UNIVERSIDADE ESTADUAL DE CAMPINAS
FACULDADE DE ENGENHARIA MECÂNICA
E INSTITUTO DE GEOCIÊNCIAS
SHARON ANDREINA ROLÓN SOLER
Study of the Influence of Wettability and Flow
Rate Changes in the Transfer Functions for
Simulation of Fractured Reservoirs
Estudo da Influência da Molhabilidade e
Vazão de Injeção nas Funções de
Transferência para Simulação de Reservatórios
Fraturados
CAMPINAS
2019
Study of the Influence of Wettability and Flow
Rate Changes in the Transfer Functions for
Simulation of Fractured Reservoirs
Estudo da Influência da Molhabilidade e
Vazão de Injeção nas Funções de
Transferência para Simulação de Reservatórios
Fraturados
Dissertation presented to the School of Mechanical Engineering and Institute of Geosciences of the University of Campinas in partial fulfillment of the requirements for the degree of Master Petroleum Sciences and Engineering in the area of Reservoirs and Management.
Dissertação apresentada à Faculdade de Engenharia Mecânica e Instituto de Geociências da Universidade Estadual de Campinas como parte dos requisitos exigidos para obtenção do título de Mestra em Ciências e Engenharia de Petróleo, na área de Reservatórios e Gestão.
Orientadora: Pesq. Dr. Erika Tomie Koroishi Blini
ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA DISSERTAÇÃO DEFENDIDA PELA ALUNA SHARON ANDREINA ROLÓN SOLER, E ORIENTADA PELA PESQ. DR. ERIKA TOMIE KOROISHI BLINI.
CAMPINAS
2019
Luciana Pietrosanto Milla - CRB 8/8129
Rolón Soler, Sharon Andreina,
R659s RolStudy of the influence of wettability and flow rate changes in the transfer functions for simulation of fractured reservoirs / Sharon Andreina Rolón Soler. – Campinas, SP : [s.n.], 2019.
RolOrientador: Erika Tomie Koroishi Blini.
RolDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica.
Rol1. Engenharia de reservatório. 2. Rochas - Fratura. 3. Molhabilidade. 4. Simulação computacional. 5. Engenharia do petróleo - Métodos de simulação. I. Blini, Erika Tomie Koroishi. II. Universidade Estadual de Campinas.
Faculdade de Engenharia Mecânica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Estudo da influência da molhabilidade e vazão de injeção nas funções de transferência para simulação de reservatórios fraturados
Palavras-chave em inglês: Reservoirs - Management Rocks - Fracture
Wettability
Computational simulation
Petroleum Engineering - Simulation Methods Área de concentração: Reservatórios e Gestão
Titulação: Mestra em Ciências e Engenharia de Petróleo Banca examinadora:
Erika Tomie Koroishi Blini [Orientador] Manuel Gomes Correia
José Sérgio de Araújo Cavalcante Filho Data de defesa: 18-07-2019
Programa de Pós-Graduação: Ciências e Engenharia de Petróleo
Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0002-1969-0684 - Currículo Lattes do autor: http://lattes.cnpq.br/7028911588690253
E INSTITUTO DE GEOCIÊNCIAS
DISSERTAÇÃO DE MESTRADO ACADÊMICO
Study of the Influence of Wettability and Flow
Rate Changes in the Transfer Functions for
Simulation of Fractured Reservoirs
Estudo da Influência da Molhabilidade e
Vazão de Injeção nas Funções de
Transferência para Simulação de Reservatórios
Fraturados
Autor: Sharon Andreina Rolón Soler
Orientadora: Pesq. Dr. Erika Tomie Koroishi Blini
A Banca Examinadora composta pelos membros abaixo aprovou esta Dissertação:
________________________________________ Pesq. Dr. Erika Tomie Koroishi Blini
DE / CEPETRO / UNICAMP
________________________________________ Pesq. Dr. Manuel Gomes Correia
DE / CEPETRO / UNICAMP
________________________________________ Prof. Dr. José Sérgio de Araújo Cavalcante Filho PETROBRAS
A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vida acadêmica do aluno.
