Papers by Abraham de Swaan

Influence of Shape and Skin of Matrix-Rock Blocks on Pressure Transients in Fractured Reservoirs
Spe Formation Evaluation, Dec 1, 1990
Summary A formulation for pressure transients in terms of the intrinsic, or core, properties of t... more Summary A formulation for pressure transients in terms of the intrinsic, or core, properties of the two media that compose the fractured reservoir establishes the influence of these properties on—and their corroboration from—the pressure/time relationship observed in well and interference tests. The influence of the following reservoir characteristics are analyzed: the area of fractures transverse to flow; the dimensions, shape, and properties of rectangular parallelepiped matrix-rock blocks; and permeability reduction in the block surfaces. A restatement of the so-called quasisteady-state intermedia flow provides Warren and Root's α and λ parameters with the physical meaning they lacked and allows direct determination of the blocks' minimum dimensions.
Teoria del desplazamiento de aceite por agua en yacimientos naturalmente fracturados
Revista del Instituto Mexicano del Petróleo, 1983
Pressure Transients in a Fractal-cluster Model of Porous Media - (SPE-164892)
Proceedings, 2013
ABSTRACT

Development of the Critical Gas Saturation
Journal of Petroleum Technology, May 1, 1981
Introduction The critical or threshold gas saturation is a specific property of the porous medium... more Introduction The critical or threshold gas saturation is a specific property of the porous medium and internal fluids. property of the porous medium and internal fluids. Compared with the experimental research on the subject, I the theory of gas saturation development to its critical value and the significance of this growth on reservoir performance have received little attention. For a volumetric reservoir - i.e., one with negligible water influx and negligible change of rock porosity with internal fluid pressure change, and with porosity with internal fluid pressure change, and with water production and expansion also considered zero - the usual procedure to compute gas saturation development is the material balance equation: Bo - Boi + Bg (Rsi- Rs) Np=------------------------ N,..............(1) Bo + Bg (Rp - Rs) with average reservoir pressure below the bubble-point pressure and Sg less than Sgc. In Eq. 1, 1 Np Rp = --- RsdNp.........................(2) Np 0 In words, only the solution gas has been produced, since the gas saturation remains insufficient for gas flow. The recovery percentage is obtained from the material balance equation; the gas saturation is obtained from the recovery percentage. Eq. 1 is an involved integral equation of a type unknown to me. Np is a self-dependent variable with the limit of the integral in the denominator - Eq. 2. In Ref. 2, Eq. 1 is not presented in the form shown but as a numerical solution procedure that simplifies the presentation. In fact, this numerical solution is an everyday affair for the practicing engineer. An alternative formula is presented. presented. JPT P. 907
Ingeniería petrolera, 2014
Este artículo presenta un método para analizar pruebas de variación de presión con rasgos diferen... more Este artículo presenta un método para analizar pruebas de variación de presión con rasgos diferentes de los más comunes que denotan geometrías de flujo lineal, radial o esférico, y que son indicadores de flujo de tipo fractal en el yacimiento. El método utiliza una solución a la respuesta de presión con flujo de tipo fractal en el dominio de Laplace e incluye los efectos de daño y almacenamiento por medio de un procedimiento usado para la descripción del flujo en geometrías compuestas. Este enfoque es más sencillo que los presentados en artículos anteriores a éste. La determinación de las propiedades fractales en aplicaciones a casos de campo da por resultado valores plausibles; para la comparación con valores reales es necesario conocer la distribución de las fracturas en el yacimiento.

Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles, Nov 18, 2014
The reservoir is described as a "supercritical cluster"; that is, an aggregate of conductive elem... more The reservoir is described as a "supercritical cluster"; that is, an aggregate of conductive elements that comprises a "backbone" of connected pores or fractures that span the zone of interest, and also a collection of "sub-critical clusters" or "dangling ends" joined to the backbone to a limited extent. The scheme resembles the usual fracture and matrix-blocks setting but both backbone and sub-clusters are of the same material and share similar petrophysical properties. Whereas the backbone is a homogeneous porous medium, the sub-critical clusters behave as fractal porous media. The backbone-cluster type of flow has been observed in laboratory experiments. The subcritical clusters were approximated as linear fractal media characterized by static and dynamic fractal exponents and also by porosity and permeability of the compound medium. One of the ends of the linear clusters is closed and the other is joined to the backbone, where the mainstream occurs. A new solution was developed for that problem. The Laplace transform in time and space was used in the mathematical scheme. The theory developed was applied to field cases of interference between wells in aquifers. The matches of computed and observed dynamic pressures show fair fits.
A Comprehensive Theory of Transient Inertial Flow in Tubing, Skin Zone and Reservoir
All Days, Mar 8, 1992
Complete analysis of well tests, from early to late time stages, is feasible with a model that in... more Complete analysis of well tests, from early to late time stages, is feasible with a model that includes in a single equation the effects of: a tubing composed by pipes of different diameters; chokes; segregated gas in the tubing; skin zone; reservoir geometry; and inertia of the flowing fluid.

