Albert Reynolds | University of Tulsa (original) (raw)
Papers by Albert Reynolds
Résumé — Caractérisation des erreurs de mesure des données sismiques 4D et des données de product... more Résumé — Caractérisation des erreurs de mesure des données sismiques 4D et des données de production par l’algorithme EM — La caractérisation des erreurs de mesure est cruciale pour l’utili-sation de l’approche de Bayes afin de conditionner les modèles de réservoir aux données dynamiques, c’est-à-dire les données sismiques 4D ainsi que les données de production, par l’history matching automatique. Dans la littérature, les erreurs de mesure pour chaque type de données sont généralement estimées en appliquant la technique de lissage sur tout le domaine des données engendrant, de ce fait, un surlissage des données (en particulier aux alentour des points pour lesquels les données réelles changent de façon drastique), ainsi qu’une surestimation des erreurs de mesure. Cet article présente une nouvelle procédure pour l’estimation de l’erreur de mesure. La méthode developpée ici, est basée sur l’algorithme EM (Expectation-Maximization) modifié, combiné à un fit polynomial mouvant. Cette mét...
Computers & Geosciences, 2013
ABSTRACT In the last decade, ensemble-based methods have been widely investigated and applied for... more ABSTRACT In the last decade, ensemble-based methods have been widely investigated and applied for data assimilation of flow problems associated with atmospheric physics and petroleum reservoir history matching. This paper focuses entirely on the reservoir history-matching problem. Among the ensemble-based methods, the ensemble Kalman filter (EnKF) is the most popular for history-matching applications. However, the recurrent simulation restarts required in the EnKF sequential data assimilation process may prevent the use of EnKF when the objective is to incorporate the history matching in an integrated geo-modeling workflow. In this situation, the ensemble smoother (ES) is a viable alternative. However, because ES computes a single global update, it may not result in acceptable data matches; therefore, the development of efficient iterative forms of ES is highly desirable. In this paper, we propose to assimilate the same data multiple times with an inflated measurement error covariance matrix in order to improve the results obtained by ES. This method is motivated by the equivalence between single and multiple data assimilation for the linear-Gaussian case. We test the proposed method for three synthetic reservoir history-matching problems. Our results show that the proposed method provides better data matches than those obtained with standard ES and EnKF, with a computational cost comparable with the computational cost of EnKF.
Reservoir Characterization—Recent Advances
International Journal for Numerical Methods in Engineering
We consider a technique to estimate an approximate gradient using an ensemble of randomly chosen ... more We consider a technique to estimate an approximate gradient using an ensemble of randomly chosen control vectors, known as Ensemble Optimization (EnOpt) in the oil and gas reservoir simulation community. In particular, we address how to obtain accurate approximate gradients when the underlying numerical models contain uncertain parameters because of geological uncertainties. In that case, 'robust optimization' is performed by optimizing the expected value of the objective function over an ensemble of geological models. In earlier publications, based on the pioneering work of Chen et al. (2009), it has been suggested that a straightforward one-to-one combination of random control vectors and random geological models is capable of generating sufficiently accurate approximate gradients. However, this form of EnOpt does not always yield satisfactory results. In a recent article, Fonseca et al. (2015) formulate a modified EnOpt algorithm, referred to here as a Stochastic Simplex Approximate Gradient (StoSAG; in earlier publications referred to as 'modified robust EnOpt') and show, via computational experiments, that StoSAG generally yields significantly better gradient approximations than the standard EnOpt algorithm. Here, we provide theoretical arguments to show why StoSAG is superior to EnOpt.
SPE Reservoir Simulation Symposium
Proceedings of SPE Annual Technical Conference and Exhibition
Abstract Type curve analysis of pressure transient data influenced by wellbore storage effects ca... more Abstract Type curve analysis of pressure transient data influenced by wellbore storage effects can yield non-unique answers due to similarity in the shapes of the curves. In order to alleviate this problem, several researchers have proposed methods for unbiased ...