To my mom, Susana, for her endless love, support and encouragement. Your life history is such a huge inspiration for me.
To my little nieces, Nicoll and María José, may my life history inspire your paths. To my husband, Rodrigo, who has been a constant source of support and encouragement through the last years. I am truly thankful for having you in my life.
Firstly, I would like to sincerely express my gratitude to Prof. Dr. Luís Lamas, for all his help, guidance and effort invested in this work. Your advices on this research and my career have been invaluable.
The second person to be acknowledged is researcher Dr. Erika Blini, who has always been supportive to me during this time. I highly appreciate Erika as an enthusiastic person deeply passionate for her work.
Additionally, I would like to thank researcher Dr. Manuel Gomes Correia, who always kept the door open to share his knowledge, having a possible solution to every problem. I also express my gratitude to PhD student Robison Saalfeld who always gave me a helping hand.
To all researchers and colleagues from the Laboratory of Miscible Methods of Recovery (LMMR) research group and friends from the Energy Department (DEP) at UNICAMP, with whom I've been sharing, working and mostly learning during the last two years.
I especially thank to the sponsor of this project: Repsol Sinopec Brasil, for the financial and technical support.
“The ultimate measure of a man is not where he stands in moments of comfort and convenience, but where he stands at times of challenges and controversy” (Martin Luther King)
A implementação bem-sucedida de um projeto de recuperação em um reservatório fraturado exige que a transferência de massa matriz-fratura seja bem compreendida. Como resultado, vários processos envolvidos na transferência de massa têm sido amplamente estudados ao longo do tempo devido ao seu impacto no meio poroso fraturado. A embebição capilar é um desses fenômenos significativos e é considerada através da molhabilidade em várias formulações de transferência de massa (também chamadas funções de transferência) como a principal força motriz entre a matriz e a fratura. Esta dissertação apresenta resultados numéricos de testes de injeção de água em duas escalas diferentes de modelos fraturados: core plug e ampliada (um quarto de 5-spot), com o objetivo de avaliar a influência da alteração de molhabilidade e vazão de injeção na transferência de massa matriz-fratura. A metodologia aplicada baseia-se em análises de sensibilidade de cenários de molhabilidade e vazão, comparando parâmetros envolvidos na interação matriz-fratura: continuidade capilar, taxa de transferência de fluidos e condutividade hidráulica do sistema de fraturas. A metodologia é dividida em três etapas. Inicialmente, modelos core plug de porosidade simples são modelados, recriando uma fratura longitudinal induzida em escala de laboratório, a fim de obter modelos precisos em termos de representatividade às alterações de molhabilidade e vazão de injeção. Em segundo lugar, modelos de dupla porosidade são construídos no intuito de analisar e comparar as respostas dos modelos em relação à transferência de massa. Como terceira etapa, modelos ampliados são criados a fim de analisar os impactos dos parâmetros de sensibilidade da transferência de massa em uma escala maior. Os resultados mostram que o aumento da preferência da rocha por água leva a maiores fatores de recuperação de petróleo a baixas e altas vazões de injeção, beneficiando-se principalmente da embebição espontânea de água. Notavelmente, a embebição espontânea nestes casos é mais significativa em cenários de baixa vazão, devido ao maior tempo de contato entre a água e a rocha. No entanto, o incremento na produção pode não ser economicamente viável, devido ao longo tempo (altos volumes porosos injetados) necessário para obter esse aumento. Em contraste, os cenários intermediários e na condição de molhabilidade ao óleo exibem baixas eficiências de varrido e deslocamento de óleo a baixas e altas vazões de injeção de água. Assim, esses cenários apresentam uma irrupção precoce da água e exibem uma tendência menos acentuada às alterações de saturação de água, quando comparadas com um cenário na condição de molhabilidade à água. Além disso, os resultados para os modelos core plug mostram um comportamento coerente em relação às distribuições de saturação de água ao longo do comprimento e o efeito da fratura induzida, validando assim seu uso. Os resultados também refletem e discutem os efeitos das formulações de meios porosos fraturados nas escalas dos dois modelos, bem como os efeitos dos fatores de forma. Desta forma, o trabalho apresenta a importância da inter-relação entre as alterações de molhabilidade e vazão de injeção com os parâmetros que controlam a transferência de massa matriz-fratura, sendo também demonstrada a significância desses parâmetros.
