Conditional Statistical Moment Equations for Dynamic Data Integration in Heterogeneous Reservoirs

2006 ◽  
Vol 9 (03) ◽  
pp. 280-288 ◽  
Author(s):  
Liyong Li ◽  
Hamdi A. Tchelepi
Keyword(s):  
Statistical Moment ◽  
Limited Data ◽  
Pressure Data ◽  
Moment Equations ◽  
Reservoir Properties ◽  
Incomplete Knowledge ◽  
Heterogeneous Reservoirs ◽  
Knowledge Uncertainty ◽  
Available Information ◽  
Permeability Field

Summary An inversion method for the integration of dynamic (pressure) data directly into statistical moment equations (SMEs) is presented. The method is demonstrated for incompressible flow in heterogeneous reservoirs. In addition to information about the mean, variance, and correlation structure of the permeability, few permeability measurements are assumed available. Moreover, few measurements of the dependent variable are available. The first two statistical moments of the dependent variable (pressure) are conditioned on all available information directly. An iterative inversion scheme is used to integrate the pressure data into the conditional statistical moment equations (CSMEs). That is, the available information is used to condition, or improve the estimates of, the first two moments of permeability, pressure, and velocity directly. This is different from Monte Carlo (MC) -based geostatistical inversion techniques, where conditioning on dynamic data is performed for one realization of the permeability field at a time. In the MC approach, estimates of the prediction uncertainty are obtained from statistical post-processing of a large number of inversions, one per realization. Several examples of flow in heterogeneous domains in a quarter-five-spot setting are used to demonstrate the CSME-based method. We found that as the number of pressure measurements increases, the conditional mean pressure becomes more spatially variable, while the conditional pressure variance gets smaller. Iteration of the CSME inversion loop is necessary only when the number of pressure measurements is large. Use of the CSME simulator to assess the value of information in terms of its impact on prediction uncertainty is also presented. Introduction The properties of natural geologic formations (e.g., permeability) rarely display uniformity or smoothness. Instead, they usually show significant variability and complex patterns of correlation. The detailed spatial distributions of reservoir properties, such as permeability, are needed to make performance predictions using numerical reservoir simulation. Unfortunately, only limited data are available for the construction of these detailed reservoir-description models. Consequently, our incomplete knowledge (uncertainty) about the property distributions in these highly complex natural geologic systems means that significant uncertainty accompanies predictions of reservoir flow performance. To deal with the problem of characterizing reservoir properties that exhibit such variability and complexity of spatial correlation patterns when only limited data are available, a probabilistic framework is commonly used. In this framework, the reservoir properties (e.g., permeability) are assumed to be a random space function. As a result, flow-related properties such as pressure, velocity, and saturations are random functions. We assume that the available information about the permeability field includes a few measurements in addition to the spatial correlation structure, which we take here as the two-point covariance. This incomplete knowledge (uncertainty) about the detailed spatial distribution of permeability is the only source of uncertainty in our problem. Uncertainty about the detailed distribution of the permeability field in the reservoir leads to uncertainty in the computed predictions of the flow field (e.g., pressure).

Dynamic Data Integration and Quantification of Prediction Uncertainty Using Statistical-Moment Equations

SPE Journal ◽  
2011 ◽  
Vol 17 (01) ◽  
pp. 98-111 ◽  
Author(s):  
P.. Likanapaisal ◽  
L.. Li ◽  
H.A.. A. Tchelepi
Keyword(s):  
Data Integration ◽  
History Matching ◽  
Dynamic Properties ◽  
Statistical Moment ◽  
Statistical Moments ◽  
Moment Equations ◽  
Inversion Algorithm ◽  
Reservoir Properties ◽  
Dynamic Data ◽  
Prediction Uncertainty

