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.
The era of discovery and subsequent exploitation of the denominated “easy oil” (part of the so called conventional reserves) is over [
These techniques, used to predict and optimize the exploitation, include laboratory and field tests (e.g. seismic 2D/3D/4C before starting operations and 4D to follow up the changes during exploitation, geostatistics) to give an idea of the conditions of the porous medium and its production performance [
phenomena occurring during operation so as to analyze and predict behavior as a function of time. The reservoir simulation can be used as well in inverse engineering problems for optimizing existing numerical models and couple the dynamic/historic data (production) in the simulation [
Generally speaking, reservoir simulation consists of three main parts: the physical characterization of a geological model describing the rock formation; a model characterizing the fluid flow and finally “well models” which describe the conditions under which fluids are injected or extracted from the reservoir [
The development of increasingly complex and detailed models requires the use of numerical techniques to solve these at reasonable times. Moreover, the representation of the properties of the porous medium and the characteristics of crude oil and natural gas at high pressures and temperatures may differ from laboratory tests, causing differences between the results and simulation. Another important topic is how to assess and properly estimate the properties of the rock formation. Geologists use statistical techniques in order to recreate the model properties of a porous medium, which are determined by several tests (e.g. seismic studies, drill core samples and even production data). However, other numerical tools are required (e.g. Monte-Carlo or Stochastic Processes) to take into account the effects of uncertainties in the model [
The aim of this review is to briefly describe the characterization of porous media and its main parameters. Then, models governing fluid flow in porous media are introduced. This consists in Darcy’s equation and the more complex compositional model for multiphase flow (where the component transfer between phases introduces nonlinearities in the mass transport equation), which is useful to describe chemical EOR operations. The second part comprises a brief explanation of the numerical methods used as well as the problems associated with these schemes (numerical dispersion and dissipation), and possible solutions using advanced numerical schemes.
A reservoir is an underground trap where different fluids (water, oil and gas) have accumulated due to a migration from the source rock where they were originated. The porous medium is generally considered of sedimentary origin and consists of a series of microchannels (about 1 - 100 microns diameter) interconnected where these fluids can flow (
The origin of crude oil prior to the migration and deposit of hydrocarbons in the porous media is also a long and complex process [
A porous rock formation is composed of a solid part, called solid matrix, and the remaining void space or microchannels whereto oil migrates [
0.35; unconsolidated sands with normal packing 0.40 and unconsolidated clays 0.60.
The permeability of the formation is a property that characterizes the ease with which fluids can flow when a pressure gradient is applied between two points. Nonetheless, reservoir rocks usually have no uniformity in their properties because of the mechanisms involved on its formation, thus the permeability will have a large dispersity in its values [
In order to develop a mathematical model of the porous medium, Corey [
The flow of reservoir fluids in porous media can be described at several different scales, from a microscopic to a macroscopic/formation scale. In order to perform large-scale reservoir simulations, a microscopic description of the flow channels would be too demanding for the computational power available and besides to characterize a reservoir rock so accurately to determine the geometry of the pore network is beyond the scope of modern techniques and equipment. A continuum scale description is then utilized, and its behavior is governed by forces acting between the different fluids and the rock formation. The goal of a reservoir continuum model is then to average both the fluids and reservoir rock [
The procedure for estimating REV dimensions and establish boundaries between microscopic and macroscopic scales is explained below using the porosity in
Then, the porosity of an element with characteristic dimension d is defined by Equation (2),
This relationship allows explaining the evaluation of the porosity as a function of the dimension d. For sizes smaller to a value dm the porosity varies significantly, with no particular pattern or trend; then, if the value of said dimension is between dm and dM, the value of the porosity plateaus and remains constant for the entire considered range. Finally, for values greater than dM, the porosity may remain constant in the case of a homogeneous medium, while in the case of an heterogeneous one, the function becomes again chaotic [
In underground processes in porous media, fluid flow involves mainly convection (advection and diffusion) of the different phases through a heterogeneous medium. The equations used to describe the flow at a microscopic level or poralscale are variations of Navier-Stokes (creeping flow) and the mass conservation law. At a macroscopic level, Darcy’s law [
Generally, three fluid phases may exist inside a reservoir (
Both microscopic and macroscopic effects control the movement of fluids in the reservoir. At the pore scale, interfacial tension (IFT) and capillary effects control the fluid behavior. Macroscopically, fluid flow is controlled by reservoir heterogeneity and mobility differences between the fluids. Viscosities, capillary pressure, IFT and mobility differences vary throughout the reservoir and depend mainly on phase saturations, their interactions and molecular compositions. Chemical components may transfer between contacting phases, altering the fluid properties of both. Interactions between the fluids or their components and the reservoir rock may also impact performance (e.g. adsorption of chemical components onto the surface of the rock altering the wettability). Thermal effects are generally very small due to the large heat capacity of the rock. However, in EOR thermal processes (steam flooding or in-situ combustion), the conservation of energy in the REV should be considered.