Palavras Chave: Reservatórios Fraturados. Transferência de Massa. Funções de Transferência. Porosidade Dupla. Embebição. Simulação de Reservatório.
Successful implementation of an oil recovery project in a fractured reservoir requires that the matrix-fracture mass transfer is well understood. As a consequence, several processes involved in the mass transfer have been widely studied along time on account of its impact on the fractured porous media. Capillary imbibition is one of these significant phenomena and is considered through wettability in several mass transfer formulations (also called transfer functions) as the main mass driving force between matrix and fracture. This dissertation provides simulation results of waterflooding in two different scales of fractured models: Core Plug models and Extended models (A quarter of 5-spot), aiming to evaluate the influence of wettability and flow rate alteration on the matrix-fracture mass transfer. The methodology applied is based on sensitivity analyzes of wettability and flow rates scenarios, comparing parameters involved in matrix-fracture mass transfer: capillary continuity, fluid transfer rate, and hydraulic conductivity of the fracture system. The methodology is divided into three main parts. Initially, single-porosity core models with an induced longitudinally fracture at laboratory scale are simulated, to obtain accurate models in terms of representative answers for wettability and flow rate changes. Secondly, dual-porosity models are constructed to analyze and compare answers regarding mass transfer. As a third step, Extended models are created attempting to analyze the impacts of sensitivity parameters of mass transfer on a larger scale. Results show that the increase of rock preference for water leads to highest oil recovery factors at low and high-water injection rates, benefiting mainly from the spontaneous imbibition of water. Notably, the spontaneous imbibition in these cases is more considerable in low-rate scenarios, due to its larger contact time with water and rock. However, the increment on production may not be economically feasible, because of the long time (high injected pore volumes) needed to get this increase. In contrast, intermediate and oil-wet scenarios exhibit low oil sweep and displacement efficiency at low and high-water injection rates. Accordingly, these scenarios reach water breakthrough quickly and exhibit a less accentuated tendency to water saturation alterations if compared with a water-wet scenario. Furthermore, results from Core Plug models show a good agreement between the water saturation distributions along the length and the effect of the induced fracture, validating its use. Results also reflect the effects of the fractured porous media formulations at both model scales as well as the effects of the shape-factors. In a numerical simulation study, this work shows the importance of close interaction between the wettability, flow rate changes, and the parameters that control matrix-fracture mass transfer. At last, the significance of these sensitive parameters is also demonstrated.
Key Word: Fractured Reservoirs. Mass Transfer. Transfer Functions. Dual Porosity. Imbibition. Reservoir Simulation.
