Finite-difference Reservoir Simulator Research Articles

A New Mechanistic Model for Wettability-Altering Surfactant Floods in Carbonates

Summary Surfactants and low-salinity brines have been shown to be effective for enhanced oil recovery in carbonate rocks through wettability alteration (WA). Oil wettability of carbonates is ascribed to the adsorbed organic acid components in oil. The removal of the adsorbed acids leads to WA. Previous experiments with wettability-altering surfactants have shown the following: WA is a slow process; acid removal is irreversible in most cases; surfactants can access the rock surface in water-wet regions and at three-phase contact lines rather than the entire rock surface; surfactant molecules become inactive after interactions with acids. Existing models/simulators do not incorporate the aforementioned observations. In this work, a multiphase, multicomponent, finite-difference reservoir simulator incorporating a new mechanistic model for WA was developed. The model captures the key physicochemical reactions between adsorbed acids and surfactant molecules and honors the four experimental evidences. The model was first tested at the core scale. The simulation results demonstrated that the model can accurately predict waterflood performance in rocks with various wettability. It can also effectively account for the influence of injection rates in surfactant flood experiments. The effectiveness of the surfactant, controlled by an interaction constant in the model, was found to be a dominant factor. The model was also tested for field-scale pilot tests. The results revealed that total quantity of chemicals injected and the injection rate have a more pronounced effect on oil recovery compared to the timing of surfactant treatment and the concentration of surfactant slug.

Modeling of low salinity waterflooding in carbonates: Effect of rock wettability and organic acid distribution

Carbonate reservoirs hold a substantial amount of our existing oil reserves. However, waterflood recovery from these reservoirs is often low due to their oil-wet nature. Recent studies have shown that oil recovery from oil-wet carbonate reservoirs can be increased by wettability alteration using low salinity brines having carefully modified ionic compositions. In this study, we developed a multiphase, multicomponent, finite-difference reservoir simulator and incorporated an improved mechanistic model to model low salinity waterflood in carbonates. The model considers key geochemical reactions relevant to low salinity waterflooding as reported by many authors recently. However, our modeling approach differs in two ways. Firstly, our model assumes that geochemical reactions of injected brines with the rock take place only on water-wet surfaces of these rocks and at the three-phase contact line; compared to other existing models that consider reactions to occur on the entire rock surface for oil-wet rocks. Secondly, previous models consider organic acids as a single species. In reality, oil is likely to have a mixture of different organic acids of different polarity, chain length, etc. In our model, we consider the oil to have a mixture of different organic acids and studied its impact on wettability alteration and oil recovery. Using this model, single phase and oil recovery experiments reported in the literature were modeled. The effect of rock wettability on geochemical reactions of low salinity brines with carbonate rocks was studied. Similarly, the effect of organic acid distribution on wettability alteration and oil recovery was investigated. Our simulation results showed that the extent of reaction of low salinity brines with carbonate rocks decreased as the rock became more oil-wet due to a decrease in reactive surface area. For example, less delay in effluent sulfate ions was observed in case of oil-wet rocks. The water-wet regions grew as a result of wettability alteration at the three-phase contact line. The initial wettability, extent of wettability alteration, and oil recovery profiles were dependent on the organic acid distribution. Furthermore, the inclusion of only key geochemical reactions (dolomitization, sulfate adsorption, and anhydrite dissolution/precipitation) were found to be reasonable for modeling various experiments reported in the literature. The proposed model will help in improving our understanding of mechanisms responsible for wettability alteration.