The governing equations for single phase flow in porous media are the conservation of mass, the Darcy equation and an equation of state (EOS). Considering the flow of a single fluid with density ρ through a REV of a porous medium the differential form of the continuity Equation (3) can be expressed as [
where f is the porosity of the rock formation, q represents the fluid source/sink term and
where
When
In the special case of incompressible rock and fluid (generally acceptable for liquid systems) the partial differential equation (PDE) simplifies to a Poisson elliptic equation.
The space in a reservoir is generally filled by both an oleous phase and brine. In addition, during secondary recovery processes, water is frequently injected in order to improve oil recovery. If the fluids are immiscible, they are referred to as phases. A two-phase system is commonly divided into a wetting and a non-wetting phase, given by the contact angle between the solid surface and the fluid-fluid interface on the microscale. For each pair of phases, one phase will wet the rock more than the other phase, and that phase will be referred to as the wetting phase (j = w). The other phase is then the non-wetting phase (j = nw). Normally, water is the wetting phase in a water-oil system, and oil is the wetting phase in an oil-gas system. In the absence of phase transitions, the saturations change when one phase displaces the other. During the displacement, the ability of one phase to move is affected by the interaction with the other phase at the pore scale. In the mathematical model, at a macroscopic scale, this effect is represented by the relative permeabilities krj (j = w, nw), which are a dimensionless scaling factor that depends on the saturation and modifies the absolute permeability to account for the rock’s reduced capability to make one phase to flow in the presence of the other. Then, the mass conservation Equation (8) for each phase yields [
And the multiphase extension of Darcy’s law is presented in Equation (9),
Together, they form the basic system of equations. Because of the interfacial tension (IFT), the pressure in the two phases will differ. This difference is called capillary pressure (
where
The volume fraction occupied by each phase is defined as the saturation of that phase. Thus, for a two-phase system, and considering no phase transitions, the sum of the saturation of both the wetting and non-wetting phases is equal to unity, as presented in Equation (12). Similar to the void space indicator function, the phase indicator piece-wise function is defined by Equation (13),
Then, the saturation of the phase j in an REV Ω0 element with characteristic dimension x0 will be defined by Equation (14),
The relative permeability of each phase depends on the phase saturations but does not depend directly on fluid flow properties [
In addition to relative permeability correlations, also analytical capillary pres- sure functions are needed. In two phase simulations it is standard to use the rela- tions provided by either Brooks-Corey or Van Genuchten [
permeability, these depend on empirical constants (e.g., if the system is oil-wet or water-wet), so several models have been developed through the years [
In two phase models it was assumed a no mass transfer condition between the phases. This assumption is valid for two phase flows of water/brine and oil, which is often the case in primary and secondary recovery mechanisms. In EOR processes, mass transfer and compositional effects are deemed essential to model accurately as they may become the driving mechanisms for the displacing process. A typical reservoir fluid may consist of several different chemical pseudo-components. Fully compositional models must be used when the fluid flow depends strongly on component transfer between phases. In fact, many EOR techniques, mainly chemical and miscible gas processes, are specifically designed to take advantage of the phase behavior of multicomponent fluid systems. Because these components may be transferred between phases (and change their composition), the basic conservation laws must be expressed for each component instead for each phase. For a chemical flooding compositional model, the governing differential equations consist of a mass conservation equation for each component, Equation (16), and Darcy’s law for each phase [
Here zi is the overall concentration of component i calculated by Equation (17), Ncomp is the number of components in the system,
In addition to the advective, directional movement of a component described by the Darcy phase velocity, components may also move due to dispersive forces. The simplest movement is molecular diffusion described by the random Brownian motion of molecules. Such motion is usually considered in reservoir simulation of negligible importance compared to other forces acting on the fluid. A more substantial phenomenon is mechanical dispersion. Narrow channel flows experience parabolic diffusion along the fronts (Taylor dispersion) and the irregular pore networks disperse the mass at a microscale (
where
The dispersion part of the tensor is significantly larger than the molecular diffusion; also, dlj is usually considerably larger than dtj and their relationship can be