Figure 2.1 Sketch of a smaller section of a fractured reservoir (HAUGEN, 2010). ... 24
Figure 2.2 Naturally fractured reservoir classification (NELSON, 2001)... 25
Figure 2.3 Main drive recovery mechanisms in fractured reservoirs (Lemonnier and Bourbiaux, 2010). ... 27
Figure 2.4 Wettability of a system between oil, water and rock (CRAIG JR, 1971). ... 28
Figure 2.5 Schematic diagram of connectivity for the dual-porosity model (Adapted from Correia, 2017). ... 35
Figure 2.6 Schematic diagram of connectivity for the dual-permeability model (Adapted from Correia, 2017). ... 37
Figure 2.7 Summary of the main studies in Transfer Functions over the past 60 years. ... 45
Figure 3.1 Summary of the methodology applied. ... 46
Figure 3.2 Synthetic capillary pressure curves for (a) WW scenario, (b) IW scenario and (c) OW scenario (FAERSTEINet al. 2011). ... 48
Figure 3.3 Synthetic relative permeability curves for (a) WW scenario, (b) IW scenario and (c) OW scenario (FAERSTEIN et al. 2011). ... 49
Figure 3.4 Relative permeability curve for a fracture system according to Romm (1966). .... 50
Figure 3.5 Design in scale of POM spacer. ... 52
Figure 3.6 Dimensions of POM spacer... 52
Figure 3.7 Geometry overview of the Core Plug model. ... 53
Figure 3.8 Overview of (a) injector (Inj-01) and (b) producers (Prod-01, Prod-02) wells... 55
Figure 3.9 Exemplification of a Dual Porosity model. ... 57
Figure 3.10 Extended model. ... 62
Figure 4.1 Recovery factor over time for all scenarios. ... 64
Figure 4.2 Recovery factor over time for all scenarios (extended time). ... 65
Figure 4.3 Recovery factor for all scenarios at PV injected. ... 66
Figure 4.4 Water cut for all scenarios at the first 5 PV injected. ... 67
Figure 4.5 Oil flow rate at the first 144 min of the simulation time for all scenarios. ... 67
Figure 4.6 Oil flow rate for all scenarios at the first 5 PV injected. ... 68
Figure 4.7 Water saturation mapuntil water breakthrough at low-rate in the scenario of (a) oil-wet, (b) intermediate-wet and, (c) water-wet scenarios... 69
Figure 4.9 Water saturation distributions along length until water breakthrough at low-rate in the scenario of (a) oil-wet (b) intermediate-wet and, (c) water-wet. ... 70 Figure 4.10 Water saturation distributions along length until water breakthrough at high-rate in the scenario of (a) oil-wet, (b) intermediate-wet and, (c) water-wet. ... 71 Figure 4.11 Comparison Core Plug vs Dual-Porosity (DP) models: Recovery factor over time for all wettability scenario at high-water rate. ... 72 Figure 4.12 Comparison Core Plug vs Dual-Porosity (DP) models: Recovery factor over time for all wettability scenario at low-water rate. ... 73 Figure 4.13 Comparison Core Plug vs Dual Porosity / Permeability (DP and DPDK) models: Recovery factor over time for water-wet scenario at low-water rate. ... 73 Figure 4.14 Comparison Core Plug vs Dual Porosity / Permeability (DP and DPDK) models: Water cutduring the first 60 min of waterflooding for water-wet scenario at high-water rate. 74 Figure 4.15 Comparison Core Plug vs Dual Porosity / Permeability (DP and DPDK) models: Oil flow rate over time for water-wet scenario at high-water rate. ... 74 Figure 4.16 Scenario 1: Water flow rate during breakthrough time for water-wet scenario at high-water rate. ... 75 Figure 4.17 Scenario 1: Water flow rate during breakthrough time for water-wet scenario at low-water rate. ... 76 Figure 4.18 Scenario 2: Recovery factor over time for the wettability scenarios: A and C at low-water rate. ... 77 Figure 4.19 Scenario 3: Recovery factor over time for the wettability scenarios: A and C at low-water rate. ... 77 Figure 4.20 Scenario 4: Recovery factor over time for the wettability scenarios: A and C at low-water rate. ... 78 Figure 4.21 Comparison 1Φ (SP) vs 2Φ (DP) models: Recovery factor over time for both wettability scenarios. ... 79 Figure 4.22 Comparison 1Φ (SP) vs 2Φ (DP) models: Recovery factor over time for
