# Upscaling of Single- and Two-Phase Flow in Reservoir Engineering

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## Book Description

This book describes fundamental upscaling aspects of single phase / two-phase porous media flow for application in petroleum and environmental engineering. Many standard texts have been written about this subject. What distinguishes this work from other available books is that it covers fundamental issues that are frequently ignored but are relevant for developing new directions to extend the traditional approach, but with an eye on application.

Our dependence on fossil energy is 80-90 % and is only slowly decreasing. Of the estimated 37 (~ 40) Gton/year anthropogenic emissions about 13 Gton/year of carbon dioxide remains in the atmosphere. An Exergy Return on Exergy Invested analysis shows how to obtain an unbiased quantification of the exergy budget and the carbon footprint. The intended audience of the book learns thus to quantify his method of optimization of recovery efficiencies supported by spreadsheet calculations.

As to single phase-one component fluid transport, it is shown how to deal with inertia, anisotropy, heterogeneity and slip. Upscaling requires numerical methods. The main application of transient flow is to find the reasons for reservoir impairment. The analysis benefits from solving the porous media flow equations using (numerical) Laplace transforms. Multi-phase flow requires the definition of capillary pressure and relative permeabilities. When capillary forces dominate we have dispersed (Buckley-Leverett flow). When gravity forces dominate we obtain segregated flow (interface models). Miscible Flow is described by a convection-dispersion equation. We give a simple proof that the dispersion coefficient can be approximated by Gelhar's relation, i.e., the product of the interstitial velocity, the variance of the logarithm of the permeability field and a correlation length.

The book will appeal most to students and researchers of porous media flow in connection with environmental engineering and petroleum engineering.

## Table of Contents

1 Dutch and World-wide Energy recovery; Exergy Return on Exergy Invested 1.1 Fraction fossil in current energy mix 1.2 Possible new developments 1.3 Exergy 1.4 Exergy Return on Exergy Invested (ERoEI) analysis 1.4.1 Exercise EROEI 1.4.2 Anthropogenic emissions versus natural sequestration 1.4.3 Exercise: trees to compensate for intercontinental flights 2 One Phase Flow 2.1 Mass Conservation 2.2 Darcy’s law of flow in porous media 2.2.1 Definitions used in Hydrology and Petroleum Engineering 2.2.2 Exercise, EXCEL naming 2.2.3 Empirical relations for permeabilit (Carman-Kozeny equation) 2.3 Examples that have an analytical solution 2.3.1 One dimensional flow in a tube 2.3.2 Exercise, Two layer sand pack 2.3.3 Exercise, Numerical model 2.3.4 Exercise, EXCEL numerical 1-D simulation 2.3.5 Radial inflow equation 2.3.6 Boundary conditions for radial diffusivity equation 2.3.7 Exercise, radial diffusivity equation 2.4 Modifications of Darcy’s law 2.4.1 Representative elementary volume 2.4.2 Exercise, Slip factor 2.4.4 Why is the flow resistance proportional to the shear viscosity? 2.4.6 Exercise, Inertia factor 2.4.7 Adaptation of Carman-Kozeny for higher flow rates 2.4.8 Exercise, Carman Kozeny 2.4.9 Anisotropic Permeabilities 2.4.10 Exercise, Matrix multiplication 2.4.11 Substitution of Darcy’s law in the mass balance equation 2.5 Statistical methods to generate heterogeneous porous media 2.5.1 The importance of heterogeneity 2.5.2 Generation of random numbers distributed according to a given distribution function 2.5.3 Log-normal distributions and the Dykstra-Parson’s coefficient 2.5.4 Exercise, Lognormal distribution functions 2.5.5 Generation of a Random Field 2.5.6 Exercise, Log-normal permeability field 2.5.7 Exercise, Average permeability field 2.6 Upscaling of Darcy’s law in heterogeneous media 2.6.1 Arithmetic, geometric and harmonic averages 2.6.2 The averaged problem in two space dimensions 2.6.3 Effective medium approximation 2.6.4 Pitfall: a correctly averaged permeability can still lead to erroneous production forecasts 2.7 Numerical upscaling 2.7.1 Finite volume method in 2 □□ D; the pressure formulation 2.7.2 The Finite Area Method; the stream function formulation 2.7.3 Finite element method (after F. Vermolen) 2.7.4 Flow calculation 2.A Finite volume method in EXCEL 2.A.1 The data sheet 2.A.2 The sheet for calculation of the X-dip averaged permeability 2.A.3 The harmonically averaged grid size corrected mobility in the x-direction 2.A.4 The geometrically averaged grid size corrected mobility in the y-direction between the central and the cell S. Sheet I contains the permeabilities 2.A.5 The sheet for the well flow potential

2.A.6 The sheet for Productivity/Injectivity indexes 2.A.7 The sheet for the wells 2.A.8 The sheet for flow calculations 2.B Finite element calculations 2.C Sketch of proof of the effective medium approximation formula 2.D Homogenization 3 Time dependent problems in porous media flow 3.1 Transient Pressure Equation 3.1.1 Boundary conditions 3.1.2 The averaged problem in two space dimensions 3.1.3 The problem in radial symmetry 3.1.4 Boundary conditions for radial diffusivity equation 3.1.5 Dimensional analysis for the radial pressure equation; adapted from lecture notes of Larry Lake 3.1.6 Solution of the radial diffusivity equation with the help of Laplace transformation 3.1.7 Laplace transformation 3.1.8 Self similar solution 3.1.9 The dimensional draw-down pressure