Summary The use of a probabilistic framework for dynamic data integration (history matching) has become accepted practice. In this framework, one constructs an ensemble of reservoir models, in which each realization honors the available (static and dynamic) information. The variations in the flow performance across the ensemble provide an assessment of the prediction uncertainty due to incomplete knowledge of the reservoir properties (e.g., permeability distribution). Methods based on Monte Carlo simulation (MCS) are widely used because of the generality and simplicity of MCS. As a black-box approach, only pre- and post-processing of conventional flow simulations are needed. To achieve reasonable accuracy in estimating the statistical moments of flow-performance predictions, however, large numbers of realizations are usually necessary. Here, we use a different, and direct, approach for model calibration and uncertainty quantification. Specifically, we describe a statistical-moment-equations (SMEs) framework for both the forward and inverse problems associated with immiscible two-phase flow. In the SME method, the equations governing the statistical moments of the quantities of interest (e.g., pressure and saturation) are derived and solved directly. We assume that statistical information and a few measurements are available for the permeability field. As for the dynamic properties, we assume that measurements of pressure, saturation, and flow rate are available at a few locations and at several times. For the forward problem, the flow (pressure and total-velocity) SMEs are solved on a regular grid, while a streamline-based strategy is used to solve the transport SMEs. We use a kriging-based inversion algorithm, in which the first two statistical moments of permeability are conditioned directly using the available dynamic data. We analyze the behaviors of the saturation moments and their evolution as they are conditioned on measurements, in both space and time. Moreover, we discuss the relationship between the widely used MCS-based Kalman-filter approach and our SME inversion scheme.

Estimation of the Location of the Boundary of a Petroleum Reservoir

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

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

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

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

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

A New Method for Estimating Average Reservoir Pressure: The Muskat Plot Revisited

2008 ◽  
Vol 11 (02) ◽  
pp. 298-306 ◽  
Author(s):  
James G. Crump ◽  
Robert H. Hite
Keyword(s):  
Direct Synthesis ◽  
New Method ◽  
Theoretical Ground ◽  
Time Behavior ◽  
Reservoir Pressure ◽  
Reservoir Properties ◽  
Pressure Derivative ◽  
Pressure Buildup ◽  
Heterogeneous Reservoirs ◽  
Fluid Properties

Summary This paper describes a new method for estimating average reservoir pressure from long-pressure-buildup data on the basis of the classical Muskat plot. Current methods for estimating average reservoir pressure require a priori information about the reservoir and assume homogeneous reservoir properties or use empirical extrapolation techniques. The new method applies to heterogeneous reservoirs and requires no information about reservoir or fluid properties. The idea of the method is to estimate from the pressure derivative the first few eigenvalues of the pressure-transient decay modes. These values are characteristic of the reservoir and fluid properties, but not of the pressure history or well location in the reservoir. The smallest eigenvalue is used to extrapolate the long-time behavior of the transient to estimate the final reservoir pressure. The second eigenvalue can be used to estimate the quality of the estimate. Numerical tests of the method show that it estimates average reservoir pressure accurately, even when the reservoir is heterogeneous or when partial-flow barriers are present. Examples with real data show that the behavior predicted by the theory is actually observed. We expect the method to have value in reservoir limits testing, in making consistent estimates of average reservoir pressure from permanent downhole gauges, and in characterizing complex reservoirs. Introduction Several different methods of interpreting pressure-buildup data to obtain average reservoir pressure have been proposed (Muskat 1937; Horner 1967; Miller et al. 1950; Matthews et al. 1954; Dietz 1965) in the past, and in recent years some new techniques have appeared in the literature (Mead 1981; Hasan and Kabir 1983; Kabir and Hasan 1996; Kuchuk 1999; Chacon et al. 2004). Larson (1963) revisited the Muskat method and put it on a firm theoretical ground for a homogeneous cylindrical reservoir. Some of the existing techniques depend on knowledge of the reservoir size and shape and assume homogeneous properties (Horner 1967; Miller et al. 1950; Matthews et al. 1954; Dietz 1965). Such methods may result in uncertain predictions when reservoir data are unavailable or reservoir heterogeneity exists. The inverse time plot by Kuchuk (1999) is essentially a modification of Horner's method (1967) and works well in reservoirs that can be treated as infinite during the time of the test. The hyperbola method proposed by Mead (1981) and further developed by Hasan and Kabir (1983) is an empirical technique, not based on fundamental fluid flow principles for bounded reservoirs (Kabir and Hasan 1996). Chacon et al. (2004) develop the direct synthesis technique, in which conventional theory is used to derive an average pressure directly from standard log-log plots. Homogeneous properties and radial symmetry are assumed. Muskat's original derivation was a wellbore storage model. Larson reinterpreted Muskat's method and derived relationships showing how Muskat's plot could be used to estimate average reservoir pressure in a cylindrical, homogeneous reservoir. This paper revisits the ideas underlying Larson's paper. Similar ideas are shown to hold for heterogeneous reservoirs of any shape. A new analysis technique replacing the Muskat plot by a plot of the pressure derivative simplifies the determination of average reservoir pressure. It is shown that parameters from analysis of a long buildup on a reservoir can be used in subsequent buildup tests to shorten the required time of the subsequent buildups. Finally, estimates for time required for a buildup in homogeneous reservoirs of any shape are given.