wettability scenarios at low-water rate. ... 80 Figure 4.23 Comparison 1Φ (SP) vs 2Φ (DP) models: Water cut over time. ... 80 Figure 4.24 Scenario 1: Water flow rate during breakthrough time for water-wet scenario at high-water rate. ... 81
Figure 4.26 Scenario 2: Recovery factor over time for the wettability scenarios: A and C at low-water rate. ... 83 Figure 4.27 Scenario 2: Recovery factor over time for the water-wet scenario at high-water rate. ... 83 Figure 4.28 Scenario 3: Recovery factor over time for the wettability scenarios: A and C at low-water rate. ... 84 Figure 4.29 Scenario 4: Recovery factor over time for the wettability scenarios: A and C at low-water rate. ... 85
Table 2.1 Comparison of shape factors reported in the literature (Adapted from
Rangel-Germán et al.2006). ... 43
Table 3.1 Summary of wettability indexes for each scenario (FAERSTEIN et al. 2011). ... 48
Table 3.2 Fluid and rock properties. ... 53
Table 3.3 PVT data fluid (IMEX®). ... 54
Table 3.4 Main information related to Dual Porosity / Permeability models. ... 57
Table 3.5 Double model cases. ... 58
Table 3.6 Scenarios generated for the sensitivity analysis. ... 58
Table 3.7 Scenario 1: Capillary continuity. ... 59
Table 3.8 Scenario 2: Matrix permeability. ... 60
Table 3.9 Scenario 3: Fracture spacing. ... 60
CEPETRO Center for Petroleum Studies
DPDK Dual-Porosity
DPDK Dual-Permeability
GK Gilman-Kazemi
IMEX® IMplicit EXplicit – Black-oil reservoir simulator provided by CMG®
IW Intermediate-wet
LMMR Laboratory of Miscible Methods of Recovery LSWI Low Salinity Water Injection
MBC Modified Brooks and Corey model MINC Multiple Interacting Continua OOIP Original Oil in Place
OW Oil-wet
POM Polyoxymethylene
PV Pore volume
REV Representative Elementary Volumes
RF Recovery factor
SP Single-Porosity
USBM United States Bureau of Mines
VMO Volume of mobile oil
WCUT Water cut
WR Warren-Root
∆𝐷 Change in the depth between grid blocks in the fracture system
∆𝑝 Change in the pressure between grid blocks in the fracture system for phase p Δ𝑆 Change in oil saturation by forced drainage
Δ𝑆 Change in oil saturation by spontaneous imbibition Δ𝑆 Change in water saturation by forced imbibition Δ𝑆 Change in water saturation by spontaneous imbibition ∆t Time step size for the simulation
θ Contact angle
𝜑 Porosity
𝜇 Viscosity
𝜎 Interfacial tension between oil and solid 𝜎 Interfacial tension between oil and water 𝜎 Interfacial tension between water and solid
𝜎 Shape factor
𝜎 Shape factor based on pressure 𝜎 Shape factor due to imbibition
𝜏 Matrix-fracture transfer function for phase p 𝛼 Hydraulic diffusivity
𝜌 Density of phase p
𝜌 Density of phase p in the fracture system A Grid-block area in the direction of flow
b Fracture width
𝐵 Formation volume factor
𝑐 Compressibility
𝐷 Depth
𝐸 Gas expansion factor
𝐼 Amott-Harvey relative displacement index Io Displacement by oil ratio
Iw Displacement by water ratio k Absolute permeability
𝑘 Grid-block relative permeability 𝑘 Equivalent isotropic permeability
L Grid-block length in the direction of flow or Fracture spacing N Number of normal sets of fractures
𝑃 Pressure
𝑃 Capillary pressure
𝑝 Volumetric average matrix pressure
𝑝 Pressure of grid blocks in the fracture system for phase p 𝑝 Pressure of grid blocks in the matrix system for phase p
𝑞 Volumetric flow rate
𝑞 Source term for phase p in the fracture system 𝑞 Source term for phase p in the matrix system
𝑄 Flow rate
𝑅 Solution gas-oil ratio
𝑆 Phase saturation
𝑆 Initial water saturation 𝑆 Maximum water saturation
𝑆 Average water saturation in the matrix
𝑇 Transmissibility of the phase p in the fracture system 𝑇 Transmissibility of the phase p in the matrix system