Multiscale Integration for Karst-Reservoir Flow-Simulation Models

Summary The significant oil reserves related to karst reservoirs in a Brazilian presalt field add new frontiers to the development of upscaling procedures to reduce time for numerical simulations. This work aims to represent karst reservoirs in reservoir simulators using special connections between matrix medium and karst medium, each modeled in different grid domains of a single-porosity flow model. This representation intends to provide a good relationship between accuracy and simulation time. The concept follows the embedded discrete-fracture model (EDFM) developed by Li and Lee (2008) and later extended by Moinfar et al. (2014); however, this work extends the approach for karst reservoirs [embedded discrete-karst model (EDKM)] by adding a representative volume through gridblocks to represent karst geometries and porosity. For the extension of the EDFM approach in a karst reservoir, we adapt the methodology to four stages: construction of a single-porosity model with two grid domains; geomodeling of karst and matrix properties, each for the corresponding grid domain; application of special connections through the conventional reservoir simulator to represent the transmissibility between the matrix and the karst medium; and calculation of transmissibility between the matrix and the karst medium. For a proper verification, we applied the EDKM methodology in a carbonate reservoir with megakarst structures, which consists of nonwell-connected enlarged conduits and greater than 300 mm of aperture. The reference model was a refined grid with karst features explicitly combined with matrix facies, including coquinas interbedded with mudstones and shales. The gridblock of the reference model measures approximately 10 × 10 × 1 m. For the simulation model, the matrix-grid domain has a gridblock size of approximately 100 × 100 × 5 m. The karst-grid domain had the same block size as the refined grid. Flow in the individual karst-grid domain or matrix-grid domain is governed by Darcy's equation, implicitly solved by a simulator. However, the transmissibility for the special connections between karst and matrix blocks is calculated as a function of open area to flow, matrix permeability, and block-center distance. The matrix properties were scaled up through conventional analytical methods. The results show that EDKM had a considerable performance regarding a dynamic matching response with the reference model, within a reduced simulation time, while maintaining a higher dynamic resolution in the karst-grid domain without using an unconstructed grid. This work aims to contribute to the extension of the EDFM approach for karst reservoirs, which can be applied to commercial finite-difference reservoir simulators. This could be a solution to reduce simulation time without disregarding the explicit representation of karst features in structured grids.

An efficient embedded discrete-fracture model for 2D anisotropic reservoir simulation

In this study, we present an embedded discrete fracture model (EDFM) for an in-house reservoir simulator that incorporates the effect of each fracture with arbitrary orientations, lengths and fracture conductivities. Instead of using the linear flow approximation for the EDFM, the interflows between local matrix grids and fracture elements are derived using boundary element method, which considers the effect of local grid geometry and boundary pressure on the flow into the fracture. The implementation is generally compatible with existing finite-difference reservoir simulators, and the accuracy of the EDFM is validated by comparing the results of analytical/semi-analytical models for fracture flow, LGR modules for fracture flow in commercial reservoir simulators and complex fracture flow in discrete fracture network using finite element method. A grid-sensitivity study is conducted to show the convergence of the new method. Moreover, the cases of multiple bi-wing fractures and complex fracture systems are presented to demonstrate the robustness, applicability and computational convenience of the novice approach for simulating the fracture flow in anisotropic reservoir.

Finite-Wellbore-Radius Solution for Gas Wells by Green's Functions

Summary Even when written in terms of a pseudopressure function, the diffusivity equation for flow of gases through porous media is, rigorously speaking, nonlinear because the viscosity/compressibility product is pseudopressure-dependent. However, several techniques and analysis procedures neglect such nonlinearity. A new methodology for constructing solutions for gas reservoirs through the Green's-function (GF) technique was recently proposed in the literature. Such methodology handles the viscosity/compressibility product variation rigorously, and it was successfully applied to solve several gas-well-test problems. In those problems, the wellbore is always represented by a line source. This work extends the theory a little further by considering a finite-wellbore-radius (FWR) boundary condition for a single vertical well producing at constant rate from an isotropic homogeneous and infinite gas reservoir. The proposed solution does not consider non-Darcy-flow effects, wellbore storage, and skin. Results from our FWR solution are compared with a commercial finite-difference reservoir simulator that shows a very close agreement. We also compared the FWR solution to the correspondent line-source solution to study the difference between the two solutions. As expected, the pseudopressure solutions by use of line-source and FWR boundary conditions do not match at early times, but they do agree at long times, which is exactly how FWR and line-source well solutions for slightly compressible fluids behave. It seems that, even for gas-well problems, the wellbore can be satisfactorily represented by a line source without significant loss of generality. The line-source assumption greatly simplifies the mathematics and the computational effort. This aspect is especially attractive for complex nonlinear gas-well problems that remain to be solved by the GF approach.