Proceedings of SPE Production Operations Symposium
... 10 or 11~ ls 11, then using the modilied Schapery's approximation , it emy to show t... more ... 10 or 11~ ls 11, then using the modilied Schapery's approximation , it emy to show that qn = l/pWD, or equivalently, 1 = PWDO !fD (23) ... (:)' = 24) &($) = 'D~D(qD Page 5. SPE 18877 WENZHONG DING, MUSTAFA ONUR AND ALBERT C. REYNOLDS 5 ...
Proceedings of SPE Annual Technical Conference and Exhibition
This paper presents new procedures for analyzing pressure buildup data obtained from drillstem te... more This paper presents new procedures for analyzing pressure buildup data obtained from drillstem tests. The new methods apply for cases where the produced fluid does not reach the surface during the flowing period so that, the flow period represents a slug test. The combined effects of variable flow rate, short producing time and changing wellbore storage generally make it producing time and changing wellbore storage generally make it difficult to apply conventional analysis methods to DST buildup data. Two new straight-line methods for analyzing buildup data presented in this work account for both the variable rate during the presented in this work account for both the variable rate during the flow period and producing time effects. If the buildup wellbore storage-skin group CSD exp(2s) is small, then shortly after shut-in, a well-defined straight line is obtained for both methods. Using the slope of the straight line obtained by either method, it is shown that the flow capacity kh), the skin factor (s) and the initial reservoir pressure (Pi) can be determined. If the group CSD exp(2s) is large, longer shut-in times are needed before the proper straight line can be obtained. However, for the latter case, it is shown that a multi-rate equivalent time can be constructed so that standard type-curve matching can be performed to obtain estimates of the flow capacity and the skin factor. A field example is presented to illustrate the applications of the proposed methods. proposed methods Introduction A drillstem test is a temporary well completion with the specific purposes of rock and fluid characterization under dynamic reservoir conditions. DST's are also used for monitoring reservoir and well condition in developed productive zones. Standard DST equipment and operational procedures have been described elsewhere in the literature. Essentially, the testing tool consists of a packer and a tester valve attached to the tail of the drill pipe packer and a tester valve attached to the tail of the drill pipe or production tubing. After the packer is set in position, the opening of the DST valve imposes a sudden pressure drop at formation face causing the reservoir fluid to be pushed into the wellbore, increasing the liquid level in the string. When the tester valve is closed for the buildup period, the wellbore storage coefficient decreases from one for a rising-liquid-level to one for fluid- compression, which may represent a reduction of two orders of magnitude in the storage coefficient. For wells in which the liquid level does not reach the surface, the DST typically shows a variable, decreasing flow rate throughout the flow period. The flow period pressure data can be analyzed by conventional multi-rate techniques, though this is rarely done in practice. The standard procedure for analyzing DST flow data has been by type-curve matching the measured pressure data with the slug type curves presented by Cooper et al. and Ramey et al. Recently, Peres et al. presented by Cooper et al. and Ramey et al. Recently, Peres et al. presented a new procedure for DST flow and slug test analysis which presented a new procedure for DST flow and slug test analysis which converts the measured data into an equivalent constant rate problem. The "converted' data can then be analyzed by standard constant rate procedures (e.g. type-curve matching, semilog straight line procedures (e.g. type-curve matching, semilog straight line analysis). Buildup analysis following a variable rate drawdown has been the object of several investigations. When the flow rate history is known, the multi-rate method of Ref. 11 can be used to analyze pressure buildup data. Horner also proposed a simplified approach in which the actual producing time is replaced by the ratio of the cumulative production to the last rate. Odeh and Selig presented expressions for correcting both the rate and flowing time in order to account for the flow rate variation on pressure buildup calculations. Dolan et al. verified that if the pressure buildup calculations. Dolan et al. verified that if the rate variation during the DST flow period is not severe, the use of the average production rate in the semilog slope of the Horner plot yields an accurate estimate of the flow capacity. Streltsova showed that for small shut-in times, the buildup response is affected by the last flow rate before shut-in. Ref. 14 also showed that for large shut-in times the Horner semilog slope does not depend upon the rate-history but only upon its average value. Several authors indicated that the rate variation effect on the shape of semilog plots can be erroneously interpreted as reservoir heterogeneity. Soliman proposed a continuous multi-rate method for DST applications, which approximates the flow rates by a polynomial expression. p. 747
Proceedings of SPE Annual Technical Conference and Exhibition
... Recently, an alternative way of generating pressure-derivative type curves for analyzing well... more ... Recently, an alternative way of generating pressure-derivative type curves for analyzing well test data was pre sented by Onur and Reynolds*'; also see, Refs. ... O at tp/CD M W. Thus, if we wish to present type cuNes based on a iog-log plot of functions involving l(pfiD)/tD, we ...
