Abstract The paper presents a practical method for calculating the matrix-fracture fluid transfer correctly, applicable under expansion, solution gas, capillary and gravitational drives and their combinations, considering water, gas and oil as displacing agents. The method is an improvement of the Reiss (1973) formulation based on time-depending recovery functions. Reiss could not consider depletion drive in combination with water and gas displacement. Kazemi et al. (1976) extended the Warren-Root shape factor approach to multi-phase cases, but without satisfying the conditions under which the single-phase solution is correct. Since that continuous efforts were made for improvement including diversification and making the shape factor time-dependent. Many publications demonstrated that these efforts are futile. Su et al. (2013) clearly postulated "a change of focus to the improvement of the transfer function would be a path to explore". The paper is an attempt in this sense with encouraging results. The suggested approach determines on fine gridded single matrix block (SMB) models recovery curves. Those are used directly in dual continuum models.

Abstract This work suggests a paradigm change. Instead of history matching an (upscaled) geological model by the reservoir engineer focusing on matching the dynamic data. The model builder should be equipped with the right tools to identify areas where the geological model cannot fulfill the dynamic requirements. The target is to make history matching obsolete. Every modification of the model must have its geological reasoning and justification. Introducing artifacts for the sole target of matching dynamic data is depreciated. Often such models lack forecasting credibility. Especially selecting infill drilling locations becomes highly uncertain. This paper presents applicable and ready to use methods which will be demonstrated on a full-field project. There are two key elements, each to be fulfilled during every timestep. At first the average (region) pressures must fit and second each well must be allowed to search for the required phases to satisfy its production targets. The first requires at least one region for which the historic pressure decline is known. At the delineation towards an outer aquifer water will be automatically injected/withdrawn to match the pressure history. The second is realized by perforations (well-grid-connections) which are allowed to move away from the well's trajectory and to search for the required phases within a user or algorithm defined drainage area. There is just one restriction. Drainage areas must not overlap. In case the drainage area does not contain the necessary fluids the geological model has to be revised. At the same time important information about fluid movements are gathered. The promoted concept was successfully applied to a North African naturally fractured carbonate reservoir having more than 500 MMstb original oil in place and a 35-years production history. The reservoir was divided into eight pressure regions. Five were selected as target for the automatic pressure match, the remaining three were control regions. Two update cycles were performed. During the first cycle the structure was corrected. Thus the phase distribution was affected so that all drainage areas could produce the required historic rates. During the second cycle local adjustments in porevolume and communication across faults were performed to increase the pressure match of the control regions. The demonstrated workflow has the potential to revolutionize the commonly practiced reservoir model creation. Up to now no similar workflow allowing to assess the geological model quality within one single simulation run was published. It is unique that pressure and production targets can be met in each timestep without corrupting the underlying geological model. Contrary the modeler is provided with the required information where and how to improve the geological model.

The Discrete Fracture Network and Dual Continuum concept are two common methods to model naturally fractured reservoirs. Each has specific limitations. The newly introduced recovery curve method is believed to be a compromise between these two current methods. In the recovery curve method, two recovery curves for water and gas displacements are used to more realistically calculate matrix-fracture mass exchanges. This paper presents a new approach to determine appropriate recovery curves by using the recovery curve method and production data. In particular, two recovery curves were determined for a column model by matching phase contact positions throughout the production data. Hybrid genetic algorithm and Neighborhood–Bayes technique were utilized for optimization and uncertainty quantification purposes respectively. Also, the distribution of the history matched models was shown by Boxplot. The results with the predicted recovery curves, showed very good agreement with the reference production data and the Bayesian credible intervals (P10–P90) of the proposed method were acceptable.

This work presents a new method based on the recovery-curve method to evaluate the insitu rock wettability in naturally fractured reservoirs. The recovery-curve method uses recovery curves to describe more realistically the matrix-fracture interactions. The capability of the proposed method was shown in two models, namely a simple column model and a realistic naturally fractured reservoir. Appropriate recovery curves were determined by history matching the movement of the water-oil contact. The slopes of the obtained recovery curves were analyzed with the method presented by Mirzaei-Paiaman etal.,12 which is referred to as the MPMS method in this study. The MPMS method has many advantages over other methods. Proper indices describing insitu rock wettability states were determined. The results are in very good agreement with those obtained from conventional methods.

