1 Rosa and Horne2 computed the exact solution in the case of an infinite homogeneous reservoir containing a single circular permeability discontinuity. Most of these analytical solutions are written in the Laplace domain. Numerical methods can treat much more general situations, but have some disadvantages: their use is cumbersome, investigation is empirical and general insights are difficult to be extracted, results are inaccurate if the time and the spatial discretization were not carefully conducted. Approximate analytical solutions can be a practical way to understand the pressure behavior in geometrically complex heterogeneous media. Kuchuk et al.3 proposed one of these approximate methods. Another popular class of approximate analytical solutions is based on the first-order approximation obtained from perturbation methods. This article is related to these first-order approximate solutions of well pressure in arbitrarily heterogeneous reservoirs. In particular, we propose an easy and general method to calculate the permeability weighting function in various flow geometries. In the next section, we define what the permeability weighting function is and review previous work in the domain. After that, we present our method to calculate the permeability weighting functions. The technique is demonstrated in three situations, including the case of flow through a horizontal well. Permeability Weighting Function The perturbation method is a well known technique by which to solve partial differential equations involving mathematical difficulties, like variable coefficients. According to this technique, we start from an easier problem, the background problem, to modify or perturb it. The full problem is approximated by the first few terms of a perturbation expansion, usually the first two terms. In our context, we start from considering a background medium with permeability k0 and with specified boundary conditions. The k0 may vary in space, i.e., k0(x→D). What is important is that the background problem has a known exact analytical solution, pD0(x→D,tD). The full problem has the same boundary conditions of the background problem but the permeability k(x→D) differs from k0(x→D) in arbitrary regions of space. Strictly speaking, k(x→D)/k0(∙xD) has to be close to 1 in order to obtain sound approximations. In practice, errors tend to be small, say less than 10%, even for relatively greater contrasts up to, say, 10 between these permeabilities, depending on the specific problem. The dimensionless well pressure of the full problem, pwD(tD), is approximated by the sum of two terms: p w D ( t D ) ≅ p w D 0 ( t D ) + p w D 1 ( t D ) , ( 1 ) where pwD0 is the solution of the background problem, which is known, and pwD1 corresponds to the effect of the variation of the permeability. This second term is computed by p w D 1 ( t D ) = ∫ − ∞ + ∞ Δ k D ( x → D ) W ( ∙ x D , t D ) d ∙ x D , ( 2 ) the terms of which will be explained. The dimensionless permeability variation ΔkD may be alternatively defined by Δ k D ( ∙ x D ) = l n ( k ( x → D ) / k 0 ( x → D ) ) , ( 3 a ) Δ k D ( ∙ x D ) = 1 − ( k 0 ( x → D ) / k ( x → D ) ) , ( 3 b ) Δ k D ( ∙ x D ) = ( k ( x → D ) / k 0 ( x → D ) ) − 1 , ( 3 c ) or other equivalent first-order approximations. These three expressions have the same first-order terms of their Taylor series, and produce very close results for k(x→D)/k0(x→D) near 1. However, these definitions are not equally robust for greater permeability contrasts.

Window-controlled CMP crosscorrelation analysis for surface waves in laterally heterogeneous media

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. EN95-EN105 ◽  
Author(s):  
Tatsunori Ikeda ◽  
Takeshi Tsuji ◽  
Toshifumi Matsuoka
Keyword(s):  
Surface Waves ◽  
Wave Velocity ◽  
Heterogeneous Media ◽  
Velocity Structure ◽  
Weighting Function ◽  
Dispersion Curves ◽  
Lateral Resolution ◽  
Heterogeneous Structure ◽  
S Wave ◽  
Data Set