SPE Formation Evaluation, 1986
ABSTRACT
SPE Formation Evaluation, 1989
ABSTRACT
Society of Petroleum Engineers Journal, 1983
A finite-conductivity vertical fracture intersecting a well produced at a constant rate or at a c... more A finite-conductivity vertical fracture intersecting a well produced at a constant rate or at a constant pressure is considered. The pressure (or rate) response is obtained from a numerical model. Two aspects of this problem are considered:variable fracture conductivity andunequal fracture wing lengths. The first part of this paper examines the influence of fracture conductivity on the well response. In practice, the fracture conductivity is a decreasing function of distance from the wellbore. If the fracture conductivity decreases monotonically with distance from the wellbore, then at late times the variable fracture conductivity solutions behave like a constant-conductivity fracture, with conductivity equal to the arithmetic average of the conductivity. At early times the response is identical to that of a constant-conductivity fracture, corresponding to the highest conductivity of the fracture. For the variable fracture conductivity case, the bilinear flow period characterized by...
Society of Petroleum Engineers Journal, 1984
This work examines interference test data in a naturally fractured reservoir. The reservoir model... more This work examines interference test data in a naturally fractured reservoir. The reservoir model examined here assumes that the reservoir can be represented by a system of horizontal fractures that are separated by the matrix. This model is identical to the deSwaan-Kazemi model. The main contribution of our work is that we combine the parameters of interest in a simple way and present solutions that can be used directly for field application. These solutions can be used to design or analyze interference tests. We also compare the solution for unsteady-state flow in the matrix with the Warren-Root model, which assumes pseudosteady-state fluid flow in the matrix. Introduction This work examines the pressure response at an observation well in a fractured reservoir. Previous works by Kazemi et al. and Streltsova-Adams have examined the pressure response based on the Warren and Root model. In this work, however, we assume unsteady-state fluid transfer from the matrix to the fracture sys...
Journal of Petroleum Technology, 1984
ABSTRACT
Journal of Petroleum Technology, 1983
Summary This paper presents new methods for analyzing pressure drawdown and buildupdata obtained ... more Summary This paper presents new methods for analyzing pressure drawdown and buildupdata obtained at wells producing naturally fractured reservoirs. The modelused in this study assumes unsteady-state fluid transfer from the matrixsystem to the fracture system. A new flow regime is identified. The discovery of this flow regimeexplains field behavior that has been considered unusual. The probabilityof obtaining data reflecting this flow regime in a field test is higher than that of obtaining the classical responses given in the literature. The identification of this new flow regime provides methods for preparing acomplete analysis of pressure data obtained from naturally fractured reservoirs. Applications to field data are discussed. Introduction In this work, we investigate the pressure response in a naturally fractured reservoir. Several models of naturally fractured reservoirs have been presented in the literature. Warren and Root and Odeh assume pseudosteady-state flow in the matri...
Journal of Petroleum Technology, 1982
A finite-conductivity vertical fracture intersecting a well is considered. The well is produced a... more A finite-conductivity vertical fracture intersecting a well is considered. The well is produced at a constant rate or at a constant pressure and then is shut in. The pressure response during the buildup period and the pressure or rate response during the production period are obtained from a numerical model. In this work the authors discuss the analysis of pressure buildup data following a short flow period. The authors present a buildup type curve to analyze pressure buildup data when the pressure (or rate) response is influenced by only the bilinear flow period. This type curve can be used to estimate the fracture flow capacity and a lower limit for the fracture half-length. It will be useful when the equivalent-time or the equivalent-drawdown concepts cannot be used. The authors discuss strengths and limitations of the equivalenttime method proposed in the literature. The theoretical basis for this procedure is provided.
