Reservoir Simulation Grids: Opportunities and Problems

Three-dimensional, multiphase flow problems are solved on static, composite grids. The composite grid consists of a global, rectangular coarse grid and a set of locally refined patches around active wells or other areas with important local flow properties. Curvilinear (locally orthogonal) refinements can be utilized to approximate radial or near-radial flow around wells if such flow is assumed to be appropriate. More general localized refinement can be utilized otherwise. Finite differencing is presented which connects the local and global grid cells in an accurate manner. Other methods for composite grid applications which have appeared in the SPE literature generate linearizations which have lost their banded structure and must utilize corresponding solution algorithms that are extremely difficult to vectorize for efficient solution. Preconditioned iterative techniques will be discussed which allow easy implementation of the composite grid techniques in existing simulators with full vectorization capabilities. Pressure calculations in a single-phase flow problem with a distribution of injection and production wells exhibit excellent agreement between numerical calculations on locally radial grids and analytical solutions. Water and gas coning results for a single well also show very good agreement between computations on a hybrid grid and a circular cylindrical grid. This indicates the potential for incorporating coning models around any well in a full field-scale application without destroying the efficiency of the original simulator.

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper SPE 188575, “A New Shortcut Work Flow in Flexible Reservoir Modeling: Introducing Structural Features Without Regridding,” by Alejandro Rodríguez Martínez and Stefano Frambati, Total, prepared for the 2017 Abu Dhabi International Petroleum Exhibition and Conference, Abu Dhabi, 13–16 November. The paper has not been peer reviewed. Current reservoir-modeling work flows are rigid, because modification to the understanding of the underlying structural model often requires a complete regeneration of the reservoir grid, which brings additional costs, delays, and incompatibilities with past calculations. This paper proposes a novel work flow for structural-features modeling that allows the introduction of faults and other structural and nonstructural features to any simulation grid without modification. Introduction The modeling of hydrocarbon reservoirs is a multiscale and multidisciplinary process that usually involves several months for advancement from seismic interpretation to reservoir simulation. One crucial moment in the life of a reservoir model is the creation of its reservoir-simulation grid. This involves the encoding of the so-called structural model (Fig. 1), a set of surfaces representing interfaces between different geological features. A given grid cell can belong only to one side of each of these surfaces, generating geometrical tension on the grid. Once the reservoir simulation grid is created, every aspect of the simulation depends on it. Several weeks or even months are spent building and quality checking the reservoir-simulation model, which is completely based on the cell division of the space determined by the reservoir grid. Simulations are then run and compared with actual data, when it is available, or used for conceptual design of production methods and structures. Other times, the model already exists and new production data are acquired. These simulations and comparisons sometimes re-veal fundamental flaws in the simulation model, such as the existence of an unseen or underestimated fault that was not included or the need to model differently the flow across a fault because a fault relay was ignored. In such cases, the engineer can decide either to ignore the issue, leaving an inaccurate model in place for the rest of the field’s life but retaining compatibility with previous simulations, or to modify the reservoir properties manually to mimic the missing feature. These manual modifications, however, are extremely costly, and the result is often nongeological and possibly inconsistent with future evolutions of the model. A final option is for the seismic-interpretation team to add the missing fault to the structural model. This may require many steps, ultimately changing the geometry and number of cells. Expected timelines will be delayed, the budget will grow, and some development decisions will have to wait for the new simulations. Because of this, in future projects, the engineer will take great care to ensure that every trace of a fault is included in the first version of the structural model. That way, transmissibility multipliers can always be set to unity and the faults can be removed de facto from the actual simulation if need be. This strategy, however, creates large, unwieldy models, often with more than 200 faults.