𝑉 Volume 1Φ Single-Porosity 2Φ Dual-Porosity 2Φ2k Dual-Permeability Subscripts/Superscripts f Fracture g Gas m Matrix o Oil w Water
1 INTRODUCTION ... 19
1.1 Motivation ... 20
1.2 Objectives ... 21
1.3 Outline of the Dissertation ... 21
2 LITERATURE REVIEW ... 23
2.1 Fractured Reservoirs ... 23
2.1.1 Classification of Fractured Reservoirs... 24
2.1.2 Characteristics of Fractured Reservoirs ... 26
2.1.3 Recovery Mechanisms ... 26
2.2 Simulation of Fractured Reservoirs ... 32
2.2.1 Fractured Porous Media ... 33
3 METHODOLOGY ... 46
3.1 Single Porosity Model Analysis ... 51
3.1.1 Core Plug Model ... 51
3.1.2 Sensitivity Analysis ... 55
3.2 Dual Porosity / Permeability Models Analysis ... 56
3.2.1 Dual Porosity / Permeability Models ... 56
3.2.2 Sensitivity Analysis ... 57
3.3 Extended Model Analysis ... 61
3.3.1 Extended Model ... 61
3.3.2 Sensitivity Analysis ... 63
4 MAIN RESULTS AND DISCUSSION ... 64
4.1 Single Porosity Model ... 64
4.2 Dual Porosity / Permeability Models ... 72
4.2.1 Comparison Core Plug model vs Dual Porosity / Permeability models ... 72
4.3 Extended Model ... 78 4.3.1 Comparison1Φ vs 2Φ ... 79 4.3.2 Scenario 1 ... 80 4.3.3 Scenario 2 and 3... 82 4.3.4 Scenario 4 ... 84 5 CONCLUDING REMARKS ... 86 5.1 General Conclusions ... 86 5.2 Future Works ... 89 REFERENCES ... 90
1
INTRODUCTION
Fractured hydrocarbon reservoirs provide over 20% of the world's oil reserves and production (FIROOZABADI, 2000). Fractured porous media are usually divided into matrix and fracture systems. The matrix system contains most of the fluid storage, but the fluid movement is slow. Fractures contain less fluid compared to the matrix and fluids flow more easily. Fractures are porous medium discontinuities with distinct capillarity and hydraulic conductivity properties that change the reservoir flow behavior as well the physical mechanisms acting in petroleum recovery (PAIVA, 2012).According to Rangel-Germán (2002), oil recovery from fractured reservoirs is generally very low, usually below 30%, and most will eventually go through a process of hydrocarbon recovery by waterflooding.
Waterflooding in fractured reservoirs depends upon the combined, nonlinear effects of hydraulic connectivity and wettability of fractures, rock matrix permeability and porosity, matrix-block size, capillary pressure, and the interfacial tension between wetting and non-wetting phases (RANGEL-GERMÁN, 2002). For instance, the presence of fractures in a porous, low permeable rock matrix provides a relatively high permeability flow path from the injector to the producer. According to Puntervold (2008), the permeability of fractures is often 50 times higher than the permeability of the matrix. It means that waterfloods are less useful because the injected water follows the least resistance path to the producer, only displacing the oil residing in the fractures, which is only a small percentage of Original Oil in Place (OOIP). In fractured and low permeable reservoirs, oil displacement from the matrix blocks by spontaneous imbibition of the injection fluid constitutes the principal drive mechanism to obtain high recovery (CHILINGAR and YEN, 1983). However, being mostly neutral to oil-wet, oil recovery from fractured by spontaneous imbibition of water is limited. The unfavorable wetting state prevents spontaneous uptake of water into the matrix due to negative capillary pressure. When capillary imbibition forces are weak, the injected water is expected to flow primarily through high permeability fractures rather than the low permeability matrix. Unless imbibition forces are strong enough to pull water into the matrix and expel nonwetting fluid, there is little mass transfer between matrix and fractures. Without imbibition, water propagates through the fracture network and does not enter the matrix. The injection fails, and oil recovery is low (AKIN et al. 2000).