In this paper, a very efficient method, called single matrix block analyzer (SMBA), has been developed to determine relative permeability and capillary pressure curves from spontaneous imbibition (SI) data. SMBA mimics realistically the SI tests by appropriate boundary conditions modeling. In the proposed method, a cuboid with an identical core plug height is considered. The equal dimensions of the cuboid in x and y directions are set such that the cylindrical core plug and the cuboid have the same shape factor. Thus, by avoiding the difficulties of the cylindrical coordinates, a representative model for the core plug is established. Appropriate grid numbers in x-y and z directions are specified to the model. Furthermore, the rock and fluid properties of SI test are set in the SMBA. By supposing forms of the oil-water capillary pressure and relative permeability and comparing the oil recovery curves of SMBA and SI data, capillary pressure and relative permeability can be determined. The SMBA is demonstrated using three experimental data with different aging times. Suitable equations are employed to represent the capillary pressure and relative permeability curves. The genetic algorithm is used as the optimization tool. The obtained results, especially for capillary pressure, are in good agreement with the experimental data. Moreover, the Bayesian credible interval (P10 and P90) evaluated by the Neighborhood Bayes Algorithm (NAB) is quite satisfactory.

The discrete fracture network (DFN) and Multiple-Continua concept are among the most widely used methods to model naturally fractured reservoirs. Each faces specific limitations. The recently introduced recovery curve method (RCM) is believed to be a compromise between these two current methods. In this method the recovery curves are used to determine the amount of mass exchanges between the matrix and fracture mediums. Two recovery curves are assigned for each simulation cell, one curve for gas displacement in the presence of the gravity drainage mechanism, and another for water displacement in the case of the occurrence of the imbibition mechanism. These curves describe matrix-fracture mass transfer more realistically and therefore can be of great use when obtaining historical production data.This paper presents the potential of the RCM within the framework of history matching of naturally fractured reservoirs to determine the appropriate recovery curves. In particular, the phase contact positions of a sector model, extracted from a real fractured reservoir, are matched throughout the production history using the RCM. Therefore, the main contribution of this work is using the historical data of contact positions to obtain a better description of the matrix-fracture communication. The distribution of an ensemble of the history matched models was illustrated by Boxplot, and the Genetic Algorithm (GA) and the Neighborhood-Bayes technique (NAB) were utilized for optimization and uncertainty quantification, respectively. The calculated recovery curves results are in good agreement with the production history of the model and the Bayesian credible interval (P10-P90) of the proposed method is satisfactory. This work also confirmed the potential applicability of the combined GA and NAB method for optimization and uncertainty quantification purposes. (C) 2014 Elsevier B.V. All rights reserved.

Fifty years ago Warren and Root have introduced the shape factor. This fundamental parameter for modeling of naturally fractured reservoirs has been discussed stormily ever since. Different definitions for shape factor have been suggested which all of them are heuristically based. Recently, Heinemann and Mittermeir mathematically derived - based on the dual-continuum theorem assuming pseudo-steady state condition- a general and proper form of the shape factor formula which can be simplified to the previously published shape factor definitions. This paper discusses the practical relevance of the Heinemann-Mittermeir formula. Its difference to the most commonly used Kazemi et al.formula is its demonstration by fine-scale single matrix block simulation. Furthermore, it is shown that the generally applied isotropy assumption can lead to significantly wrong results. Consequently, the generalized Heinemann-Mittermeir shape factor formula is recommended to be routinely practiced in the industry for more accurate results. The paper tries to present a proper realization of the nature of the shape factor as well as presentation of detailed mathematical and practical approaches for measuring all the required values in order to determine the shape factor for individual matrix rock pieces from outcrops of fractured formations. Performing those measurements routinely is regarded as essential parameter for its usability.