This paper describes application of Project INTERSECT, a next generation highly scalable reservoir simulator on real large scale field models. High resolution reservoir simulation is required to better define and describe fluid flow and enable improved field development and tactical operational planning. Massively parallel computing techniques overcome limitations of problem size and space resolution. This paper demonstrates that large-scale simulation models can be performed on commodity hardware taking advantage of evolution in multicpu hardware architecture and software engineering. This allows both geologists and reservoir engineers to include more realistic geologic and engineering detail for better and more reliable production optimization. Intense computer simulation is essential for effective reservoir management. The advances in reservoir characterization techniques and the industry drive towards the ‘smart oilfield’ with rapid model updates will require more efficient model processing to achieve timely field operational decisions. Parallel reservoir simulators have the potential to solve larger, more realistic problems than previously possible. The size and application of reservoir simulation problems have been limited by the availability of computing hardware, reservoir simulation architecture and of solution methods for solving large-scale heterogeneous problems. The next generation reservoir simulator demonstrates that key modeling challenges has been overcome by a software architecture and capability to model more realistic subsurface and surface models. Applications of the new reservoir simulator illustrates how typical reservoir engineering options such as local grid refinement, local grid coarsening, multilateral wells and aquifer modeling affect the overall parallel performance and scalability using highly heterogeneous large-scale models. Application of new modeling techniques highlight increased accuracy of modeling results and more reliable field development planning and reservoir management decisions.

In this paper three different schemes for the finite-difference approximation of ( ) terms with irregular spacings of the grid points are investigated. Several methods of selecting the grid spacing are considered, and numerical results are presented for some carefully selected examples. presented for some carefully selected examples. Results indicate that one of the most commonly used finite-difference schemes may unnecessarily result in large discretization errors; the other two schemes discussed result in greater accuracy and are no more difficult to use. Introduction Finite-difference methods are used widely for solving partial-differential equations encountered in petroleum reservoir simulation. In many practical problems it is necessary to refine the grid in certain problems it is necessary to refine the grid in certain parts of the reservoir, thus resulting in an irregular parts of the reservoir, thus resulting in an irregular grid spacing for the system. For example, the local refinement of grid is usually necessary in the study of the flow behavior near a well, i.e., the case of coning problems. On the other hand, it is usually advantageous to make the grid locally coarse over aquifers and large gas caps. In spite of the importance of irregular grids in reservoir simulation and other engineering problems, little is known about the effect of various schemes on the over-all discretization error. Most literature on the subject of irregular grid spacings is concerned with the treatment of irregular boundaries. Here, the properties of several difference approximations to properties of several difference approximations to the second-order operator Au = [ ] are investigated. Operator A is encountered in all diffusion-type equations, such as multiphase flow in porous media and heat conduction with temperature-dependent conductivity. For simplicity, we limit our study to one space variable in this study. All results reported here can be extended readily to any number of space variables.

In this research study, we conducted a coupled thermal-stress-fluid flow numerical model on the UTF Phase A SAGD project in order to investigate the fundamental geomechanical behaviour involved in the SAGD process, and to gain insight into the reservoir response to temperature and pore pressure changes. The numerical simulation is carried out by using a self-developed coupled finite element model which incorporates our proposed strain-induced permeability model. The obtained simulation results were compared with the measured data. Introduction Thermal recovery processes involve coupling between heat transfer, multiphase flow and stress/deformation, which has become an increasingly important subject in the petroleum field(1). Particularly, the coupling is crucial in problems such as borehole stability, hydraulic fracturing and injection/production induced deformation of the ground surface during the thermal recovery process in heavy oil or oil sand reservoirs. Numerical modelling of the coupled processes is historically carried out in the areas of geomechanics modelling and reservoir simulation. The former is to compute the stress-strain behaviour; therefore, the deformation. The latter is to essentially model the multiphase flow and heat transfer in porous media. Each of these disciplines simplifies the part of the problem that is not of primary interest. These approaches are unacceptable in situations where the coupling is strong and the changes of porosity and permeability cannot be accounted for by rock compressibility alone. Gutierrez and Lewis(2) extend Biot's theory to multiphase fluid flow in deformable porous media. Based on their formulation, they conclude that the coupling between the geomechanics and the multiphase flow occurs simultaneously. Thus, fully coupled system equations of deformations, multiphase flow and heat transfer should be solved simultaneously. Development of such kinds of fully coupled geomechanics-multiphase flow-heat transfer simulators needs tremendous effort, since the existing FEM geomechanics codes and FDM reservoir simulators cannot be used. Settari and Mourits(3) present an approach to couple the stress-strain behaviour to multiphase flow-heat transfer using porosity as a coupling parameter. The geomechanics module and the thermal reservoir simulator are used in a staggered manner. Pore pressure and temperature changes are calculated from the thermal reservoir simulator and transferred to the geomechanics module. The stress and the displacement changes are then calculated in the geomechanics simulation. An iterative algorithm is used to ensure that the porosity calculated from the geomechanics module is the same as that from the thermal reservoir simulator. The staggered technique employed to solve the coupled system equations allows for the use of the existing geomechanics codes in conjunction with a standard reservoir simulator. Currently, most of the commercial coupled geomechanics-multiphase flow-heat transfer simulators are developed in this way. The disadvantage of these kinds of coupled simulators is that the thermal reservoir module, usually developed using the finite difference method (FDM), cannot accommodate the full permeability tensor, since they adopt the standard discretization scheme, such as 5-spot for 2D problems and 7-spot for 3D problems. In this paper, the development of a coupled geomechanics-multiphase flow-heat transfer simulator using the finite element method (FEM) is described with the use of Galerkin's least squares (GLS) technique(4) to stabilize the saturation equation.