Double-porosity models are generally used in fractured reservoir simulation and have been implemented in the prominent commercial reservoir simulators (LEMONNIER and
BOURBIAUX, 2010; PAIVA, 2012; CORREIA et al. 2016). They are modeled through a dual-porosity (2Φ) or dual-permeability formulations (2Φ2k). In the dual-dual-porosity formulation, fractures are entirely responsible for flow between blocks and flow to wells, while the dual-permeability formulation allows some fluid movement between matrix blocks (GILMAN and KAZEMI, 1993).
The rate of mass transfer between the rock matrix and fractures is significant, and the physical processes acting in recovery are represented in double-porosity models by matrix-fracture transfer functions that incorporated shape factors. Commercial simulators have their transfer function implementations, and as a result, different kinetics and final recoveries are obtained.
The difficulty of modeling and unify the transfer functions behavior comes from the fact that matrix and fracture interact in a specific manner depending on parameters such as fluid viscosity, rock properties, fracture properties, capillary pressure, relative permeability, and fracture continuity. These parameters should be analyzed by coupling the effects of both the matrix and the fracture to get a better understanding of the fractured porous media.
Motivation
Significant investments in exploratory activities in the Brazilian Offshore Coasts have been generated discoveries, some in fractured reservoirs, leading to the growing interest of the petroleum industry in determining the role that fractures play in the production of hydrocarbons. Some of these discoveries have shown fractured reservoirs with different physical processes involved in the oil and gas recovery, representing the interaction in the matrix-fracture mass transfer. Studies related to these physical processes have been widely published in the last 50 years (BARENBLATT et al. 1960; WARREN and ROOT, 1963; KAZEMI et al. 1976; GILMAN and KAZEMI, 1983; THOMAS et al. 1983; UEDA et al. 1989; COATS, 1989; DE SWAAN, 1990; CHANG, 1993; LIM and AZIZ, 1995; PEÑUELA et al. 2002; RANGEL-GERMÁN and KOVSCEK, 2003), reveling capillary imbibition as one of these main mechanisms which governed the matrix-fracture mass transfer.
The efficiency of capillary imbibition in fractured reservoirs is thus strongly influenced by the wettability of the system (ZHOU et al. 2000). While conventional reservoirs are often considered water-wet, fractured reservoirs tend to be more intermediate to oil-wet (TREIBER et al. 1972; CHILINGAR and YEN, 1983; CUIEC, 1984), altering directly the traditional way to model the fractured porous media.
Most of the publications on the mass transfer address field theoretical aspects and present field experimental results analysis (HORIE et al. 1988; BOURBIAUX and KALAYDJIAN, 1990; FIROOZABADI and HAUGE, 1990; LABASTIE, 1990; FIROOZABADI and MARKESET, 1992; HUGHES, 1995; LE GALLO et al. 1997). However, there are not numerous references regarding the use of numerical simulation at laboratory scale to evaluate the impacts of wettability on the mass transfer.
Consequently, as a contribution, this work presents an advance to fill knowledge gaps, justify and reconcile divergences found in the literature related to the modelling of fractured porous media at laboratory scale, and provide meaningful numerical insights in order to a better understanding.
Objectives
The objective of this work is to study the influence of wettability and injection flow rate in the mass transfer between matrix and fracture of fractured reservoirs through numerical simulation.
In order to meet this goal, the work is carried out through three purposes. The first purpose focuses on constructing numerical models at laboratory scale based on the porous media formulations, whose matrix-fracture system presents characteristics of a fractured reservoir rock. The second aims to assess the efficiency of oil sweeping and displacement in the numerical models, constructed initially, through a parametric analysis assuming different scenarios of wettability and injection flow rate. Finally, the third task is to evaluate the effects of imbibition and injection rate on oil recovery through continuous waterflooding.
Outline of the Dissertation
The dissertation is organized into five chapters, as follows.