Development of an Efficient Embedded Discrete Fracture Model for 3D Compositional Reservoir Simulation in Fractured Reservoirs

Summary Many naturally fractured reservoirs around the world have depleted significantly, and improved-oil-recovery (IOR) processes are necessary for further development. Hence, the modeling of fractured reservoirs has received increased attention recently. Accurate modeling and simulation of naturally fractured reservoirs (NFRs) is still challenging because of permeability anisotropies and contrasts. Nonphysical abstractions inherent in conventional dual-porosity and dual-permeability models make them inadequate for solving different fluid-flow problems in fractured reservoirs. Also, recent technologies for discrete fracture modeling may suffer from large simulation run times, and the industry has not used such approaches widely, even though they give more-accurate representations of fractured reservoirs than dual-continuum models. We developed an embedded discrete fracture model (DFM) for an in-house compositional reservoir simulator that borrows the dual-medium concept from conventional dual-continuum models and also incorporates the effect of each fracture explicitly. The model is compatible with existing finite-difference reservoir simulators. In contrast to dual-continuum models, fractures have arbitrary orientations and can be oblique or vertical, honoring the complexity of a typical NFR. The accuracy of the embedded DFM is confirmed by comparing the results with the fine-grid, explicit-fracture simulations for a case study including orthogonal fractures and a case with a nonaligned fracture. We also perform a grid-sensitivity study to show the convergence of the method as the grid is refined. Our simulations indicate that to achieve accurate results, the embedded discrete fracture model may only require moderate mesh refinement around the fractures and hence offers a computationally efficient approach. Furthermore, examples of waterflooding, gas injection, and primary depletion are presented to demonstrate the performance and applicability of the developed method for simulating fluid flow in NFRs.

Joint inversion of controlled-source electromagnetic and production data for reservoir monitoring

We present a fluid-flow constrained inversion approach for integrating controlled-source electromagnetic data and production data. In this approach, we assumed that the reservoir model has been well defined from a priori knowledge obtained from other independent measurements such as seismic and/or well-logs. Our objective was to reconstruct the permeability distribution and the shape and location of the flooding front. A finite-difference reservoir simulator was used to model the water-flooding process to simulate the time-dependent production data as well as the temporal and spatial distributions of water saturation and salt concentration, which were then transformed into the reservoir resistivity distribution using a petrophysical relationship. A finite-difference frequency-domain electromagnetic forward-modeling code was then employed to simulate the controlled-source electromagnetic response. The permeability distribution was reconstructed using a multiplicative-regularized Gauss-Newton algorithm for jointly inverting controlled-source electromagnetic and production data. From the water saturation distribution, we can identify the shape and location of the fluid-front resulting from the recovery process.

Two-dimensional flow towards a guarded downhole sampling probe: An experimental study

The guarded-probe concept for downhole fluid sampling is analogous to that of focused electrodes for borehole electrical measurements. The guard part of the probe surrounds the sample probe and draws fluid from the near-wellbore region of the rock formation where pore fluid is contaminated with drilling fluid filtrate (invasion fluid). Fluid entering the central sample probe comes from points deeper in the formation, where contamination is lower. We report experiments performed to study the guard probe concept in two dimensions. For these experiments, glass beads contained in a thin cell formed the porous medium. The original, uncontaminated pore fluid was an optical index-matching oil. The invasion fluid was the same oil with a dissolved dye. A camera recorded the displacement of the dyed fluid by the clear fluid and the dye concentration was measured in the flow lines through which fluid was withdrawn. The experiments verify that in 2 dimensions a guarded probe produces clean samples of original pore fluid faster than would a simple probe. The experiments were successfully modeled, without adjustable parameters, by means of a finite-difference numerical reservoir simulator and by computations based on an approximate velocity field given by a complex variable solution of the Laplace equation.