The paper presents a discussion on the issues related to the interaction between geomechanics and reservoir simulation in deformable hydrocarbon reservoirs. Geomechanics is important in order to account for rock deformations due to pore pressure and temperature changes resulting from production and fluid injection. Rock deformation can affect the permeability and pore compressibility of the rock. In turn, the pore pressures will be vary due to changes in the pore volume. Geomechanics is also required in order to account for the effect of the non-pay rock surrounding the reservoir on the overall reservoir compressibility and the loads transmitted to the reservoir by the weight of the overburden rock. The paper gives the formulation and finite element discretization of Biot's equations for multi-phase fluid flow in deformable porous media. Based on this formulation, it is argued that geomechanical response and multi-phase fluid flow are fully-coupled processes in that pore pressure changes affect rock mechanical response and vice-versa, and that the two processes occur simultaneously. By contrasting Biot's equations and its discretization to the corresponding multiphase fluid flow equations used in reservoir simulations, it is shown that reservoir simulators neglect or simplify important geomechanical aspects that can have impact on reservoir productivity. This is attributed to the fact that the only rock mechanical parameter involved in reservoir simulations is pore compressibility. This parameter is not sufficient in representing aspects of rock behaviour such as stress path dependency and dilatancy, which require a full constitutive relation. Furthermore, the pore pressure changes due to the applied loads from the non-pay rock cannot be accounted for by simply adjusting the pore compressibility. Example problems are shown in order to illustrate the value of coupling geomechanics to reservoir simulators. Finally, the practical benefits of using coupled geomechanics and reservoir simulations are discussed. P. 439

Distinguished Author Series articles are general, descriptive representations that summarize the state of the art in an area of technology by describing recent developments for readers who are not specialists in thetopics discussed. Written by individuals recognized as experts in the area, these articles provide key references to more definitive work and presentspecific details only to illustrate the technology. Purpose: to informthe general readership of recent advances in various areas of petroleumengineering. "Today, desktop computers and a wide range of reservoir. simulation systemsprovide the engineer with an economic means to solve complex reservoir.engineering problems with efficiency." Introduction This paper describes the state of the art in reservoir simulation for thenonexpert, focusing primarily on advances that have occurred since a similarpaper was published by Coats1 in 1982. For completeness, a historyof simulation is given first, followed by a brief and simple review of thesimulation process. Advances in simulation technology are then described, highlighting those in reservoir description and simulation of naturallyfractured reservoirs, hydraulic fracturing, and horizontal wells. The future ofreservoir simulation is discussed last. Many papers have been presented since Coats' review in 1982. Rather thanrepeat the exhaustive list provided in the SPE indices, a reduced set ofpertinent references is given. I offer my apologies in advance to the authorsof the many excellent papers contributed during the past decade that are notmentioned here. History of Simulation In the broadest sense, reservoir simulation has been practiced since thebeginning of petroleum engineering. By the late 1940's, the potential ofreservoir simulation was recognized and several companies embarked on thedevelopment of both analog and numerical models to enhance the existinganalytical solutions, material balances, and ID Buckley-Leverett displacementcalculations. By the late 1950's, major commitments to fundamental research on thenumerical solution of flow equations began to bear fruit. The results werecrude, but certainly useful, computer programs for simulating reservoirs. Theseprograms represented a major advance and used the solution of sets offinite-difference equations to describe 2D and 3D, transient, multiphase flowin heterogeneous-porous media. For the first time, reservoir engineers couldsolve complex problems. The driving force for development of numerical-simulation systems was theability to predict the future performance of a reservoir under various "whatif" conditions so that better reservoir-management decisions could be made. This force still drives the use of reservoir simulation today. Current businessincentives for virtually all reservoir-simulation applications fall into one ormore of the following categories: economics; credibility and reliability;decision-making; arbitration and unitization; performance monitoring;nonelective studies resulting from safety, environmental, or regulatoryconcerns; and communications and personnel utilization.2 Throughout the 1960's, reservoir-simulation development was devoted largelyto three-phase, black-oil reservoir problems. Recovery methods that weresimulated included pressure depletion and various forms of pressuremaintenance. Many computer programs were developed to address most of thegeneral reservoir problems encountered. These programs operated on mainframecomputers and used decks of cards for input. During the 1970's, the picture changed remarkably because of theproliferation of research in EOR processes, advances in numerical-solutiontechniques, and the decreased size and increased speed of computers. Mathematical simulators were developed that extended beyond conventionalprimary and secondary recovery into such areas as chemical flooding, steamflooding, and in-situ combustion. Research during this period resulted insignificant advances in characterizing the physics of hydrocarbon displacementsunder the influences of temperature, chemical agents, and complexmulticomponent phase behavior. Simulators had to reflect chemical adsorptionand degradation, reaction kinetics, interfacial-tension-reduction effects, andmore complex equilibrium phase behavior. The range of reservoir-simulation application continued to expand during the1980's. Reservoir description, finally becoming a major topic on its own, advanced to the use of geostatistics to describe heterogeneities and to providereservoir definition not expressed in reservoir models. The technology to modelnaturally fractured reservoirs, including compositional effects, was alsodeveloped, with extensions into the simulation of hydraulic fracturing andhorizontal wells. and application to such complex processes as reservoirmonitoring.