SPE Reservoir Evaluation & Engineering, 1998
In recent years, the numerical Laplace transformation of sampled-data has proven to be useful for... more In recent years, the numerical Laplace transformation of sampled-data has proven to be useful for well test analysis applications. However, the success of this approach is highly dependent on the algorithms used to transform sampled-data into Laplace space and to perform the numerical inversion. In this work, we investigate several functional approximations (piecewise linear, quadratic, and log-linear) for sampled-data to achieve the "forward" Laplace transformation and present new methods to deal with the "tail" effects associated with transforming sampled-data. New algorithms that provide accurate transformation of sampled-data into Laplace space are provided. The algorithms presented can be applied to generate accurate pressure-derivatives in the time domain. Three different algorithms investigated for the numerical inversion of sampled-data. Applications of the algorithms to convolution, deconvolution, and parameter estimation in Laplace space are also presen...
SPE Formation Evaluation, 1989
SPE Formation Evaluation, 1997
ABSTRACT
Journal of Petroleum Technology, 1985
Summary Recently, a new flow regime has been identified for pressure drawdown or buildup data obt... more Summary Recently, a new flow regime has been identified for pressure drawdown or buildup data obtained from a well pressure drawdown or buildup data obtained from a well producing a naturally fractured reservoir. The identification producing a naturally fractured reservoir. The identification of this new, intermediate-time flow regime means that pressure data may exhibit as many as three semilog pressure data may exhibit as many as three semilog straight lines. A virtually complete analysis of well-test pressure data is possible when the data exhibit two or more pressure data is possible when the data exhibit two or more of the three possible semilog straight lines. In this work, we present type curves that can be used to obtain a virtually complete analysis of pressure data when only one of the three semilog straight lines is reflected by the data. One model that frequently has been used assumes that a naturally fractured reservoir is equivalent to a system of plane horizontal frac...
Résumé — Caractérisation des erreurs de mesure des données sismiques 4D et des données de product... more Résumé — Caractérisation des erreurs de mesure des données sismiques 4D et des données de production par l’algorithme EM — La caractérisation des erreurs de mesure est cruciale pour l’utili-sation de l’approche de Bayes afin de conditionner les modèles de réservoir aux données dynamiques, c’est-à-dire les données sismiques 4D ainsi que les données de production, par l’history matching automatique. Dans la littérature, les erreurs de mesure pour chaque type de données sont généralement estimées en appliquant la technique de lissage sur tout le domaine des données engendrant, de ce fait, un surlissage des données (en particulier aux alentour des points pour lesquels les données réelles changent de façon drastique), ainsi qu’une surestimation des erreurs de mesure. Cet article présente une nouvelle procédure pour l’estimation de l’erreur de mesure. La méthode developpée ici, est basée sur l’algorithme EM (Expectation-Maximization) modifié, combiné à un fit polynomial mouvant. Cette mét...
Computers & Geosciences, 2013
ABSTRACT In the last decade, ensemble-based methods have been widely investigated and applied for... more ABSTRACT In the last decade, ensemble-based methods have been widely investigated and applied for data assimilation of flow problems associated with atmospheric physics and petroleum reservoir history matching. This paper focuses entirely on the reservoir history-matching problem. Among the ensemble-based methods, the ensemble Kalman filter (EnKF) is the most popular for history-matching applications. However, the recurrent simulation restarts required in the EnKF sequential data assimilation process may prevent the use of EnKF when the objective is to incorporate the history matching in an integrated geo-modeling workflow. In this situation, the ensemble smoother (ES) is a viable alternative. However, because ES computes a single global update, it may not result in acceptable data matches; therefore, the development of efficient iterative forms of ES is highly desirable. In this paper, we propose to assimilate the same data multiple times with an inflated measurement error covariance matrix in order to improve the results obtained by ES. This method is motivated by the equivalence between single and multiple data assimilation for the linear-Gaussian case. We test the proposed method for three synthetic reservoir history-matching problems. Our results show that the proposed method provides better data matches than those obtained with standard ES and EnKF, with a computational cost comparable with the computational cost of EnKF.