FRACTAL GEOMETRY, RESERVOIR CHARACTERISATION AND OIL RECOVERY

1991 ◽  
Vol 31 (1) ◽  
pp. 377 ◽  
Author(s):  
I.J. Taggart ◽  
H.A. Salisch
Keyword(s):  
Fluid Flow ◽  
Oil Recovery ◽  
Large Scale ◽  
Dominant Factor ◽  
Property Value ◽  
Reservoir Properties ◽  
Reservoir Heterogeneity ◽  
Heterogeneous Reservoirs ◽  
Analysis Of Results ◽  
Point Data

Reservoir heterogeneity is a dominant factor in determining large-scale fluid flow behaviour in reservoirs. Engineering estimates of oil production rates need to acknowledge and incorporate the effect of such heterogeneities. This work examines the use of fractal-based scaling techniques aimed at characterising heterogeneous reservoirs for simulation purposes. Well log data provide suitable fine-scale information for estimating the fractal dimension of reservoirs as well as providing known end- point data for interwell property value interpolation. Fractal techniques allow this interpolation to be performed in a manner which reproduces the same correlation structure as that found in the original well logs. Conditional simulation in these property fields allows the interaction between reservoir heterogeneity and fluid flow to be studied on a range of scales up to the interwell spacing. Analysis of results allows the calculation of effective reservoir properties which characterise the reservoir in terms of large-scale performance.

The paper describes methods of studying the texture properties of rocks from photographs of a whole core using variograms and matrices of co-occurrence. The practical goal of our research is to obtain curves based on images of rocks; for example, core photographs, microresistivity formation images (FMI), and X-ray computerized tomography (CT), which make it possible to segment images depending on the difference of textural characteristics of different areas of the image. Such texture attributes—separately or along with color information—can be used in the tasks of automatic lithotyping, as well as in modeling the petrophysical properties of rocks. Our practical experience in building predictive models from core photographs has shown that the use of versatile information improves the quality of modeling. In some cases, the classification of lithotypes and the prediction of reservoir properties is difficult and leads to erroneous results if we do not use all the available information. In practice, UV images of the core are not always available. There are also daylight core photos, when the color characteristics are not very informative. In such cases, including imaging types such as X-ray CT, texture attributes can be of great help in predictive model building

2021 ◽  
Author(s):  
I.A. Seleznev ◽  
D.O. Makienko ◽  
V.V. Abashkin ◽  
A.A. Chertova ◽  
A.F. Samokhvalov
Keyword(s):  
Model Building ◽  
Practical Experience ◽  
Color Information ◽  
Reservoir Properties ◽  
X Ray ◽  
The Difference ◽  
Available Information ◽  
Texture Attributes
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