Grid orientation effect (GOE) is the appearance of preferential flow along grid coordinate directions in numerical reservoir simulation. GOE is most evident in simulations with strong adverse mobility ratios, such as immiscible gas injection and steamfloods. Motivated by previous work, an eleven-point finite difference formulation for multiphase flow is investigated and found to reduce errors for steamfloods using structured grids. The eleven-point formulation is implemented in a parallel, fully-implicit reservoir simulator with thermal, black-oil and compositional formulations, and the implementation supports both local grid refinement (LGR) and dual-porosity, dual-permeability (DPDK) modeling. Systematic tests are performed for compositional steamflood cases with different grid resolutions and grid coordinate angles between wells. A comparison of seven and eleven-point formulation results, using different grid scales and hybrid unstructured grids, demonstrate that the eleven-point scheme is effective in mitigating GOE and can leverage the benefits of structured LGR and DPDK options. Using grid-refinement as a means of reducing GOE is case dependent and is not always successful. Additional results suggest that using grid refinement with local application of the eleven-point scheme around only the injector does help mitigate GOE with increased computational efficiency, but GOE is not reduced as well as when the eleven-point scheme is used in the entire grid-system.

Carbon dioxide (CO2) injection at supercritical conditions using the huff-n-puff process has been proven to be an effective resource to mitigate the production impairment generated by near wellbore condensate banking in rich gas condensate fields. This approach decreases the density of condensate and at the same time increases the density of the gas, thereby generating a reduction in the interfacial tension between the two co-existing phases. This study evaluates the impact of CO2 injection under the huff-and-puff process on gas recovery enhancement using reservoir simulation. A compositional reservoir simulator was utilized to perform CO2 injection at supercritical conditions. Several sector models with local grid refinement (LGR) were constructed to evaluate the impact of CO2 injection at supercritical conditions (huff-n-puff) on the condensate banking in wells with rich gas condensate production. The evaluation considered numerous scenarios of injecting supercritical CO2 at different decreasing reservoir pressures in combination with sensitivities of CO2 injected volume, gas production rate, and soaking time duration. The results indicated a significant reduction of condensate saturation around the wellbore leading to improvements in gas productivity. In addition, an optimum CO2 injection frequency required to maintain low condensate saturation in the near wellbore region was established. Overall, the results indicated that injecting CO2 at supercritical conditions in carbonate reservoirs has the potential to mitigate the condensate banking problem. A positive correlation between reservoir depletion status and gas recovery was also established from the study: optimum treatment results are obtained at condensate saturation above 20%, and at reservoir pressures below the dew point. Condensate banking around the wellbore can significantly limit gas/condensate production and recovery. The injection of CO2 to overcome this problem has been more commonly used on sandstones while its use on carbonates is still under testing and development. This paper presents a reservoir simulation study using huff-n-puff CO2 injection in a carbonate reservoir to enhance well production and mitigate condensate formation around the wellbore region.