expressed as a function of the Peclet number [
If a compositional model is formed by a system of Ncomp components and Np phases, there are a total of
In case the flow cannot be considered isothermal, or the recovery process involves the addition of considerable amounts of energy to the reservoir, an extra condition and variable must be introduced to the system. The conservation of energy, Equation (22), and its dependent variable, the temperature, are then added to the system. The major difference with respect to the other equations is that the energy is also conducted by the rock formation, and not only between the phases. If the local thermal equilibrium concept is applied, the temperature in the REV for all the phases and the porousmedium is considered to be the same and the energy equation is as follows [
where Uj is the specific internal energy, Cs the specific heat capacity of the rock, Hj the enthalpy of the phase,
A production/injection well is a vertical (or vertical/horizontal in case of horizontal wells), open hole through which fluid can flow in and out of the reservoir, according to the strategies or its degree of maturity. These are cemented and then perforated along specific intervals (multi-zone wells). The primary function of production wells is to extract hydrocarbons and later on, the water/chemical products injected as part of EOR processes. On the other hand, injection wells can be used for disposal of certain fluid (e.g. CO2 storage) as well as to inject chemical solutions so as to increase the recovery efficiency, sweeping the oil towards production wells. These wells are controlled through surface facilities (e.g. choke valves, Christmas trees) (
The main purpose of a well model is to represent the flow in the wellbore and provide equations that serve as input for the mass conservation and Darcy equations, to calculate the flow rate of each component being injected or produced. Generally, the bottomhole pressure is significantly different from the average pressure in the perforated grid blocks. Modeling injection and production of fluids using point sources causes numerical problems in the flow field, so the concept of Productivity/Well Index (PI) was introduced in the form
Peaceman [
where s is the skin factor, rw is the well radius and r0 is the effective block radius at which the steady-state pressure equals the computed block pressure. The Peaceman model has been also extended to horizontal wells and it was modified to take into account non-square grids, boundary blocks and non-Darcy effects [
Reservoir flow problems can be highly complex, consisting of many different physical effects when it comes to EOR processes. The analysis of all these phenomena can be achieved, up to some extent, by laboratory experiments or field tests at small scale, but these tend to be expensive to conduct may not be extrapolated to the whole reservoir. In order to solve this problem, mathematical models became progressively more important. Using these along with analytical solutions, engineers provided basic performance predictions so as to modify production strategies.
Several numerical formulations are employed to solve the non-linear systems of equations. The most stable approach is a fully implicit solution technique in pressure and saturation/concentration, but this generally leads to large, ill-conditioned matrices. Another scheme broadly utilized in compositional formulation consists, in order to reduce the level of implicitness, to solve the pressure equation system implicitly (which can be viewed as an overall volume balance) plus a sequence of (Ncomp − 1) components conservation equations [
The aim of this section is the derivation and explanation of the numerical schemes to be used for solving differential equations presented above, as well as also explain the reasons for the occurrence and possible numerical solutions of certain phenomena that affect simulation results [
Using a finite-difference approach the continuous domain is transformed into a discrete representation with a finite number of points in both a spatial (i) and temporal (
where
Even though the centered scheme has a higher order of precision, it generates unstable results when applied to the advection equation (wave transport equation). Hence, numerical methods using only points that are “upwind” of the wavefront are employed [
The inclusion of diffusion phenomena in the description of a fluid flow leads to non-trivial complications in the numerical solution of the mass conservation equations. From an analytical point of view, the resulting equations are no longer purely hyperbolic PDE’s but rather mixed hyperbolic-parabolic PDE’s. This means that the numerical method used to solve them must necessarily be able to cope with the parabolic part of the equations. For the diffusive term, the second order derivative is usually discretized using a centered scheme yielding Equation (29),
The final upwind discretized equation (FTUS-Forward in Time, Upwind in Space) and its matrix form for the advective-diffusive system are then presented in Equations (30) and (31), respectively.
