A rigorous interporosity flow equation incorporating a time-dependent shape factor is derived and validated for improved dual-porosity modeling of naturally fractured gas-condensate reservoirs. The equation expresses theinterporosity molar rate in terms of the pseudo-pressure gradient in thematrix, fracture surface area, matrix permeability, and a variable matrix-blockshape factor. This approach can accommodate the flow directed from matrix tofractures in a simulator that represents the permeability of the interconnected fractures by a tensor inside each grid-block. This feature distinguishes thismodel from the popular sugar-cube approach to modeling naturally fracturedreservoirs. Compositional simulation is performed to verify the flow equation using thetime-dependent shape factor. Numerical experiments with various matrix-blocksizes indicate that the shape factor varies with time but converges to values derived by Lim and Aziz (1995). The average matrix-pressure location in the matrix block shifts from near the fracture face to the block center as thefluid flows from the matrix into the fracture. This phenomenon indicates that neglecting the time dependency of the shape factor can introduce significant errors in numerical simulation of naturally fractured reservoirs. The time dependency can contribute significantly to fluid production once the pressure front moves through the reservoir along the highly permeable fracture network.This phenomenon is not considered in present commercial simulators that usepseudo-steady state factors. The model equations are expressed in dimensionless form so they can bereadily integrated into current simulators. The dimensionless time used in the proposed method includes pseudo-functions that capture multiphase effects ofthe gas-condensate systems. Applications to single- and multiphase black-oils ystems are also discussed. Introduction Numerical simulation of naturally fractured reservoirs has received significant attention and its application has increased in recent years withthe advent of highly efficient computers. Much of the research on naturally fractured reservoir modeling has focused on accurately representing thematrix-fracture fluid transfers. Various mechanisms, including gravity andcapillary effects,1–3 reinfiltration and capillary continuity of the matrix blocks,4–7 and cocurrent/counter current imbibition phenomena,8–10 have been extensively investigated. However, in spiteof the great level of current model sophistication, the highly anisotropic andheterogeneous nature of a fractured formation makes fractured reservoir modeling a challenging task, frequently with uncertain results inforecasting. Typically, numerical simulation of naturally fractured reservoirs assumesthere are two continua, matrix and fractures, within each grid-block. The flow equations are written for each system with a matrix/fracture transfer function to relate the loss or gain of matrix fluids to or from the fracture. For single-phase fluid flowing through an interconnected fracture system, thefollowing governing equation applies: Equation 1 where the fluid transfer rate per unit volume of rock, q, is commonly calculated as a function of the pressure difference between the matrix and fracture systems, matrix flow capacity and matrix geometry considered through aconstant shape factor.

The use of permeability tensors is required when modelling fluid flow in anisotropic and heterogeneous reservoirs presenting multiple zones of directional permeability, or those categorized as naturally fractured reservoirs. A general procedure for characterizing complex reservoirs utilizing their permeability tensor is being developed by integrating data and methods from different disciplines. Permeability tensors for geologically defined fracture patterns are derived, and finally these small-scale descriptors are incorporated into a reservoir simulation program capable of handling full tensor permeabilities. The application and convenience of the method presented in this paper is illustrated with a field example from a naturally fractured reservoir. Introduction For many years, substantial research has been conducted in the areas of geosciences and engineering in order to characterize naturally fractured reservoirs. Numerous approaches have been presented to properly overcome this difficult task. Geoscientists have focused their research towards understanding the process of fracturing (rock mechanics), and the subsequent description of fracture characteristics such as density and orientation. Engineers, on the other hand, have focused their attention to the description of the fluid flow in the fracture systems, and in the development of accurate models (reservoir simulators) to reproduce the history, and predict the hydrocarbon production, for these complex systems. One of the first matrix-fracture models was presented by Warren and Root(1), who presented an idealized sugar cube model with two classes of porosity, a primary porosity that is intergranular, and a secondary porosity that is induced by fractures. Even though the sugar cube model has been widely accepted as the forerunner of the modern interpretation of dual-porosity systems, its limitations in describing the behaviour of some complex fractured reservoir systems have been observed. It is now well known that almost all fracture systems are much more complicated than the suggested model of three orthogonal sets of uniform fractures. One of the most important factors that has been identified as a necessary addition to improve the overall description of such complex reservoirs is the definition of a nine-component permeability tensor for the fracture system. This tensor is used to model fluid flow in complex reservoirs with multiple zones of directional permeability, where the orientation and magnitude of the principal permeabilities may vary between different zones in the reservoir. Snow(2, 3) studied the convenience of using mathematical equivalents of parallel plate openings to simulate fractures dispersed in orientation, distributed in aperture, and of arbitrary spacing. Models for fractured media which contained any number of planar conductors of any orientation and any fine aperture were presented. A key assumption was that all of the conduits have smooth parallel plane walls of indefinite extent (infinite fractures), and an rbitrary aperture. As a result, a permeability tensor could be obtained by superposition of contributions due to the fractures, and due to the permeable matrix. Long et al.(4, 5) addressed the more realistic scenario of finite or discrete fracture systems, where properties such as shape, orientation and location of the fractures in an impermeable matrix were considered to be random variables.

