Reservoir Simulation - Heriot-Watt University Course

Introduction and Case Studies; Basic Concepts in Reservoir Engineering

• be able to describe what is meant by a simulation model, saying what analytical models and numerical models are.
• be familiar with what specifically a reservoir simulation model is.
• be able to describe the simplifications and issues that arise in going from the description of a real reservoir to a reservoir simulation model.
• be able to describe why and in what circumstances simple or complex reservoir models are required to model reservoir processes.
• be able to list what input data is required and where this may be found.
• be able to describe several examples of typical outputs of reservoir simulations and say how these are of use in reservoir development.
• know the meaning of all the highlighted terms - or terms referred to in the Glossary - in Chapter 1 e.g. history matching, black oil model, transmissibility, pseudo relative permeability etc.
• be able to describe and discuss the main changes in reservoir simulation over the last 40 years from the 60's to the present - and say why these have occurred.
• know in detail and be able to compare the differences between what reservoir simulations can do at the appraisal and in the mature stages of reservoir development.
• have an elementary knowledge of how uncertainty is handled in reservoir simulation.
• know all the types of reservoir simulation models and what type of problem or reservoir process each is used to model.
• know or be able to work out the equations for the mass of a phase or component in a grid block for a black oil or compositional model.

BASIC CONCEPTS IN RESERVOIR ENGINEERING

• be familiar with the meaning and use of all the usual terms which appear in reservoir engineering such as, Sw, So, Bo, Bw, Bg, Rso, Rsw, cw, co, cf, kro, krw, Pc etc.
• be able to explain the differences between material balance and reservoir simulation.
• be aware of and be able to describe where it is more appropriate to use material balance and where it is more appropriate to use reservoir simulation.
• be able to use a simple given material balance equation for an undersaturated oil reservoir (with no influx or production of water) in order to find the STOOIP.
• know the conditions under which the material balance equations are valid.
• be able to write down the single and two-phase Darcy Law in one dimension (1D) and be able to explain all the terms which occur (no units conversion factors need to be remembered).
• be aware of the gradient (—) and divergence (—.) operators as they apply to the generalised (2D and 3D) Darcy Law (but these should not be committed to memory).
• know that pressure is a scalar and that the pressure distribution, P(x, y, z) is a scalar field; but that —P is a vector.
• know that permeability is really a tensor quantity with some notion of what this means physically (more in Chapter 7).
• be able to write out the 2D and 3D Darcy Law with permeability as a full tensor and know how this gives the more familiar Darcy Law in x, y and z directions when the tensor is diagonal (but where we may have kx π ky π kz).
• be able to write down and explain the radial Darcy Law and know that the pressure profile near the well, DP(r), varies logarithmically.

RESERVOIR SIMULATION MODEL SET-UP

Simulation Input

• Identify what questions the simulation is expected to address.
• Identify what data is required as input to perform the desired calculations.
• Format data correctly, taking account of keyword syntax and required units.

Simulation Output

• Select required output of calculations.
• Quality check output data to check for errors in input.
• Identify purpose of each output file and use post-processors to analyse data.

Analysis of Results

• Identify impact of reservoir engineering principles in calculation performed.
• Identify numerical effects and impact of grid block size and orientation on results.
• Perform simple upscaling calculation to address numerical diffusion.

CHAPTER 4

GRIDDING AND WELL MODELLING

• understand and be able to describe the basic idea of gridding and of spatial and temporal discretisation.
• be aware of all the main types of grid in 1D, 2D and 3D used in reservoir simulation and be able to describe examples of where it is most appropriate to use the different grid types.
• be able to give a short description with simple diagrams of the phenomena of numerical dispersion and grid orientation and to explain how these numerical problems can be overcome.
• be familiar with more sophisticated issues in gridding such as the use of local grid refinement (LGR), distorted, PEBI and corner point grids
• given a specific task for reservoir simulation, the student should be able to select the most appropriate grid dimension (1D, 2D, 3D) and geometry/structure (Cartesian, r/z, corner point etc.).
• be able to discuss the issues of grid fineness/coarseness (i.e. how many grid blocks do we need to use) in terms of some examples of what can happen if an inappropriate number of grid blocks are used in a reservoir simulation calculation.
• be able to describe the basic ideas behind streamline simulation and to compare it with
  conventional reservoir simulation in terms of its advantages and disadvantages.
• be familiar with the different types of average used for single phase , two phase relative permeabilities (krp) and for mp and Bp (p = o, w, g phase) when calculating the block to block flows (Qp) in a reservoir simulator.
• be able to describe the physical justification for using the upstream value of two phase relative
  permeabilities when calculating the block to block flows (Qp) in a reservoir simulator.
• understand the origin of all the pressure drops that are experienced by the reservoir fluids from deep in the reservoir, through to the wellbore and then to the surface facilities and beyond
• know what a well model is and what productivity index (PI) is, including knowing the radial Darcy Law and how this gives a mathematical expression for PI for single phase flow (know the expression from memory).
• be able to describe the main issues in relating the pressure in the reservoir, Pe, at some drainage radius, re, to an average grid block pressure and how this leads to the Peaceman formula (Dr = 0.2 Dx) which is then used to calculate PI.
• be able to extend PI to the concept of multi-phase flow to calculate PIw and PIo.
• describe a well model for a multi layer system where there is two phase flow into the wellbore.
• understand and be able to describe the various types of well constraint that can be applied e.g. injection volume constrained wells, well flowing pressure constraints and voidage replacement constraints.