Impact on Scale Management of the Engineered Depressurization of Waterflooded Reservoirs: Risk Assessment Principles and Case Study

Summary Increased oil and particularly gas production may be achieved in waterflooded reservoirs by stopping further water injection and depressurizing the reservoir to release solution gas. Pressure depletion may be accelerated by backproducing injected brines. However, there is the possibility that these brines may cause formation damage by mobilizing fines or deposition of inorganic scales. Scale deposition in production wells may also occur as a result of pressure depletion, with calcite scales being precipitated when the system drops below the CO2 bubblepoint pressure. This paper discusses the assessment and prediction of scale-related formation damage problems that are likely to occur during depressurization of a case study field. The potential for the specific problem arises from the formation of barium sulfate scale as a result of mixing of injected and formation brines during production. Data used in this study include well brine chemistries and an existing finite-difference reservoir simulation model of the field depressurization, which was used to calculate the mixing of injected and formation brines and the movement of the mixing and temperature fronts during waterflooding and subsequent depressurization. This study has determined that the behavior of the scaling potential for each well in this field is different. Also, the degree of scaling, both deep within the reservoir where it does the least damage and around the wellbore (for both injectors and producers) where it may adversely affect production, can be predicted by detailed modeling using both conventional and reaction flow simulations. Former injectors converted to water production or infill wells drilled in the aquifer for pressure depletion may experience an increase in the scaling potential that significantly impacts the economics of the project because of the need for extensive prevention (inhibition) treatments. The increased scaling potential in these wells is a result of the dynamics of brine mixing in the reservoir, the lowering of reservoir temperature in the vicinity of injection wells during waterflooding, and the large volumes of water required to be produced to achieve depressurization. The magnitude of the scaling problem and the economic impact are lower for the production wells because of lower water production rates and higher temperatures.

Optimizing Multiple-Field Scheduling and Production Strategy With Reduced Risk

Developing large E&P assets requires long-term commitments of capital that are tied to decisions on facilities, wells, scheduling, and production strategy. The decisions often must be made in the project-planning phase when large uncertainties exist that can lead to project risks. We present an optimization system and method that enables finding more-optimal reservoir-planning and management-decision alternatives under conditions of uncertainty, such that the associated risks can be managed. The system integrates a global. stochastic search-optimization algorithm, finite-difference reservoir simulation, and economics. The optimization problem is posed with the business goal stated as the objective and includes economic, reservoir, and production constraints and statistical requirements. The optimization is illustrated in an example E&P asset with multiple oil fields produced through a common surface-pipeline network with uncertainties in reservoir volume, fluid quality, deliverability, and costs. Decision solutions seeking to maximize the mean net present value (NPV) while managing risk arc presented. The example includes multiperiod decisions on scheduling of reservoir units, number of wells, and production-process capacity.

Coupled Vadose Zone and Atmospheric Surface‐Layer Transport of Carbon Dioxide from Geologic Carbon Sequestration Sites

Geologic CO2 sequestration is being considered as a way to offset fossil fuel–related CO2 emissions to reduce the rate of increase of atmospheric CO2 concentrations. The accumulation of vast quantities of injected CO2 in geologic sequestration sites may entail health and environmental risks from potential leakage and seepage of CO2 into the near‐surface environment. We are developing and applying a coupled subsurface and atmospheric surface‐layer modeling capability built within the framework of the integral finite difference reservoir simulator TOUGH2. The overall purpose of the modeling studies is to predict CO2 concentration distributions under a variety of seepage scenarios and geologic, hydrologic, and atmospheric conditions. These concentration distributions will provide the basis for determining aboveground and near‐surface instrumentation needs for CO2 sequestration monitoring and verification, as well as for assessing health, safety, and environmental risks. A key feature of CO2 is its large density (ρ = 1.8 kg m−3) relative to air (ρ = 1.2 kg m−3), a property that may allow small leaks to cause concentrations in air above the occupational exposure limit of 4% in low‐lying and enclosed areas such as valleys and basements where dilution rates are low. The approach we take to coupled modeling involves development of T2CA, a TOUGH2 module for modeling the multicomponent transport of water, brine, CO2, gas tracer, and air in the subsurface. For the atmospheric surface‐layer advection and dispersion, we use a logarithmic vertical velocity profile to specify constant time‐averaged ambient winds, and atmospheric dispersion approaches to model mixing due to eddies and turbulence. Initial simulations with the coupled model suggest that atmospheric dispersion quickly dilutes diffuse CO2 seepage fluxes to negligible concentrations, and that rainfall infiltration can cause CO2 to return to the subsurface as a dissolved component in infiltrating rainwater.