CMP crosscorrelation (CMPCC) analysis of surface waves enhances lateral resolution of surface wave analyses. We found the technique of window-controlled CMPCC analysis, which applies two kinds of spatial windows to further improve the lateral resolution of CMPCC analysis. First, a spatial weighting function given by the number of crosscorrelation pairs is applied to CMPCC gathers. Because the number of crosscorrelation pairs is concentrated near the CMP, the lateral resolution in extracting dispersion curves on CMPs can be improved. Second, crosscorrelation pairs with longer receiver spacing are excluded to further improve lateral resolution. Although removing crosscorrelation pairs generally decreases the accuracy of phase velocity estimations, the required accuracy to estimate phase velocities is maintained by considering the wavenumber resolution defined for given receiver configurations. When applied to a synthetic data set simulating a laterally heterogeneous structure, window-controlled CMPCC analysis improved the retrieval of the lateral variation in local dispersion curves beneath each CMP. We also applied the method to field seismic data across a major fault. The window-controlled CMPCC analysis improved lateral variations of the inverted S-wave velocity structure without degrading the accuracy of S-wave velocity estimations. We discovered that window-controlled CMPCC analysis is effective in improving lateral resolution of dispersion curve estimations with respect to the original CMPCC analysis.

The Research of Production Split Using Well Logging and Well Testing Method

2014 ◽  
Vol 915-916 ◽  
pp. 1438-1442
Author(s):  
Xi Wei
Keyword(s):  
Flow Characteristics ◽  
Well Logging ◽  
Test Method ◽  
Well Testing ◽  
Well Test ◽  
New Production ◽  
Testing Method ◽  
Split Method ◽  
Real Fluid ◽  
General Method

The most popular method to split the production is mainly based on permeability and effective thickness of reservoir. In this general method, the permeability comes from the well logging interpretation. But this permeability can not reflect the flow characteristics of real fluid in the reservoir. In this paper, a new production split method is proposed considering the well logging method, the well test method and relative permeability curve. It is proved to be reliable and effective after cases study. It is worth to promote in oilfields.

A General Method for Spectrometer Design in the Scoff Approximation

1976 ◽  
Vol 34 ◽  
pp. 538-539
Author(s):  
J. R. Fields
Keyword(s):  
Electron Microscope ◽  
Transmission Electron Microscope ◽  
Energy Analysis ◽  
Incident Beam ◽  
Scanning Transmission Electron Microscope ◽  
Chemical Identification ◽  
Scanning Transmission Electron ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
General Method

The energy analysis of electrons scattered by a specimen in a scanning transmission electron microscope can improve contrast as well as aid in chemical identification. In so far as energy analysis is useful, one would like to be able to design a spectrometer which is tailored to his particular needs. In our own case, we require a spectrometer which will accept a parallel incident beam and which will focus the electrons in both the median and perpendicular planes. In addition, since we intend to follow the spectrometer by a detector array rather than a single energy selecting slit, we need as great a dispersion as possible. Therefore, we would like to follow our spectrometer by a magnifying lens. Consequently, the line along which electrons of varying energy are dispersed must be normal to the direction of the central ray at the spectrometer exit.

Phase behavior of cationic and anionic surfactants at low concentrations

1992 ◽  
Vol 50 (1) ◽  
pp. 694-695
Author(s):  
E. Naranjo
Keyword(s):  
Phase Behavior ◽  
Structural Characterization ◽  
Dynamic Systems ◽  
Anionic Surfactants ◽  
Intermolecular Forces ◽  
Stable Form ◽  
Surfactant Mixtures ◽  
Low Concentrations ◽  
Cationic And Anionic Surfactants ◽  
General Method

Equilibrium vesicles, those which are the stable form of aggregation and form spontaneously on mixing surfactant with water, have never been demonstrated in single component bilayers and only rarely in lipid or surfactant mixtures. Designing a simple and general method for producing spontaneous and stable vesicles depends on a better understanding of the thermodynamics of aggregation, the interplay of intermolecular forces in surfactants, and an efficient way of doing structural characterization in dynamic systems.

Caricaturing as a general method to improve poor face recognition: Evidence from low-resolution images, other-race faces, and older adults.

2019 ◽  
Vol 25 (2) ◽  
pp. 256-279 ◽  
Author(s):  
Amy Dawel ◽  
Tsz Ying Wong ◽  
Jodie McMorrow ◽  
Callin Ivanovici ◽  
Xuming He ◽  
...  
Keyword(s):  
Older Adults ◽  
Face Recognition ◽  
Low Resolution ◽  
General Method ◽  
Low Resolution Images
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