Abstract Recently Mittermeir (2014) introduced a new material balance (MB) method applicable to naturally fractured dual porosity reservoirs which correctly considers the matrix-fracture fluid transfer too. The MB model is built by two columns, representing the matrix and the fracture continua. The overall MB is calculated as usual, determining the relation between pressure, OOIP, production and water influx. On top of that the fluid distribution in the fracture and matrix will be considered also. In the fracture full phase separation is assumed and the actual phase contacts define three sections. In the bottom section the fracture injects water by capillary imbibition, in the top section the gravitation forces the gas to invade the matrix and in the middle section expansion drive is active. The fracture-to-matrix injection rates are determined from the relation recovery factor versus injected pore volume (called recovery curves). They are derived from laboratory and field data or estimated on the basis of sound engineering judgment. The paper presents the calculation scheme and a successful application to the Sabah field (Libya) that had more than 500 MMstb original oil in place and a 35-year history of production. The new MB method matches both the reservoir pressure and the positions of the phase contacts. It also provides aquifer and matrix-fracture fluid transfer models. For the first time ever it becomes possible to realistically – this means by fully considering the governing recovery mechanisms and thus the matrix-fracture transfer – calculate material balance for naturally fractured dual porosity reservoirs.

Kazemi et al. (SPE Reserv Eng 7(2):219–227, 1992 ) suggested an empirical matrix-fracture transfer function, verified based on experimental data of Mattax and Kyte (Trans AIME 225(15):177–184, 1962 ), to model fluid flow in naturally fractured dual porosity petroleum reservoirs using a dual-porosity numerical simulator. Their generalized shape factor should be valid for all possible irregular matrix blocks. The factor is calculated based on the volume of the matrix block, the surface open to flow in all directions and the distances of these surfaces to the centre of the matrix block. The summation is done over all open surfaces of a matrix block. Kazemi et al. ( 1992 ) showed that for rectangles and cylinders the formula reduces to the well-known forms of the shape factor. By the time, many authors indicated the validity of the formula, but no theoretical proof was offered for that so far. This study derives the Kazemi et al. ( 1992 ) shape factor using control volume finite difference discretization on the fracture-matrix dual continuum. The matrix blocks are handled as Voronoi polyhedra. The derivation is given for both isotropic and tensorial matrix permeability. Based on this derivation the authors conclude that the Kazemi et al. (SPE Reserv Eng 7(2):219–227, 1992 ) formula is exact under pseudo-steady-state conditions within the dual continuum mathematical concept of natural fractured dual porosity systems.

Abstract This work presents an automatic history matching approach called Target Pressure and Phase Method or TPPM. In a conventional history matching approach the model wells are fixed and one seeks for a reservoir model, in which the historical rates, well pressures, WC and GOR are provided over the entire time. The presented method does the opposite. The reservoir model is given and the computer automatically places wells, which can do what they should do – namely correctly reproduce the measured data. These pseudo wells are perhaps at different locations and connect to other layers as the real wells but the overall material transfer will be correct for the entire time period. If such wells cannot be placed, then the static model is fundamentally wrong and must be replaced. Stochastic realisations can be screened at this level already. Supplementary, water is added into the model boundaries, ensuring that the average pressure in any chosen region closely follows the historical pressure. After the global match has succeeded, the pseudo wells are shifted towards the real ones and the reservoir and perforation properties will be tuned step by step, partly automatically partly manually. The history match is completed after all pseudo wells have been replaced by the model wells. The method is implemented in PRS, a fully developed not commercial user friendly tool. It can be used standalone but also as a pre-processor to ECLIPSEc. The tool needs some command lines added to the ECLIPSE input only. PRS writes out a modified SCHEDULE file containing the actual settings of the pseudo wells and the parameters for the Fetkovich and Carter-Tracy analytical aquifer models (replacing the boundary injections) for the next ECLIPSE run. The paper describes the concept, the corresponding algorithms. The applicability is demonstrated on a small scale example and additionally with a successful field case.