Analysis of Well Tests in Multiple Fractured Reservoirs - Field Case Applications
All Days, Feb 1, 2000
A simplified analytical model of pressure transients in fractured reservoirs with matrix-rock blo... more A simplified analytical model of pressure transients in fractured reservoirs with matrix-rock blocks of different sizes and/or properties was formulated and applied to field cases. In the applications, relations between traits in the pressure vs. time plots and rock properties are used to determine rock block sizes and block-size distributions from actual well tests. Introduction The mathematical description of flow in fractured reservoirs with rock-blocks of different properties has been an obvious development after the models of media with a single type of block. References 1 to 6 present such models. However, these papers present involved formulations of the block-to-fracture flow. As a result their application in the analysis of actual well tests is not straightforward. Flow in fractured media is accurately represented as flow in the connected fractures medium plus a block-to-fracture component that depends on the blocks, or matrix, properties. In the present paper a simple formulation of such block-fracture interaction as presented in previous papers7,8 describing flow in single block fractured media is extended to the case with multiple rock blocks. Theory For multiple-block fractured media the interaction of blocks and fractures is governed by the relative frequency of occurrence (or just frequency) of a block type in the pay and by the block properties included in the block function of flow. The flow equation in radial (cylindrical) geometry with a multiple block to fracture interaction term is1,Equation 1 The Laplace transform of the former equation is,Equation 2 Block's function of flow. The flow from or into blocks is described approximately by an exponential function, see Ref. 8, for a unit pressure variation at the blok's surface.Equation 3 or, in the Laplace domain,Equation 4 a is the ratio of block surface to block volume divided by a fraction of the block minimal dimension8,Equation 5 the value of a for other parallelepiped shapes, such as prisms or finite slabs, have values intermediate between the two values shown in Eq. 5. The block function of flow reproduces closely results of exact functions of flow from parallelepipeds; but it is simple for computations and also for linking block properties to traits in the plots of transient pressure response vs. time. Block's function of flow. The flow from or into blocks is described approximately by an exponential function, see Ref. 8, for a unit pressure variation at the blok's surface.Equation 3 or, in the Laplace domain,Equation 4 a is the ratio of block surface to block volume divided by a fraction of the block minimal dimension8,Equation 5 the value of a for other parallelepiped shapes, such as prisms or finite slabs, have values intermediate between the two values shown in Eq. 5. The block function of flow reproduces closely results of exact functions of flow from parallelepipeds; but it is simple for computations and also for linking block properties to traits in the plots of transient pressure response vs. time.

Transient Fluid Flow through Composite Geometries
The equations describing the transient flow of a fluid with low compresibility through a series o... more The equations describing the transient flow of a fluid with low compresibility through a series of connected porous media with contrasting geometries and/or properties were coupled into one single vector equation. That equation describes pressure and flow velocity at any point in that composite medium and in the Laplace domain. Numerical symmetric transforms are used to obtain solutions in terms of time. A distinction is made between actual composite geometries in which flow is delimited by actual boundaries in different configurations; and virtual composite geometries, in which flow geometries are observed in pressure plots though no physical boundaries guide the flow. The formulation was validated through applications to cases presenting composite geometry flow patterns that have been analyzed previously through approaches different from the one presented. P. 9

Journal of Petroleum Science and Engineering, Feb 1, 1993
This paper presents three functions that describe approximately the flow of fluids between matrix... more This paper presents three functions that describe approximately the flow of fluids between matrix-rock blocks and fractures in a fractured reservoir. The different types of flow described are: transient pressure-difference flow, gravitational drainage of oil and water-oil imbibition interchange. The functions were obtained through simplifications of the mathematical formulations of the three types of matrix-fracture flow interchanges. They were validated through comparison with several results of experiments, analytical models and numerical simulations of flow. The simple mathematical models of matrix-fracture flow presented can be easily implemented in numerical simulators of fractured reservoirs with the benefit of including actual petrophysical properties and dimensions of the fractured medium, in contrast with widely used inaccurate formulations involving parameters.