Reservoir Characterization—Recent Advances
International Journal for Numerical Methods in Engineering
We consider a technique to estimate an approximate gradient using an ensemble of randomly chosen ... more We consider a technique to estimate an approximate gradient using an ensemble of randomly chosen control vectors, known as Ensemble Optimization (EnOpt) in the oil and gas reservoir simulation community. In particular, we address how to obtain accurate approximate gradients when the underlying numerical models contain uncertain parameters because of geological uncertainties. In that case, 'robust optimization' is performed by optimizing the expected value of the objective function over an ensemble of geological models. In earlier publications, based on the pioneering work of Chen et al. (2009), it has been suggested that a straightforward one-to-one combination of random control vectors and random geological models is capable of generating sufficiently accurate approximate gradients. However, this form of EnOpt does not always yield satisfactory results. In a recent article, Fonseca et al. (2015) formulate a modified EnOpt algorithm, referred to here as a Stochastic Simplex Approximate Gradient (StoSAG; in earlier publications referred to as 'modified robust EnOpt') and show, via computational experiments, that StoSAG generally yields significantly better gradient approximations than the standard EnOpt algorithm. Here, we provide theoretical arguments to show why StoSAG is superior to EnOpt.
SPE Reservoir Simulation Symposium
Proceedings of SPE Annual Technical Conference and Exhibition
Abstract Type curve analysis of pressure transient data influenced by wellbore storage effects ca... more Abstract Type curve analysis of pressure transient data influenced by wellbore storage effects can yield non-unique answers due to similarity in the shapes of the curves. In order to alleviate this problem, several researchers have proposed methods for unbiased ...
Proceedings of SPE Production Operations Symposium
... 10 or 11~ ls 11, then using the modilied Schapery's approximation , it emy to show t... more ... 10 or 11~ ls 11, then using the modilied Schapery's approximation , it emy to show that qn = l/pWD, or equivalently, 1 = PWDO !fD (23) ... (:)' = 24) &($) = 'D~D(qD Page 5. SPE 18877 WENZHONG DING, MUSTAFA ONUR AND ALBERT C. REYNOLDS 5 ...
Proceedings of SPE Annual Technical Conference and Exhibition
This paper presents new procedures for analyzing pressure buildup data obtained from drillstem te... more This paper presents new procedures for analyzing pressure buildup data obtained from drillstem tests. The new methods apply for cases where the produced fluid does not reach the surface during the flowing period so that, the flow period represents a slug test. The combined effects of variable flow rate, short producing time and changing wellbore storage generally make it producing time and changing wellbore storage generally make it difficult to apply conventional analysis methods to DST buildup data. Two new straight-line methods for analyzing buildup data presented in this work account for both the variable rate during the presented in this work account for both the variable rate during the flow period and producing time effects. If the buildup wellbore storage-skin group CSD exp(2s) is small, then shortly after shut-in, a well-defined straight line is obtained for both methods. Using the slope of the straight line obtained by either method, it is shown that the flow capacity kh), the skin factor (s) and the initial reservoir pressure (Pi) can be determined. If the group CSD exp(2s) is large, longer shut-in times are needed before the proper straight line can be obtained. However, for the latter case, it is shown that a multi-rate equivalent time can be constructed so that standard type-curve matching can be performed to obtain estimates of the flow capacity and the skin factor. A field example is presented to illustrate the applications of the proposed methods. proposed methods Introduction A drillstem test is a temporary well completion with the specific purposes of rock and fluid characterization under dynamic reservoir conditions. DST's are also used for monitoring reservoir and well condition in developed productive zones. Standard DST equipment and operational procedures have been described elsewhere in the literature. Essentially, the testing tool consists of a packer and a tester valve attached to the tail of the drill pipe packer and a tester valve attached to the tail of the drill pipe or production tubing. After the packer is set in position, the opening of the DST valve imposes a sudden pressure drop at formation face causing the reservoir fluid to be pushed into the wellbore, increasing the liquid level in the string. When the tester valve is closed for the buildup period, the wellbore storage coefficient decreases from one for a rising-liquid-level to one for fluid- compression, which may represent a reduction of two orders of magnitude in the storage coefficient. For wells in which the liquid level