BP operates the Cusiana volatile oil field and the Cupiagua gas condensate field in the Andean Mountains foothills province of Colombia (Fig. 1.). In 1992, a permanent seismic network of ten surface stations was installed in Cusiana and Cupiagua to obtain data for seismic hazard models necessary for the design of field infrastructure. The network is now in its sixteenth year of continuous operation. Currently, an average of 1000 microseismic events per month is recorded. The resulting seismological dataset is of high quality covering a range of seismic magnitudes down to about 1.0 on the Ritcher scale.Over time, the Cusiana-Cupiagua Seismic Network (CCSN) has been used for different purposes. During the past few years, it has become increasingly evident that the network and its data is an invaluable asset for evaluation of conditions relevant to production/injection operations within the reservoirs and adjacent areas.From the reservoir characterization and production operation standpoints, microseismic monitoring (also known as passive seismic) has had two main applications in Cusiana and Cupiagua: (i) to identify production/injection induced high transmissibility pathways and their temporal variations, and (ii) to image the orientation, extension, complexity, and temporal growth of hydraulic fractures. This paper is focused on the first of these applications: how microseismicity has been used as a surveillance tool to track movement of reservoir fluids away from the wellbore. A short description of the seismic network is provided. Then, the methodology for data interpretation is discussed. Finally, partial results are presented showing how microseismicity monitoring is being applied to: (i) assess transmissibility changes due to stress and pore pressure changes through time, (ii) identify potential reactivation of pre-existing weak planes, and (iii) calibrate numerical models to improve history matches.Analysis of the data shows a strong correlation between reservoir dynamics and production induced microseismicity in Cusiana and Cupiagua Fields with great potential as a surveillance tool for improved reservoir characterization and management. IntroductionOil and gas production and injection change the pore pressure and the stress state in the reservoir. These changes give rise to a change in volume of both reservoir fluids and reservoir rock. The volumetric behavior of the reservoir fluid is controlled by the fluid composition and the change in the pore pressure and is not the subject of this paper. The volumetric response of the reservoir rock depends on the mechanical properties of the rock material (matrix and pre-existing fractures) and the combined effect of changes in pore pressure and stress state.Conventional reservoir engineering incorporates the implicit assumption that the local stress state within the reservoir remains constant with time. Thus, no deformation of matrix and natural fractures, caused by stress changes, take place during the reservoir producing life. In this case, reservoir dynamics are...