All Days, Jun 10, 2013
The reservoir is described as a "supercritical cluster"; that is, an aggregate of conductive elem... more The reservoir is described as a "supercritical cluster"; that is, an aggregate of conductive elements that comprises a "backbone" of connected pores or fractures that span the zone of interest, and also a collection of "sub-critical clusters" or "dangling ends" joined to the backbone to a limited extent. The scheme resembles the usual fracture and matrix-blocks setting but both backbone and sub-clusters are of the same material and share similar petrophysical properties. Whereas the backbone is a homogeneous porous medium, the sub-critical clusters behave as fractal porous media. The backbone-cluster type of flow has been observed in laboratory experiments. The subcritical clusters were approximated as linear fractal media characterized by static and dynamic fractal exponents and also by porosity and permeability of the compound medium. One of the ends of the linear clusters is closed and the other is joined to the backbone, where the mainstream occurs. A new solution was developed for that problem. The Laplace transform in time and space was used in the mathematical scheme. The theory developed was applied to field cases of interference between wells in aquifers. The matches of computed and observed dynamic pressures show fair fits.
A Three-Phase Model For Fractured Reservoirs Presenting Fluid Segregation
A model has been developed to describe the behavior of naturally fractured reservoirs with black ... more A model has been developed to describe the behavior of naturally fractured reservoirs with black oil in which high transmissibility in the fractures and low oil production rates allow the gravitational segregation of the gas, oil and water phases. The presented formulation results in a system of three simultaneous integro-differential equations where the dependent variables are: the depths of the gas-oil and water-oil contacts, and the pressure at the datum. Observed and calculated variables, obtained applying the model, are in good agreement throughout the forty-five years long production history of an actual reservoir. 21 refs.

Theory of Waterflooding in Fractured Reservoirs
Society of Petroleum Engineers Journal, Apr 1, 1978
This paper presents a new theory of the incompressible flow of two fluids (water displacing oil) ... more This paper presents a new theory of the incompressible flow of two fluids (water displacing oil) in a fractured porous material composed of two distinct media - matrix blocks of low transmissibility embedded in a highly transmissible medium. This general description includes heterogeneous porous media not necessarily of the fractured type. The theory accounts for an important fact not considered in framer analytical model found in the literature. The blocks downstream in a reservoir subject to waterflood are exposed to a varying water saturation resulting from the water imbibition of the upstream blocks. Expressions for the water-oil ratio and the cumulative-oil production are derived, allowing a complete economic evaluation of a fractured-reservoir waterflood project. Comparison of experimental curves reported in the literature with curves obtained using this theory show a good fit. Introduction Imbibition is a most important mechanism of oil production in the waterflooding of fractured production in the waterflooding of fractured reservoirs. Using the action of capillary forces, it allows the recovery of oil from the interior of blocks that cannot be reached by the externally applied gradients of the waterflood. Previous papers assume a function to describe the time rate of exchange of oil and water for a single matrix block. In a lineal reservoir, a water table advances as water is injected with the matrix blocks progressively exposed to water, depending on their position. The oil released by the matrix blocks is assumed transferred instantly to the water-oil interphase,. In this way, the oil production is an added function of individual block contributions. An analytical approach to this problem, and a numerical model, use the problem, and a numerical model, use the simplifying assumption of a water front. This may be a sound description in the presence of vertical high-transmissivity fractures where oil may segregate readily, but in fractures with a discrete transmissivity, it is expected that water imbibition and the simultaneous release of oil by these blocks will give rise to a varying saturation in the fractures that will affect the imbibition rates of the downstream blocks. Braester's analytical approach assumes relative permeabilities of both wetting and nonwetting permeabilities of both wetting and nonwetting phases, intermediate between the fracture's and the phases, intermediate between the fracture's and the matrix's relative permeabilities; these intermediate permeabilities are impossible to measure. The permeabilities are impossible to measure. The model also uses an approximation of the fluid interchange between fractures and blocks. The model may be used for predictions after finding parameters to match observed oil and water parameters to match observed oil and water productions. productions. Kleppe and Morse conducted laboratory experiments on matrix blocks surrounded by fractures and numerical simulations (with rather coarse numerical grids) of Braester's laboratory system. Their numerical simulation computations agree well with the experimental results. This numerical formulation is exact or causalistic; capillary pressures and relative permeabilities are computed pressures and relative permeabilities are computed at every grid block. Their experimental and numerical results are used to test the theory presented here. presented here. Another numerical formulation assumes an approximation for the fluid interchange between fractures and matrix blocks. This approximate formulation did not try to reproduce the exact formulation results of Kleppe and Morse, nor their laboratory experiments. The theory presented here analitically accounts for varying saturations in the fractures by introducing a convolution. A somewhat similar approach -was used successfully to describe the transient one-phase flow in a fractured reservoir. THEORY An outline of the subject theory (developed in the Appendix) includes the following assumed mechanisms and their corresponding mathematical expressions. SPEJ P. 117
A Comprehensive Theory of Transient Inertial Flow in Tubing, Skin Zone and Reservoir
All Days, 1992
Complete analysis of well tests, from early to late time stages, is feasible with a model that in... more Complete analysis of well tests, from early to late time stages, is feasible with a model that includes in a single equation the effects of: a tubing composed by pipes of different diameters; chokes; segregated gas in the tubing; skin zone; reservoir geometry; and inertia of the flowing fluid.
Análisis de pruebas de presión en yacimientos fracturados fractales
Este articulo presenta un metodo para analizar pruebas de variacion de presion con rasgos diferen... more Este articulo presenta un metodo para analizar pruebas de variacion de presion con rasgos diferentes de los mas comunes que denotan geometrias de flujo lineal, radial o esferico, y que son indicadores de flujo de tipo fractal en el yacimiento. El metodo utiliza una solucion a la respuesta de presion con flujo de tipo fractal en el dominio de Laplace e incluye los efectos de dano y almacenamiento por medio de un procedimiento usado para la descripcion del flujo en geometrias compuestas. Este enfoque es mas sencillo que los presentados en articulos anteriores a este. La determinacion de las propiedades fractales en aplicaciones a casos de campo da por resultado valores plausibles; para la comparacion con valores reales es necesario conocer la distribucion de las fracturas en el yacimiento