does not reach the surface, the DST typically shows a variable, decreasing flow rate throughout the flow period. The flow period pressure data can be analyzed by conventional multi-rate techniques, though this is rarely done in practice. The standard procedure for analyzing DST flow data has been by type-curve matching the measured pressure data with the slug type curves presented by Cooper et al. and Ramey et al. Recently, Peres et al. presented by Cooper et al. and Ramey et al. Recently, Peres et al. presented a new procedure for DST flow and slug test analysis which presented a new procedure for DST flow and slug test analysis which converts the measured data into an equivalent constant rate problem. The "converted' data can then be analyzed by standard constant rate procedures (e.g. type-curve matching, semilog straight line procedures (e.g. type-curve matching, semilog straight line analysis). Buildup analysis following a variable rate drawdown has been the object of several investigations. When the flow rate history is known, the multi-rate method of Ref. 11 can be used to analyze pressure buildup data. Horner also proposed a simplified approach in which the actual producing time is replaced by the ratio of the cumulative production to the last rate. Odeh and Selig presented expressions for correcting both the rate and flowing time in order to account for the flow rate variation on pressure buildup calculations. Dolan et al. verified that if the pressure buildup calculations. Dolan et al. verified that if the rate variation during the DST flow period is not severe, the use of the average production rate in the semilog slope of the Horner plot yields an accurate estimate of the flow capacity. Streltsova showed that for small shut-in times, the buildup response is affected by the last flow rate before shut-in. Ref. 14 also showed that for large shut-in times the Horner semilog slope does not depend upon the rate-history but only upon its average value. Several authors indicated that the rate variation effect on the shape of semilog plots can be erroneously interpreted as reservoir heterogeneity. Soliman proposed a continuous multi-rate method for DST applications, which approximates the flow rates by a polynomial expression. p. 747
Proceedings of SPE Annual Technical Conference and Exhibition
... Recently, an alternative way of generating pressure-derivative type curves for analyzing well... more ... Recently, an alternative way of generating pressure-derivative type curves for analyzing well test data was pre sented by Onur and Reynolds*'; also see, Refs. ... O at tp/CD M W. Thus, if we wish to present type cuNes based on a iog-log plot of functions involving l(pfiD)/tD, we ...
SPE Formation Evaluation, 1986
ABSTRACT
SPE Formation Evaluation, 1989
ABSTRACT
Society of Petroleum Engineers Journal, 1983
A finite-conductivity vertical fracture intersecting a well produced at a constant rate or at a c... more A finite-conductivity vertical fracture intersecting a well produced at a constant rate or at a constant pressure is considered. The pressure (or rate) response is obtained from a numerical model. Two aspects of this problem are considered:variable fracture conductivity andunequal fracture wing lengths. The first part of this paper examines the influence of fracture conductivity on the well response. In practice, the fracture conductivity is a decreasing function of distance from the wellbore. If the fracture conductivity decreases monotonically with distance from the wellbore, then at late times the variable fracture conductivity solutions behave like a constant-conductivity fracture, with conductivity equal to the arithmetic average of the conductivity. At early times the response is identical to that of a constant-conductivity fracture, corresponding to the highest conductivity of the fracture. For the variable fracture conductivity case, the bilinear flow period characterized by...
Society of Petroleum Engineers Journal, 1984
This work examines interference test data in a naturally fractured reservoir. The reservoir model... more This work examines interference test data in a naturally fractured reservoir. The reservoir model examined here assumes that the reservoir can be represented by a system of horizontal fractures that are separated by the matrix. This model is identical to the deSwaan-Kazemi model. The main contribution of our work is that we combine the parameters of interest in a simple way and present solutions that can be used directly for field application. These solutions can be used to design or analyze interference tests. We also compare the solution for unsteady-state flow in the matrix with the Warren-Root model, which assumes pseudosteady-state fluid flow in the matrix. Introduction This work examines the pressure response at an observation well in a fractured reservoir. Previous works by Kazemi et al. and Streltsova-Adams have examined the pressure response based on the Warren and Root model. In this work, however, we assume unsteady-state fluid transfer from the matrix to the fracture sys...