Coupled Vadose Zone and Atmospheric Surface-Layer Transport of Carbon Dioxide from Geologic Carbon Sequestration Sites

Geologic CO 2 sequestration is being considered as a way to offset fossil fuel–related CO 2 emissions to reduce the rate of increase of atmospheric CO 2 concentrations. The accumulation of vast quantities of injected CO 2 in geologic sequestration sites may entail health and environmental risks from potential leakage and seepage of CO 2 into the near-surface environment. We are developing and applying a coupled subsurface and atmospheric surface-layer modeling capability built within the framework of the integral finite difference reservoir simulator TOUGH2. The overall purpose of the modeling studies is to predict CO 2 concentration distributions under a variety of seepage scenarios and geologic, hydrologic, and atmospheric conditions. These concentration distributions will provide the basis for determining aboveground and near-surface instrumentation needs for CO 2 sequestration monitoring and verification, as well as for assessing health, safety, and environmental risks. A key feature of CO 2 is its large density (ρ = 1.8 kg m −3 ) relative to air (ρ = 1.2 kg m −3 ), a property that may allow small leaks to cause concentrations in air above the occupational exposure limit of 4% in low-lying and enclosed areas such as valleys and basements where dilution rates are low. The approach we take to coupled modeling involves development of T2CA, a TOUGH2 module for modeling the multicomponent transport of water, brine, CO 2 , gas tracer, and air in the subsurface. For the atmospheric surface-layer advection and dispersion, we use a logarithmic vertical velocity profile to specify constant time-averaged ambient winds, and atmospheric dispersion approaches to model mixing due to eddies and turbulence. Initial simulations with the coupled model suggest that atmospheric dispersion quickly dilutes diffuse CO 2 seepage fluxes to negligible concentrations, and that rainfall infiltration can cause CO 2 to return to the subsurface as a dissolved component in infiltrating rainwater.

Use of reservoir simulation for optimizing recovery performance

A three-dimensional finite-difference reservoir simulator, integrated in an EOR expert system, has been used to determine the reservoir management and production strategies to optimize the oil recovery from a carbonate reservoir. The reservoir is a candidate for an EOR process or otherwise subject to abandonment. After screening the reservoir for an appropriate EOR process on the basis of its properties, it was determined that miscible carbon-dioxide injection is the most suitable process. The management strategies involved studying different design parameters to maximize the project profitability. The injection techniques that were tested in this study include: (1) water-alternating-gas (WAG), (2) simultaneous water-alternating-gas (SWAG) injection, and (3) gas injection in the bottom of the reservoir with water injection in the reservoir top. All simulation runs were conducted using permeability fields that have been conditioned with core data taken from wells in the field. Specific strategy that was used includes the use of horizontal injectors in conjunction with vertical producers. This well configuration has been shown to yield the best oil recovery compared to other well configurations. Simulation results show that the most economical method to produce this reservoir was to simultaneously inject water at the reservoir top and gas at the reservoir bottom. This production strategy has resulted in better sweep efficiency and therefore high oil recovery and good economics.

Efficient modeling of nonconventional wells with downhole inflow control devices

Complex well configurations coupled with downhole inflow control devices offer great potential for the efficient production of oil reservoirs. With systems of this type, chokes can be set to isolate a zone or to provide a more uniform inflow profile, which acts to delay the breakthrough of water and gas. While it is possible to apply existing finite difference reservoir simulators, such models can be time consuming to build and the accuracy of the results depends on the grid and the well model. To obtain the correct well model (well index) for the finite difference simulator, a reference solution (analytical or numerical) is needed. In this work, semianalytical solution methods (based on Green's functions), appropriate for modeling the performance of nonconventional wells operating under single phase flow conditions, are developed and applied to model these complex well configurations. The approach entails a fully coupled nonlinear formulation that accounts approximately for reservoir heterogeneity as well as for pressure drops in the annulus, tubing and downhole chokes. Numerical results for a variety of cases are presented, including examples involving nonconventional wells operating with downhole chokes in a highly heterogeneous fluvial reservoir. Comparisons with finite difference results are presented where appropriate and demonstrate the general level of accuracy of the semianalytical procedure. The use of the method for the determination of optimal choke settings is demonstrated. The semianalytical model developed here can also be used for the calculation of well index.