AbstractThe reservoir, located in the Sirte Basin, Libya was discovered in the early 1960s. Since then, more than 65 wells have been drilled. By the end of the year 2008, more than 160 MMstb undersatured oil have been produced from the Upper Cretaceous. Water cut has risen to 65% and average reservoir pressure is still 1000 psi above bubble point. The individual well production history is highly influenced by water coning.The field is geologically complex; heterogeneous, contains numerous large and smaller normal faults sometimes associated with fractures, leading to local permeability enhancements. Average gross thickness is about 300 ft. Recovery is mainly (80%) by the strong edge water influx and by expansion (20%).The objective of the History Match (HM) was to accurately determine the distribution of the reminder movable oil, in order to enable reliable performance predictions of existing and future wells. To meet this goal, the total amount of produced fluids on a well-by-well basis as well as the average local pressure must be matched within close limits.Conventional HM techniques failed due to the complex geology and production history. Especially the highly variable water cut could not be matched satisfyingly. For this reason, a new HM method and workflow, called Target Pressure and Phase Method (TPPM), was applied. TPPM ensures, that the calculated average reservoir pressure and the calculated amount of produced liquids match the corresponding measured values at every point in time. This is achieved by regulation of water influx from artificial boundaries and automatic distribution of the inflow along the well trajectories. Thus, the engineer can immediately concentrate on matching the well-by-well performance leading to a considerable reduction in project duration. Additionally, TPPM gave excellent results in HM. Thus reliable future field operation scenarios can be designed and calculated. The presented workflow and method is generally applicable, especially for reservoirs under strong water influx.

Abstract This paper presents a methodology for creating and running single matrix block models with a reservoir simulator. The small cell sizes lead to numerical problems that were successfully solved. Additionally recovery curves derived from numerous simulation runs under varying reservoir conditions are presented. These curves enable estimating matrix recoveries without running a simulation by only defining rock type and production conditions. The investigations are based on undersaturated oil produced by water displacement. Imbibition forces, gravity drainage and capillary continuity are allowed for. Emphasis is put on the influence of wettability, matrix shape factor and permeability on recovery. The common approach for simulating naturally fractured reservoirs is the dual continuum concept. Not so for the single matrix block model, in which a matrix block and the surrounding fractures are discretized by gird cells. The size of the resulting grid model equals the extensions of a matrix block described by the shape factor in a dual continuum simulation model. Representing both the fracture and the matrix by grid cells makes a matrix-fracture transfer function obsolete, enabling a detailed investigation of matrix-fracture interaction. Consequently, numerical transfer functions and correlations for assessing recovery factors can be derived for certain reservoir conditions. Naturally fractured reservoirs are a challenging task for the E&P industry for old and new oil fields. Many of the newly developed reservoirs happen to be fractured and therefore they are becoming more and more subject of extensive studies. In such systems most of the oil is stored in matrix blocks acting as fluid sources/sinks to the surrounding fracture planes in which fluid transport and production/injection takes place. Consequently, the matrix recovery factor is of special interest, and efficient and accurate methods to estimate the matrix depletion efficiency and matrix-fracture interaction should be found.

Summary This paper presents an incomplete LU (ILU) factorization technique coupled with generalized conjugate-gradient acceleration especially designed for linear equations resulting from locally refined grids. The factorization is based on a special ordering scheme. This repeated red/black (RRB) ordering can cope naturally with grid irregularities introduced by local refinement. The scheme is also very effective on regular grids, where the condition number (a measure of the convergence rate) of the preconditioned linear system increases asymptotically with the number of gridblocks more slowly than it does for conventional ILU preconditionings. For large linear systems, fewer iterations are needed to reach convergence. This makes the method highly competitive compared with other techniques even on regular grids, although the new ordering scheme requires more data handling than standard orderings. Results for several idealized test cases [implicit-pressure, explicit-saturation (LMPES) equations] show the new method to be faster than standard iterative methods.