An accurate interporosity flow equation incorporating a time-dependent shape factor is derived and verified for improved dual-porosity modeling of waterflooding in naturally fractured reservoirs. This equation expresses the interporosity exchange rate in terms of the oil phase pressure gradient in the matrix, fracture surface area, fluid effective permeability at the matrix/fracture interface, fluid viscosity and a variable matrix-block shape factor. This approach can accommodate the flow directed from matrix to fractures while representing the permeability of interconnected fractures as a tensor. The model equations are expressed in dimensionless form for convenient integration into present numerical simulators for accurate simulation of waterflooding in naturally fractured reservoirs. These features distinguish this model from the conventional sugar-cube modeling of naturally fractured reservoirs. Fine-grid numerical simulation of a matrix block is performed to verify the flow equation using the time-dependent shape factor. Numerical experiments with various size matrix blocks indicate the shape factor varies with time and converges to the steady-state shape factor value reported in previous studies for single-phase flow. The shape factor for single-phase flow converges to a steady-state value at a speed proportional to the reciprocal of total compressibility, while the shape factor for two-phase flow converges at a speed proportional to the slope obtained from the capillary pressure curve evaluated at the average water saturation present at the matrix/fracture interface. Therefore, the single-phase shape factor converges much more rapidly to its steady-state value than the two-phase shape factor. This study demonstrates that neglecting the time-dependency of the shape factor introduces significant errors in fractured reservoir simulation, and time-dependent shape factors are needed because of the variation of the water saturation front location in the matrix block, which moves from the fracture face to the block-center as the water flows from the fracture into the matrix. Introduction Most naturally fractured reservoirs are characterized by initial high production rates that later drop to much lower stable flow rates. The later period of production is controlled by the matrix/fracture interaction through the interporosity flow rate. During this period, the low-permeability matrix replenishes fractures with hydrocarbon fluids, which are produced from the high conductivity pathways at the wellbore.1 Fluid expansion is the primary recovery mechanism during primary recovery. Successful implementation of a secondary or tertiary recovery project in a naturally fractured reservoir requires that the matrix/fracture interaction be well understood. Some of the mechanisms investigated in order to estimate and predict the interporosity flow rate include gravity and capillary effects,2–4 reinfiltration and capillary continuity throughout the matrix blocks,5–8 and cocurrent and/or countercurrent imbibition phenomena.9–11 In waterflooding, capillary imbibition is generally the most important recovery mechanism.12

Based on an analytical analysis of the steady-state theory and its application to well test interpretation in gas-condensate reservoirs, two simplified models are proposed when relative permeability curves are not available. They allow the reservoir engineer to have relative permeability curves as functions of bottomhole pressure from constant composition experiments. Step-by-step procedures explained in the paper rely on the suitability of the steady-state assumptions. Comparisons are made using actual field data and discussions on applicability are also presented. One of the methods is applicable when very low capillary number can be assumed, and immiscible relative permeability curves can be used for pressure transient interpretation. This method is recommended to be used in pressure buildup data analysis. The other proposed method is recommended for pressure drawdown data analysis and is based on the assumption that miscible flow regime controls the fluid flow in the near wellbore region. It is concluded that, although it depends on several factors, miscible relative permeability curves are generally good approximation for pressure drawdown data analysis. An additional method is also discussed. It relies on the ability to simulate properly the near-wellbore fluid flow with accurate fluid and flow models. Using output data from a compositional simulator, it was shown that it is possible to numerically model fluid bank impairment, while analytically, it is possible to obtain reservoir properties, usually shadowed in current methods for condensate buildup. The proposed numerical method requires detailed information and can be impractical to apply at early stages of gas-condensate production. On the other hand, the two analytical methods are shown to give good estimates of permeability, mechanical skin factor and reservoir initial pressure at early stages when relative permeability data are not available or are not sufficiently reliable. Introduction The analysis of pressure transient in gas-condensate wells is strongly influenced by the presence of a liquid phase in the near wellbore region at reservoir pressures below the saturation pressure (Fig. 1). Simplified models have been considered based on comparisons between analytical and numerical solutions of the diffusivity equation for these systems. Aanonsen1 introduced a pseudo-pressure function to linearize the diffusivity equation inspired by the single-phase pseudo-pressure function of Al-Hussainy et al.2 Jones et al.3–7 developed a methodology of well test data analysis in gas-condensate wells. Their approach considers the application of the steady-state theory and prior knowledge of relative permeability curves as functions of pressure. Among others, Thompson and Reynolds8 recognized the drawback of requiring such information for practical field analysis. They proposed a more simplified analytic solution based on the single-phase pseudo-pressure function and the radial extent of the liquid zone. Penuela9 showed that is possible to obtain accurate reservoir parameter values, such as absolute permeability, if numerical and analytical solutions are combined. A two-phase pseudo-pressure and pseudo-time functions can be constructed using the relative permeability-saturation-pressure relationship from the output of a numerical simulator. Then, using analytical solutions from a fully linearized diffusivity equation, reservoir parameters can be obtained without assuming steady-state9. Penuela9 accurately predicts the saturation distribution by considering relative permeability dependency on capillary number and inertial effects. However, initial values of the absolute reservoir permeability and mechanical skin factor are required for rapid convergence of the iterative process. This paper reviews the analytical methods that require relative permeability curves and presents alternative two-phase approaches when they are not available. If the relative permeability curves are available, this paper shows how pressure drawdown and buildup data can also be analyzed by applying Penuela's9 comprehensive methodology.