Fractured wells are frequently encountered in petroleum engineering. They can be hydraulically fractured wells to improve oil production or wells intersecting natural fractures. Due to the presence of the fractures, the flow behavior and the pressure distribution in the vicinity of the fractured well are complex and the numerical modelling for this kind of wells is complicated. In this paper, we will discuss the numerical methods for the fractured well modelling by reservoir simulators. Introduction The study of fractured wells has received more attention in well test analysis for a long time. Some analytical or semi-analytical formulas have been developed for several simple cases, such as a fully penetrating hydraulically fractured vertical well, a longitudinal fractured horizontal well or a transverse fractured horizontal well. But these formulas cannot be used for the flow simulation for a field study or for the study of the complex wells and fractures. To handle these cases, a reservoir simulator can be used. During this decade, several numerical simulation studies for fractured wells have been published. In all these studies, fine gridblocks are used around the wells and the fractures. The requirement of grid refinement and the impacts of the grid discretization on the well performance is particularly discussed by Hegre. Although using grid refinement gives accurate simulation results, it is expensive and prohibitive for a full field simulation. In well test analysis, an equivalent wellbore radius rw, eq is introduced for the infinite conductivity fracture. However, this equivalent wellbore radius rw, eq can be used for a reservoir simulator only if the fracture is entirely included in one gridblock and rw, eq is smaller than the equivalent wellblock radius r0. So, it is necessary to study the techniques for the fractured well modelling with coarse grids. In this paper, we will propose a new method for the simulation of fractured wells. This method is based on the well model rather than on the gridding technique. For the well modelling in reservoir simulation, a numerical productivity index (PI) is usually introduced to relate the wellbore pressure, the well block pressure and the well flow rate, since the singularity around the well causes different values between the wellbore pressure and the wellblock pressure. In most of the well models, the well trajectory and its position are not really considered. So these well models are limited for the complex well modelling, and are inappropriate for the fractured well modelling. Recently, a new kind of well model was proposed, which takes into account well configurations by using equivalent transmissibilities to improve the flow calculation in the well vicinity. This well model will be extended to model fractured wells in this paper. In this paper, we will study particularly high conductivity fractures, and these fractures can be considered as a part of the well. To establish a well model, it is necessary to know a steady-state or pseudo-steady state pressure distribution around the well. In section 2, the integral formulae are introduced to represent the pressure distribution for a steady-state problem. This formula is general and is valid for any well configuration and any kind of fractures. In particular, we will present some applications to hydraulically fractured wells. Modelling of Fractured Wells A steady-state flow problem is considered to get the pressure solution around the wells and the fractures. This solution will be used for numerical PI and equivalent transmissibility calculations. The integral formulas are used for the pressure representation. These formulas were introduced for complex well modelling and are also suitable for fractured well modelling. P. 813

The paper presents a discussion on the issues related to the interaction between rock deformation and multiphase fluid flow behaviour in hydrocarbon reservoir simulations. Pore pressure and temperature changes resulting from production and fluid injection induce rock deformations which should be accounted for in reservoir modelling. Reservoir rock deformation can affect the permeability and pore compressibility of the rock. In turn, the pore pressures will vary due to changes in the pore volume. The deformation of the surrounding non-reservoir rocks (under-, side- and over- burdens) should also be taken into account in order to assess their effects on the overall reservoir compressibility as well as the loads transmitted to the reservoir by the weight of the overburden. The paper presents the formulation of Biot's equations for multi-phase fluid flow in deformable porous media. Based on this formulation, it is argued that rock deformation and multi- phase fluid flow are fully-coupled processes which should be accounted for simultaneously. However, the coupled equations can be decoupled to a classical hydraulic diffusivity equation for pre-defined simple stress paths. In the general case, by contrasting Biot's equations and its finite element discretization to the corresponding multiphase fluid flow equations used in classical reservoir simulations, it is shown that reservoir simulators neglect or simplify important geomechanical aspects that can have impact on reservoir productivity. This is attributed to the fact that the only rock mechanical parameter involved in reservoir simulations is the pore compressibility. This parameter is not sufficient in representing aspects of rock behaviour such as stress path dependency and dilatancy, which require a full constitutive relation. Furthermore, the pore pressure changes due to the applied loads from the non-pay rock cannot be accounted for by simply adjusting the pore compressibility. Example problems are shown in order to illustrate the importance of coupling flow and deformation in reservoir simulators. P. 63