Implementation of a Flux-Continuous Finite-Difference Method for Stratigraphic, Hexahedron Grids

Summary In this paper we present a 3D flux-continuous finite-difference formulation designed for flow simulation of models with nonorthogonal hexahedron grids with general tensor permeability. Our development follows that of Aavatsmark et al.,1 but we do not operate in transformed space. The new 27-point discretization formula has been implemented in a finite-difference reservoir simulator. This stencil has many desirable properties, including collapsing into a consistent form in two dimensions. We demonstrate that there are many practical situations in which neglecting the influence of nonorthogonality and general tensors results in firstorder errors in flow predictions. A rigorous implementation for this 27-point difference operator as a control-volume finite-difference method determines the upwinding of convection terms associated with multiphase computations. Results and issues associated with implementation of this operator in a conventional finite-difference reservoir simulator are discussed. As an alternative to directly solving the linear matrix associated with the 27-point stencil of the flux-continuous operator, we examine iterative methods that split the matrix into a 7-point stencil part and a remainder. The 7-point stencil part is solved by a direct or iterative method, with the remainder part updated from the previous timestep or iteration. This split operator may permit retention of the linear solver for the standard 7-point formulation while retaining nonorthogonal grid and tensor information.

OPTIMIZING OIL RECOVERY IN A CARBONATE RESERVOIR THROUGH INTEGRATED RESERVOIR SIMULATION STUDIES

ABSTRACT A three-dimensional finite-difference reservoir simulator, integrated in an EOR expert system, has been used to determine the reservoir management and production strategies to optimize the oil recovery from a carbonate reservoir. The reservoir is a candidate for an EOR process or otherwise subject to abandonment. After screening the reservoir for an appropriate EOR process on the basis of its properties, it was determined that miscible carbon-dioxide injection is the most suitable process. The management strategies involved studying different design parameters to maximize the project profitability. The injection techniques that were tested in this study include: (1) water-alternating-gas (WAG), (2) simultaneous water-alternating-gas (SWAG) injection, and (3) gas injection in the bottom of the reservoir with water injection in the reservoir top. All simulation runs were conducted using permeability fields that have been conditioned with core data taken from wells in the field. Specific strategy that was used includes the use of horizontal injectors in conjunction with vertical producers. This well configuration has been shown to yield the best oil recovery compared to other well configurations. Simulation results show that the most economical method to produce this reservoir was to simultaneously inject water at the reservoir top and gas at the reservoir bottom. This production strategy has resulted in better sweep efficiency and therefore high oil recovery and good economics.

Identification of Minimum Numbers of Layer Required to Model Multilayer Reservoirs

ABSTRACT This paper presents new work to identify the minimum number of layers to model multilayered, commingled reservoirs and show the same well response as the complete systems. The pressure response of a well in a multilayer reservoir may be the same as that from a slightly different reservoir with fewer layers than the original reservoir. We investigated constant rate production and constant bottomhole pressure production in a hydraulically fractured and unfractured wells with varing drainage areas. We also analyzed the behavior during the transient, transition, and boundary-dominated flow periods. We used a commercial finite-difference reservoir simulator to perform the study.

Finite Difference Simulation of Geologically Complex Reservoirs With Tensor Permeabilities

Summary The gridblock permeabilities used in reservoir simulation are commonly determined through the upscaling of a fine scale geostatistical reservoir description. Though it is well established that per- meabilities computed in this manner are, in general, full tensor quantities, most finite difference reservoir simulators still treat permeability as a diagonal tensor. In this paper, we implement a capability to handle full tensor permeabilities in a general purpose finite difference simulator and apply this capability to the modeling of several complex geological systems. We formulate a flux con- tinuous approach for the pressure equation by use of a method analogous to that of previous researchers (Edwards and Rogers1; Aavatsmark et al.2), consider methods for upwinding in multiphase flow problems, and additionally discuss some relevant implemen- tation and reservoir characterization issues. The accuracy of the finite difference formulation, assessed through comparisons to an accurate finite element approach, is shown to be generally good, particularly for immiscible displacements in heterogeneous sys- tems. The formulation is then applied to the simulation of upscaled descriptions of several geologically complex reservoirs involving crossbedding and extensive fracturing. The method performs quite well for these systems and is shown to capture the effects of the underlying geology accurately. Finally, the significant errors that can be incurred through inaccurate representation of the full per- meability tensor are demonstrated for several cases.