3.2 Pressure build up 3.2.2 Time derivatives of pressure response 3.2.3 Practical limitations of pressure build up testing 3.3 Formulation in a bounded reservoir 3.4 Non-Darcy flow 3.A About Boundary condition at r = reD 3.A.1 Exercise, Stehfest algorithm 3.B Rock compressibility 3.B.1 Physical model 3.B.2 Mass balance in constant control volume 3.C Equations disregarding the grain velocity in Darcy’s law 3.D Superposition principle 3.E Laplace inversion with the Stehfest algorithm [223] 3.F EXCEL numerical Laplace inversion programme 3.F.1 Alternative inversion techniques 4 Multi-Phase Flow 4.1 Capillary Pressure function 4.1.1 Interfacial tension and capillary rise 4.1.2 Exercise, Laplace formula 4.1.3 Exercise, Young’s law 4.1.4 Application to conical tube; Relation between capillary pressure and saturation 4.1.5 Relation between the pore radius and the square root of the permeability divided by the porosity. 4.1.6 Non-dimensionalizing the capillary pressure 4.1.7 Exercise, Ratio grain diameter / pore throat diameter 4.1.8 Three phase capillary pressures 4.1.9 Experimental set up and measurements of capillary pressure 4.1.10 Cross-dip capillary equilibrium 4.1.11 Exercise, Capillary desaturation curve 4.2 Relative permeabilities 4.2.1 Exercise, Brooks-Corey rel-perms 4.2.2 LET relative permeability model 4.2.3 Estimate of the LET parameters 4.2.4 Exercise, Residual oil and Rel-perm 4.3 Theory of Buckley-Leverett 4.3.1 Exercise, Vertical upscaling relative permeability 4.4 Material balance 4.4.1 Solutions of the theory of Buckley-Leverett 4.4.2 Equation of motion (Darcy’s Law) and the fractional flow function 4.4.3 Analytical solution of the equations

4.4.4 Construction of the analytical solution; requirement of the entropy condition 4.4.5 Exercise, Buckley Leverett profile with EXCEL 4.4.6 Derivation of the shock condition 4.4.7 Analytical calculation of the production behavior 4.4.8 Exercise, Buckley Leverett production file 4.4.9 Exercise, Analytical Buckley Leverett production curve 4.4.10 Determination of relative permeabilities from production data and pressure measurements 4.4.11 Determination of the relative permeabilities by additional measurement of the pressure drop 4.5 Finite volume approach to obtain the finite difference equations for the Buckley Leverett problem 4.5.1 Exercise, Numerical solution of Buckley Leverett problem 4.6 Vertical equilibrium as a basis for upscaling of relative permeabilities and fractional flow functions 4.6.1 Dake’s Upscaling procedure for relative permeabilities 4.6.2 Exercise, Sorting factor dependence 4.6.3 Hopmans’s formulation 4.7 Physical Theory of Interface Models 4.7.1 Derivation of interface equation of motion and productions for segregated flow 4.7.2 Stationary interface (Mobility number < Gravity number +1) 4.7.3 Exercise, Interface angle calculations 4.7.4 Production behavior for stationary solution, i.e., M < G + 1 4.8 Non-stationary interface 4.8.1 The volume balance in the form of an interface equation 4.8.2 Dietz-Dupuit-approximation 4.8.3 Approximate Equilibrium Equation 4.8.4 Derivation of flow rate Qwx from Darcy’s law 4.8.5 Quasi Stationary Solution of the Dietz-Dupuit Equation for M < G + 1 4.8.6 Exercise, Shock solution versus interface angle solution 4.8.7 Analytical Solutions 4.8.8 Analytical expressions for the interface as a function of position in the reservoir 4.8.9 Analytical expressions for the production behavior 4.8.10 Summary of analytical procedure for interface models 4.8.11 Exercise, Advantage of M □ G + 1 4.A Numerical approach for interface models 4.A.1 Exercise. Behavior for M > G + 1 4.B Numerical approaches for Buckley Leverett and interface models implemented with EXCEL 4.B.1 Simple sheet for Buckley-Leverett model 4.C Numerical Diffusion for first order upstream weighting scheme 5 Dispersion in porous media 137 5.1 Introduction 5.2 Molecular diffusion only 5.3 Solutions of the convection diffusion equation 5.3.1 Injection in a linear core 5.3.2 Taylor’s problem in a cylindrical tube 5.4 Derivation of the dispersion equation 5.5 Statistics and dispersion 5.5.1 Random walk models 5.6 Variance of concentration profile and dispersion 5.7 Dispersivity and the velocity autocorrelation function 5.8 Exercise, Numerical/Analytical 1D dispersion 5.9 Exercise, Gelhar relation 5.10 Numerical aspects 5.A Higher order flux functions for higher order schemes 5.B Numerical model with the finite volume method 6 Notions List of Symbols

## Author(s)

### Biography

Hans Bruining is a professor emeritus in geoenvironmental engineering of the Technical University of Technology of Delft, which is ranked 10 as one of the top technical institutes worldwide in Engineering. He holds a PhD degree from the University of Amsterdam. He is the founder of the Dietz-De Josselin de Jong laboratory. His special interests are the environmental aspects of fossil fuel recovery, enhanced oil recovery and theory and experiments of complex flow processes in porous media. He is review chairman of the society of Petroleum Engineering Journal (SPEJ). He is the recipient of the SPE Distinguished Achievement Award for Petroleum Engineering Faculty (2012). The international award recognizes superiority in classroom teaching, excellence in research, significant contributions to the petroleum engineering profession and/or special effectiveness in advising and guiding students.