In this research study, we conducted a coupled thermalstress-fluid flow numerical modeling on the UTF Phase A SAGD project in order to investigate the fundamental geomechanical behavior involved in the SAGD process and to gain insight into the reservoir response to temperature and pore pressure changes. The numerical simulation is carried out using a self-developed coupled finite element model which incorporates our proposed strain-induced permeability model. The obtained simulation results were compared with the measured data. Introduction Thermal recovery processes involve coupling between heat transfer, multiphase flow and stress/deformation, which has become an increasingly important subject in the petroleum field(1). Particularly, the coupling is crucial in problems such as borehole stability, hydraulic fracturing and injection/production induced deformation of the ground surface during the thermal recovery processes in the heavy oil or oil sand reservoirs. Numerical modeling of the coupled processes is historically carried out in the areas of geomechanics modeling and the reservoir simulation. The former is to compute the stress-strain behavior, therefore the deformation; the latter is to essentially model the multiphase flow and heat transfer in porous media. Each of these disciplines simplifies the part of the problem that is not of primary interest. These approaches are unacceptable in situations where the coupling is strong and the changes of porosity and permeability cannot be accounted by rock compressibility alone. Gutierrez and Lewis(2) extend Biot's theory to multiphase fluid flow in deformable porous media. Based on their formulation, they conclude that the coupling between the geomechanics and the multiphase flow occurs simultaneously. Thus, fully coupled system equations of deformations, multiphase flow and heat transfer should be solved simultaneously. Development of such kinds of fully coupled geomechanics-multiphase flow-heat transfer simulators needs tremendous effort, since the existing FEM geomechanics codes and the FDM reservoir simulators cannot be used. Settari and Mourits(3) present an approach to couple the stress-strain behavior to multiphase flow, heat transfer using porosity as a coupling parameter. The geomechanics module and the thermal reservoir simulator are used in a staggered manner. Pore pressure and temperature changes are calculated from the thermal reservoir simulator and transferred to the geomechanics module. The stress and the displacement changes are then calculated in the geomechanics simulation. An iterative algorithm is used to ensure that the porosity calculated from the geomechanics module is the same as that from the thermal reservoir simulator. The staggered technique employed to solve the coupled system equations allows for the use of the existing geomechanics codes in conjunction with a standard reservoir simulator. Currently, most of the commercial coupled geomechanics-multiphase flow-heat transfer simulators are developed in this way. The disadvantage of these kinds of coupled simulators is that the thermal reservoir module, usually developed using finite difference method (FDM), cannot accommodate the full permeability tensor, since they adopt the standard discretization scheme such as 5-spot for 2-D problems and 7-spot for 3-D problems. In this paper, the development of a coupled geomechanicsmultiphase flow-heat transfer simulator using finite element method (FEM) is described with the use of the Galerkin's least squares (GLS) technique(4) to stabilized the saturation equation.

This paper presents non-intrusive, efficient stochastic approaches for predicting uncertainties associated with petroleum reservoir simulations. The Monte Carlo simulation method, which is the most common and straightforward approach for uncertainty quantification in the industry, requires to perform a large number of reservoir simulations and is thus computationally expensive especially for large-scale problems. We propose an efficient and accurate alternative through the collocation-based stochastic approaches. The reservoirs are considered to exhibit randomly heterogeneous flow properties. The underlying random permeability field can be represented by the Karhunen-Loeve expansion (or principal component analysis), which reduces the dimensionality of random space. Two different collocation-based methods are introduced to propagate uncertainty of the reservoir response. The first one is the probabilistic collocation method that deals with the random reservoir responses by employing the orthogonal polynomial functions as the bases of the random space and utilizing the collocation technique in the random space. The second one is the sparse grid collocation method that is based on the multi-dimensional interpolation and high-dimensional quadrature techniques. They are non-intrusive in that the resulting equations have the exactly the same form as the original equations and can thus be solved with existing reservoir simulators. These methods are efficient since only a small number of simulations are required and the statistical moments and probability density functions of the quantities of interest in the oil reservoirs can be accurately estimated. The proposed approaches are demonstrated with a 3D reservoir model originating from the 9th SPE comparative project. The accuracy, efficiency, and compatibility are compared against Monte Carlo simulations. This study reveals that, compared to traditional Monte Carlo simulations, the collocation-based stochastic approaches can accurately quantify uncertainty in petroleum reservoirs and greatly reduce the computational cost.