The effects of mechanical properties and selection of completion interval upon the created and propped fracture dimensions in layered reservoirs

A finite-difference, hydraulic fracture treatment simulator that computes fracture dimensions in a layered reservoir is presented in this paper. Each layer can have different mechanical and fluid flow properties. The model allows initiation of the fracture in multiple producing intervals simultaneously. This pseudo-three-dimensional model has been used extensively to study the effects of reservoir mechanical properties, particularly stress distribution, Young's modulus, and fracture toughness of the reservoir layers on the height of the fracture. Effects of fluid properties such as apparent viscosity and injected fluid volume on fracture height have also been investigated. The use of this model for different scenarios illustrates the effects of mechanical properties on fracture dimension and proppant transport. Several examples have been generated for reservoirs with multiple producing zones, in which the fracture is initiated in one or several intervals simultaneously, so that one can illustrate the effects of perforation placement upon the distribution of proppant within the fracture. These examples illustrate that the location of the perforations can substantially affect the final proppant profile. The propped fracture geometry is used in a single-phase finite-difference reservoir simulator to show the production increase and long-term production performance for different scenarios.

A Generalized Approach to Reservoir Material Balance Calculations

Abstract This work completes the search for a generalized material balance equation. No restrictions are placed on reservoir fluid compositions. Unlike the conventional material balance equation, the new equation specifically accounts for volatilized-oil and, therefore, is uniquely applicable to the full range of fluids, including volatile-oils and gas-condensates. Equally important, the new material balance equation retains the same simplicity with which the conventional material balance equation has become known. The new, generalized material balance equation is featured here by showing how it is used to predict reservoir performance and estimate reserves in example volatile-oil and gas-condensate reservoirs. In comparison, the conventional material balance equation is shown to lead to erroneous results. The new, generalized material balance equation leads to an improved method of reservoir performance analysis. Introduction Material balance calculations are a useful method of reservoir performance analysis. They are routinely used to estimate oil and gas reserves and predict future reservoir performance. Schilthuis(l), in 1936, was among the first to formulate and apply material balances. Inherent in his and others' use of material balances were the following assumptions:at most, there are two hydrocarbon phases: oil and gas,at most, there are two hydrocarbon components: stock-tank oil and separator (surface) gas,the reservoir oil-phase consists of stock-tank oil and separator-gas,the reservoir gas-phase consists of only separator-gas and no stock-tank oil, andthere are no compositional gradients within the system. Assumptions 1 and 2 define the popular two-hydrocarbon- component formulation. Assumption 3 effectively accounts for the effects of dissolved- or solution-gas.Assumption 4 ignores the possibility of volatilized-oil and. therefore, restricts application to black-oils and dry-gases and precludes application to volatile-oils, gas-condensates, or wet-gases. Volatilized-oil is the stock-tank oil content of the free reservoir gas-phase, Assumption 5 means that the system is effectively treated as a tank with no gradients; accordingly, Coats(2) and others have sometimes referred to this type of model as a tank or zero-dimensional model. Despite their apparent oversimplicity, early tank models found widespread use in reservoir engineering applications(3–16). As time progressed, more sophisticated material balance models evolved, each striving for greater generality. Eventually, following the advent of digital computers, the material balance equations were discretized and the first generation of multi-dimensional, black-oil, finite-difference reservoir simulators were developed in the 1960's (17–19). These models, like their zero-dimensional counterparts, could effectively simulate black-oil and dry-gas reservoirs but could 1I0t model volatile-oil and gas-condensate reservoirs. To address this limitation, Cook et al 20 in 1974 introduced a volatile-oil, finite-difference simulator. Their formulation retained the simplicity of two hydrocarbon components, but allowed for volatilized-oil. Consequently, their formulation was applicable to the full range of reservoir fluids. The resulting formulation was applicable to the full range of reservoir fluids. Coats(21) later, in 1986, verified the applicability of the black- and volatile-oil approaches in reservoir simulation by showing agreement with a fully-compositional, equation-of-state (EOS) reservoir simulator.