THE FLOW EQUATIONS

• Know that the central principles in deriving the flow equations in porous media are (a) to apply
  material balance to the flow and accumulation terms; and (b) to then apply Darcy’s Law.
• Know how to apply material balance to a control volume with flow and accumulation terms, recognising the difference between mass flow rates and volumetric flow rates.
• Understand be able to describe the basic physics of single phase compressible flow through porous media.
• Be able to derive the (pressure) equation for single phase compressible flow.
• Know physically, in the context of the single phase compressible flow equation, what a non- linear partial differential equation (PDE) is and why it is difficult to solve.
• Be able to work through all of the simplifying assumptions on the single phase compressible flow equation to arrive at the much simpler pressure equation (linear PDE) for slightly compressible flow involving the hydraulic diffusivity, Dh =k/(mfcf)
• Understand the extension of the single phase pressure equation to 2D.
• Appreciate (but not memorise) the use of the gradient (—) and divergence (—.) operators from vector calculus in writing the multi-phase flow equations.
• Be able to apply conservation + Darcy’s law in the two phase case to arrive at the two phase flow equations for compressible fluids and rock.
• Follow the argument on how the full two phase pressure equation is derived.
• Be able to identify the relation of each of the terms in the two phase pressure equation to viscous, gravity and capillary forces.
• Be able to reproduce (in words and diagrams) the outline solution scheme for the two phase pressure equations for both the full equations and for the simplified pressure and saturation equations.

NUMERICAL METHODS IN RESREVOIR SIMULATION

• write down from memory simple finite difference expressions for derivatives, (∂P/∂x), (∂P/∂t) and (∂2P/∂x2) explaining your spatial (space) and temporal (time) notation (); for (∂P/∂x), the student should know the meaning of the forward difference, the backward difference and the central difference and the order of the error associated with each, O(Dx) or O(Dx2).
• apply finite difference approximations to a simple partial differential equation (PDE) such as the diffusion equation and explain what is meant by an explicit and an implicit numerical scheme.
• write a simple spreadsheet to solve the explicit numerical scheme for a simple PDE for given
  boundary and initial conditions and be able to describe the effect of time step size, Dt.
• show how the implicit finite difference scheme applied to a simple linear PDE leads to a set of linear equations which are tridiagonal in 1D and pentadiagonal in 2D.
• derive the structure of the pentadiagonal A-matrix in 2D for a given numbering scheme going from (i, j) notation to m-notation where m is an ordered numbering scheme e.g. for the natural numbering scheme, m = (j - 1).NX + i
• describe a solution strategy for the non-linear single phase 2D pressure equation where the fluid and rock compressibility (and density, r, and viscosity, m) are functions of the dependent variable, pressure, P(x,y,t).
• write down the discretised form of both the pressure and saturation equation for two-phase flow given the governing equations (in simplified form in 1D), and be able to explain why these lead to sets of non-linear algebraic equations.
• outline with an explanation and a simple flow chart the main idea behind the IMPES solution strategy for the discretised two-phase flow equations.
• write down the expanded expressions for a set of linear equations which, in compact form are written A.x = b, where the matrix A is an nxn matrix of (known) coefficients (aij; i = rows, j = columns), b is a vector of n (known) values and x is the vector of n unknowns which we are solving for.
• explain clearly the main differences between a direct and an iterative solution method for the set of linear equations, A.x = b.
• write down the algorithm for a very simple iterative scheme for solving A.x = b, and be able to describe the significance of the initial guess, x(0) , what is meant by iteration (and iteration counter, n), the idea of convergence of x(n) as n Æ •; and be able to comment on the number of iterations required for convergence, Niter.
• explain how to apply (without derivation) the Newton-Raphson method for solving a single non-linear algebraic equation, ; Newton-Raphson scheme =>
• extend the application of the Newton-Raphson to sets of non-linear algebraic equations such as those arising from the discretisation of the two-phase pressure and saturation equations; and S and P are the (unknown) vectors of saturation and pressure; (given the Newton-Raphson expression, ).
• understand, but not be able to reproduce the detailed derivation of, the more mathematical explanation of numerical dispersion.

PERMEABILITY UPS CALING LEARNING OBJECTIVES:

• Appreciate why upscaling is necessary.
• Know how to calculate effective permeability in simple models by averaging.
• Understand how to perform numerical upscaling of single-phase flow.
• Be aware of the effects of heterogeneity on two-phase flow.
• Realise the limitations of applying single-phase upscaling to a two-phase problem.
• Know how to carry out steady-state, capillary-equilibrium upscaling for two-phase flow.
  • Become familiar with two-phase dynamic upscaling (the Kyte and Berry Method), and understand the advantages and disadvantages of applying dynamic upscaling.
  • Understand how to upscale around a well.
  • Appreciate that permeability is a full tensor property.
  • Know how to upscale from the core-scale to the scale of a geological model, taking account of fine-scale structure and capillary effects.