Journal of Petroleum Technology, 1984
ABSTRACT
Journal of Petroleum Technology, 1983
Summary This paper presents new methods for analyzing pressure drawdown and buildupdata obtained ... more Summary This paper presents new methods for analyzing pressure drawdown and buildupdata obtained at wells producing naturally fractured reservoirs. The modelused in this study assumes unsteady-state fluid transfer from the matrixsystem to the fracture system. A new flow regime is identified. The discovery of this flow regimeexplains field behavior that has been considered unusual. The probabilityof obtaining data reflecting this flow regime in a field test is higher than that of obtaining the classical responses given in the literature. The identification of this new flow regime provides methods for preparing acomplete analysis of pressure data obtained from naturally fractured reservoirs. Applications to field data are discussed. Introduction In this work, we investigate the pressure response in a naturally fractured reservoir. Several models of naturally fractured reservoirs have been presented in the literature. Warren and Root and Odeh assume pseudosteady-state flow in the matri...
Journal of Petroleum Technology, 1982
A finite-conductivity vertical fracture intersecting a well is considered. The well is produced a... more A finite-conductivity vertical fracture intersecting a well is considered. The well is produced at a constant rate or at a constant pressure and then is shut in. The pressure response during the buildup period and the pressure or rate response during the production period are obtained from a numerical model. In this work the authors discuss the analysis of pressure buildup data following a short flow period. The authors present a buildup type curve to analyze pressure buildup data when the pressure (or rate) response is influenced by only the bilinear flow period. This type curve can be used to estimate the fracture flow capacity and a lower limit for the fracture half-length. It will be useful when the equivalent-time or the equivalent-drawdown concepts cannot be used. The authors discuss strengths and limitations of the equivalenttime method proposed in the literature. The theoretical basis for this procedure is provided.
SPE Reservoir Evaluation & Engineering, 1998
In recent years, the numerical Laplace transformation of sampled-data has proven to be useful for... more In recent years, the numerical Laplace transformation of sampled-data has proven to be useful for well test analysis applications. However, the success of this approach is highly dependent on the algorithms used to transform sampled-data into Laplace space and to perform the numerical inversion. In this work, we investigate several functional approximations (piecewise linear, quadratic, and log-linear) for sampled-data to achieve the "forward" Laplace transformation and present new methods to deal with the "tail" effects associated with transforming sampled-data. New algorithms that provide accurate transformation of sampled-data into Laplace space are provided. The algorithms presented can be applied to generate accurate pressure-derivatives in the time domain. Three different algorithms investigated for the numerical inversion of sampled-data. Applications of the algorithms to convolution, deconvolution, and parameter estimation in Laplace space are also presen...
SPE Formation Evaluation, 1989
SPE Formation Evaluation, 1997
ABSTRACT
Journal of Petroleum Technology, 1985
Summary Recently, a new flow regime has been identified for pressure drawdown or buildup data obt... more Summary Recently, a new flow regime has been identified for pressure drawdown or buildup data obtained from a well pressure drawdown or buildup data obtained from a well producing a naturally fractured reservoir. The identification producing a naturally fractured reservoir. The identification of this new, intermediate-time flow regime means that pressure data may exhibit as many as three semilog pressure data may exhibit as many as three semilog straight lines. A virtually complete analysis of well-test pressure data is possible when the data exhibit two or more pressure data is possible when the data exhibit two or more of the three possible semilog straight lines. In this work, we present type curves that can be used to obtain a virtually complete analysis of pressure data when only one of the three semilog straight lines is reflected by the data. One model that frequently has been used assumes that a naturally fractured reservoir is equivalent to a system of plane horizontal frac...