In Equation (30) the order of accuracy is expressed as function of both independent variables
Method | Order | Finite-Difference Form |
---|---|---|
Purely Advective | ||
Upwind (FTUS) | ||
Centered (FTCS) | ||
Lax-Friedrichs | ||
Lax-Wendroff | ||
Beam-Warming | ||
Purely Diffusive | ||
Leapfrog | ||
Crank-Nicholson | ||
Dufort-Frankel |
advective or diffusive 1D equation.
A key factor in all numerical schemes is the issue of how treating the solution on the boundaries of the spatial grid as the time evolution proceeds. Two types of conditions are generally used in reservoir simulation to describe whether the Darcy velocity or the pressure of a phase at the boundaries. These are: Dirichlet type conditions, when the values of the relevant quantity are imposed at the boundaries of the grid (these values can be either functions of time or be held constant), and Von Neumann type conditions, when the values of the derivatives of the relevant quantity are imposed.
The exact solution of the discretized equation satisfies a PDE different from the one being solved. This difference is represented by the local truncation error (LTE) of
Type | Difference Stencil | Order |
---|---|---|
Backward | ||
Backward | ||
Centered | ||
Backward | ||
Backward | ||
Centered | ||
Centered |
the numerical scheme. The LTE can be expressed as a function of higher order derivatives [
The procedure to calculate this error and assess its contribution to the numeric solution is straight-forward. It consists in performing an expansion in a double Taylor series around a single point
Introducing these terms from Equations (33), (34) and (35) in the numerical scheme presented in Equation (32) it yields,
Rearranging Equation (36) to split the original PDE and the truncation error,
Now the temporal derivatives in Equation (37) are transformed in space derivatives. Furthermore, using Courant and Peclet dimensionless groups presented in Equation (38) the LTE for the method is derived in Equation (39),
The numerical scheme does not solve the original PDE, but a modified PDE with extra terms of higher order derivatives. The extra term containing the second order derivative is interpreted as a numerical diffusion, additional to the physical coefficient D. As long as the Cr < 1 condition is met, the numerical solution will produce an artificial smearing given by the term
One way to reduce these numerical errors is by means of additional, artificial factors which stabilize or decrease the previously seen effects. As an example, the streamline diffusion method consists in adding a term of artificial diffusion to counteract the added terms by the numerical scheme; the non-oscillatory shock- capturing methods, TVD (Total Variation Diminishing) or flux-limiters [
Following a procedure similar to the previous scheme it renders Equation (41),
As shown, the first diffusive term in the LTE has disappeared leaving the dispersive term as the main source of error in this method. Since this term is negative, the spurious oscillations occur behind steep fronts. It is worth mentioning that a more accurate numerical scheme is not necessarily a preferable one. As an example, the upwind and the Lax-Friedrichs methods are both dissipative, thelatterisgenericallymoredissipativedespitebeingofhigherorderaccuracyinspace.
In the previous section two of the most common numerical schemes utilized were introduced, and the advantages or disadvantages of each were studied and inferred. While the upwind scheme can handle steep gradients, it is very diffusive and moreover a first order scheme; on the other hand, Lax-Wendroff is a second order scheme, less diffusive but presents serious problems when sharp gradients are present in the system. Therefore, new numerical schemes were published coupling low- and high-resolution methods, taking advantage of the mentioned characteristics [
The idea behind this concept is then to write the fluxes as a function of low- and high-resolution numerical schemes as in Equation (43), using a proportionality factor.
The proportionality factor, also called flux limiter function introduced in Equation (44), depends on the ratio of consecutive gradients in the numerical mesh, this is,
Using the FTUS and Lax-Wendroff in Equation (45) as low- and high-resolution schemes respectively, the flux is calculated according to Equation (46),
Finally, the discretized advection Equation (47) is written in terms of the flux limiter parameter,
Equation (48) resembles the FTUS scheme with a modified Courant number [
This is valid for positives values of r, when the following two conditions in In Equation (50) are met (
Further, more restrictive constraints are applied in order to make the scheme second order in accuracy (
This review concludes with the study of three concepts related with numerical simulation. The first to be defined is the consistency of a numerical scheme: Given a PDE in its operator form
While the concept of consistency associates the original PDE with the discretized equation, it is necessary but is not sufficient to ensure that the numerical results converge to the exact solution. It should also be ensured that the numerical results of the discretized equation converge to the exact results of the discretized equation. This concept may seem trivial, but numerical errors introduced during simulation
Type | Flux Limiter Function | Reference |
---|---|---|
Superbee | Roe [ | |
Minmod | Roe [ | |
Van Leer | Van Leer [ | |
Van Albada | Van Albada [ | |
Koren | Koren [ | |
CHARM | Zhou [ | |
MUSCL | Van Leer [ |
can grow boundless, amplifying the errors until the system eventually collapses.