C oalbed methane (CBM) is well recognised as an important natural gas resource. However, producing it commercially has proven to be a challenging process, primarily due to the complex nature of its gas storage and flow mechanisms. Although CBM reservoirs are naturally fractured due to the presence of face and butt cleats, gas stor­age and saturation depend on the adsorption capacity of the coal matrix. The ability to predict permeability variations and its evolu­tion in a CBM reservoir, which is affected by several dynamic CBM properties and operating conditions, also define the design, optimization, and analysis of the gas recovery process. As the permeability is naturally low, interventions are necessary in order to achieve commercially viable gas production rates. CBM are also dual-porosity media where the vast majority of the gas is stored in the low permeability coal matrix by sorp­tion. The flow to production wells, however, occurs through the coal’s natural fracture system, that stores relatively small amounts of gas, because coal matrix practically has no perme­ability. The result is that properties of the coal matrix have the greatest effect on the estimates of gas in place and recovery. It is well understood that the key parameter controlling gas flow in CBM reservoirs are cleat networks, gas storage, desorp­tion and permeability anisotropy (coal shrinkage and swelling). While the role of reservoir simulation is well under­stood for conventional oil and gas reservoirs and its related techniques for prediction and production optimisation, for unconventional reservoirs (CBM, Shale, etc.) current reservoir simulation software needs to be further developed and adapted to represent the performance of these types of fields and gener­ate more acceptable results. For example, the basic governing fluid flow regime in conventional reservoir simulations is limited to Darcy flow. For unconventional reservoirs, however, another flow regime term comes into the equation that reflects the flow of gas in the coal matrix and stands for diffusion. This new flow regime is not very well defined in the current simulators and needs further studies to appropriately address the production performance of a reservoir. Furthermore, the concept of multi-phase flow in reservoirs is well defined by the introduction of relative permeability equations in the flow equations. However, measuring relative permeability in coal is relatively complex (Ham and Kantas, 2008) due to its friability, heterogeneity, stress dependence and porous morphology. Young et al. (1992), for example, noticed that the derived relative permeability from a history match does not resemble the laboratory measurement. This phenomenon could be due to the change of matrix structure on phase flow in addition to wettability and capil­larity. In other words, both phases may flow through different passages over time due to heterogeneity. Thus dynamic rela­tive permeability (phase saturation) is required to accurately predict recovery from a CBM reservoir. As the focus on CBM has increased in the past decades, so has the quest for more effective technologies to help bring the reserves in such complicated conditions into production and to identify sweet spots. Reservoir modelling and reservoir simulation are technologies that are likely to play a key role in achieving this. Firstly, however, it is necessary to look at the challenges of CBM property variations.

The exploitation of an oil field is a complex and multidisciplinary task, which demands a lot of prior knowledge, time, and money. A good reservoir characterization is deemed essential in the accomplishment of Enhanced Oil Recovery (EOR) processes in order to estimate accurately the properties of the porous medium affecting the flow properties. Several techniques at a field scale are currently being used to determine these properties, which are time and money consuming. But these alone do not guarantee the success of the project. Reservoir simulation and numerical techniques were then included in the pre-development and follow-up studies as an effective tool to determine the productivity and future behavior of the oil field. As the computational power increased, more advanced and detailed models were developed, including different chemical and physical phenomena. But alongside this process, there was an active research in the area of reservoir simulation, improving the accuracy and efficiency of the numerical schemes used for the flow, transport, and energy equations. The aim of this review is to address the topics described. Firstly, the origin of an oil recovery process, the economic factors and field tests involved are introduced. Secondly, the oil and porous medium origin and characterization as well as an introduction to the fundamental concepts and equations are associated to reservoir simulation. Finally, a brief description and analysis of the techniques are used in reservoir simulation employing finite difference methods, their downsides and possible ways to overcome these problems.