Analysis of Well Tests in Multiple Fractured Reservoirs-Field Case Applications
… International Petroleum Conference and Exhibition in …, 2000
A simplified analytical model of pressure transients in fractured reservoirs with matrix-rock blo... more A simplified analytical model of pressure transients in fractured reservoirs with matrix-rock blocks of different sizes and/or properties was formulated and applied to field cases. In the applications, relations between traits in the pressure vs. time plots and rock properties are used to determine rock block sizes and block-size distributions from actual well tests. Introduction The mathematical description of flow in fractured reservoirs with rock-blocks of different properties has been an obvious development after the models of media with a single type of block. References 1 to 6 present such models. However, these papers present involved formulations of the block-to-fracture flow. As a result their application in the analysis of actual well tests is not straightforward. Flow in fractured media is accurately represented as flow in the connected fractures medium plus a block-to-fracture component that depends on the blocks, or matrix, properties. In the present paper a simple formulation of such block-fracture interaction as presented in previous papers7,8 describing flow in single block fractured media is extended to the case with multiple rock blocks. Theory For multiple-block fractured media the interaction of blocks and fractures is governed by the relative frequency of occurrence (or just frequency) of a block type in the pay and by the block properties included in the block function of flow. The flow equation in radial (cylindrical) geometry with a multiple block to fracture interaction term is1,Equation 1 The Laplace transform of the former equation is,Equation 2 Block's function of flow. The flow from or into blocks is described approximately by an exponential function, see Ref. 8, for a unit pressure variation at the blok's surface.Equation 3 or, in the Laplace domain,Equation 4 a is the ratio of block surface to block volume divided by a fraction of the block minimal dimension8,Equation 5 the value of a for other parallelepiped shapes, such as prisms or finite slabs, have values intermediate between the two values shown in Eq. 5. The block function of flow reproduces closely results of exact functions of flow from parallelepipeds; but it is simple for computations and also for linking block properties to traits in the plots of transient pressure response vs. time. Block's function of flow. The flow from or into blocks is described approximately by an exponential function, see Ref. 8, for a unit pressure variation at the blok's surface.Equation 3 or, in the Laplace domain,Equation 4 a is the ratio of block surface to block volume divided by a fraction of the block minimal dimension8,Equation 5 the value of a for other parallelepiped shapes, such as prisms or finite slabs, have values intermediate between the two values shown in Eq. 5. The block function of flow reproduces closely results of exact functions of flow from parallelepipeds; but it is simple for computations and also for linking block properties to traits in the plots of transient pressure response vs. time.

Transient Fluid Flow through Composite Geometries
International Petroleum Conference and Exhibition of …, 1998
The equations describing the transient flow of a fluid with low compresibility through a series o... more The equations describing the transient flow of a fluid with low compresibility through a series of connected porous media with contrasting geometries and/or properties were coupled into one single vector equation. That equation describes pressure and flow velocity at any point in that composite medium and in the Laplace domain. Numerical symmetric transforms are used to obtain solutions in terms of time. A distinction is made between actual composite geometries in which flow is delimited by actual boundaries in different configurations; and virtual composite geometries, in which flow geometries are observed in pressure plots though no physical boundaries guide the flow. The formulation was validated through applications to cases presenting composite geometry flow patterns that have been analyzed previously through approaches different from the one presented. P. 9
Influence of Shape and Skin of Matrix-Rock Blocks on Pressure Transients in Fractured Reservoirs
SPE Formation Evaluation
ABSTRACT
The Convolution Integral in Reservoir Physics
Thermal Recovery of Viscous Oils Using Steam Soak
Uploads
Papers by Abraham de Swaan