This is ensured introducing the concept of numerical stability. To understand this, a system of equations written in the form of modified PDE Equation (31) is analyzed. Perturbations at the baseline as well as at a generic time
where
Equation (52) exposes a central issue in the stability analysis: the question of whether amplification matrices have their powers uniformly bounded. In order for the numerical errors to not be amplified during the simulation, a stability restriction is then defined
where
These three concepts can be summarized in the following theorem: Given a properly posed initial-value problem and a finite difference approximation to it, that satisfies the consistency criterion, stability is the necessary and sufficient condition for convergence. This theorem, known as the “Lax-Richtmyer equivalence theorem” or Fundamental Theorem of Numerical Analysis (
related. In general, proving that the numerical scheme adopted is stable will validate that the discretized equations represent the PDE as well as the numerical errors in the simulation are bounded at all times [
Reservoir simulation is a branch of engineering that emerged in recent years since oil companies needed to justify and evaluate E&P investments. The former is not related to only one discipline, but includes the assistance and collaboration of various specialists so as to characterize a reservoir, estimate its profitability and give the “green light” to the project development phase. Oil reservoirs are geological traps where oil migrated and remained for long periods of time. An accurate determination of their physical characteristics has not yet been developed and statistical tools are used along with complex field tests to extrapolate these properties. In addition, oil is not a homogeneous, pure fluid but is composed by a large group of different components which may alter its properties. Hence, a set of variables must be previously evaluated and studied in order to get feasible results. The research and development of new exploration technologies to reduce the model uncertainties are deemed essential. These, along with production studies at reservoir scale on pilot wells will allow performing accurate history matching analysis. Due to the current oil reserve conditions and future production estimates, these new technologies should also consider their applicability also in non-conventional oil reservoirs (e.g. oil sands, tight oil, shale oil), or in geographical areas with harsh conditions (e.g. off-shore platforms).
Fluid flow simulations are performed using two different approaches: the Navier- Stokes equations throughout a complex network of microchannels in the porous medium; or the assumption of a continuum with averaged properties using Darcy’s equation, rendering a system independent of the geometry at a microscopic scale. The first is only circumscribed to specific laboratory tests and has limited application in field studies. This is due to several factors, among them the uncertainties associated with the poral geometry in the field as well as high computational costs required to solve the system of equations. However, this approach might be useful in the design of new chemicals while being evaluated at a microscopic level. These studies should then be supplemented with scale reservoir simulations and field tests using Darcy’s equation.
Subsequently, the mathematical tools for reservoir simulation using Finite Difference Methods (FDM) were presented. The errors introduced by these schemes as well as possible solutions to tackle these problems have been addressed. The numerical convergence of a system of PDE’s is a critical aspect that must be taken into account in order to limit numerical errors. In addition, the continuous increase in the complexity of numerical models has demanded a proportional increase of computational power to obtain results in a reasonable time frame. The development of new schemes of higher orders of accuracy as well as models dealing with the non-linearities present in the simulation could reduce either the computational requirements or the numerical errors produced. Besides the FDM’s discussed in this review, other numerical schemes of higher complexity are used and offer certain advantages, such as the capability of dealing with complex geometries or geological faults. These techniques are, among others: Finite Element Methods (FEM), Finite Volume Methods (FVM), Immersed Boundary (IB), hybrid methods, etc. However, these advantages are related to the degree of certainty in the definition of the physical boundaries and properties of the reservoir. Then, the development of the aforementioned technologies and the application of these methods are strongly connected and will allow increasing the computational efficiency and reliability of reservoir simulations.
P. D. gratefully acknowledges the support of the Erasmus Mundus EURICA scholarship program (Program Number 2013-2587/001-001-EMA2) and the Roberto Rocca Education Program.
Druetta, P., Tesi, P., De Persis, C. and Picchioni, F. (2016) Me- thods in Oil Recovery Processes and Re- servoir Simulation. Advances in Chemical Engineering and Science, 6, 399-435. http://dx.doi.org/10.4236/aces.2016.64039