Chapter 1 presents a brief introduction, motivation, and objectives.
Chapter 2 includes a literature review, including fundamental concepts for the understanding of the proposed study. Theoretical and practical aspects of wettability, injection flow rate, simulation of fractured reservoirs and matrix-fracture transfer functions are discussed.
Chapter 3 describes the methodology suggested, the models studied and the properties of these models on which the methodology is applied, including an application case at a larger
scale used for verification. This chapter aims to contribute and enrich to the literature, filling existing gaps and proposing a new study in terms of methodology and knowledge.
Chapter 4 shows the results obtained in the study and their analyses and discussion. Chapter 5 summarizes the conclusions and suggests themes for futures works.
2
LITERATURE REVIEW
In this chapter, an overview of the main concepts of (1) fractured reservoirs, (2) simulation of fractured porous media and (3) transfer functions are presented.
Fractured Reservoirs
The occurrence of fractured reservoirs over the world is well acknowledged. A quite significant portion of world oil reserves is commonly assumed to lie in fractured reservoirs; for instance, Firoozabadi (2000) estimates more than 20%. This number stems from the shares of carbonate reservoirs, 40%, and of primary age reservoirs, 13%, in the world oil reserves estimated by Perrodon (1980) based on large oil fields. These areas were defined as fields with oil reserves of more than 500 million barrels (around 70 million tons). Examples of sizeable oil-in place hydrocarbon reservoirs include the West Texas Carbonates, the North Sea Chalks, and the Asmari Limestones in Iran. These fields generally have active aquifers associated with them and will most eventually go through a process of secondary hydrocarbon recovery by waterflooding (RANGEL-GERMÁN, 2002).
Now, considering the share of carbonate reservoirs in world oil reserves plus the hydrocarbon reserves found in the siliciclastic reservoirs of Paleozoic age, one can reasonably assume that around half of oil reserves are found in reservoirs where the question of a possible fracture impact on productivity and recovery is worth being addressed. Although the impact will be turn out, effective or sensitive for only a fraction of them, depending on such prominent properties as fracture density, connectivity, and conductivity (FIROOZABADI, 2000).
It is well accepted that regardless of fractures origin, morphology or distribution in a fractured reservoir, they usually act as chief paths for the fluids flowing throughout the reservoir. It is also accepted that in most cases the matrix system represents the significant storage volume of the reservoir fluids and that the fluid flow in fractured porous media are controlled by the interaction between the matrix and the fracture systems. Water propagation and multiphase flow in fractured reservoirs depend upon the combined, nonlinear effects of hydraulic connectivity and wettability of fractures, rock matrix permeability and porosity, matrix-block size, capillary pressure, and the interfacial tension between the resident and imbibing phases (RANGEL-GERMÁN, 2002).However, the main difficulty in modeling fluid flow in fractured rock is to describe the anisotropy and heterogeneity of the porous media. Beyond the question of considering the heterogeneity of the fractured porous media, the concern
is also assessing to which extent the presence of fractures influences production strategy and final oil recovery.
The interest in the impact of fractures on production and recovery has grown during the last decade and efforts have been made to forecast potentially-related problems of production. One heuristic reason was the occurrence of unexpected production behavior, such as early water breakthroughs, in some fields initially considered as non-fractured. Nowadays, provided a minimum amount of data, the detection of fractures and the evaluation of their possible flow impact are carried out earlier in field life to optimize well placement and to choose and appropriate recovery method (FAERSTEIN, 2011).
A published survey from Allan and Sun (2003) indicates that the ultimate recovery factor for 56 fractured oil reservoirs with reliable data ranges from less than 10% to 60-70%, an estimated range already reported by Firoozabadi (2000).Analyzing recovery data into more detail, the authors conclude that such scattered values are not only the result of reservoir characteristics, as the fracture network, the aquifer drive, and the matrix properties, but also the consequence of adequate or inadequate recovery methods and production management. That is, reserves maximization for fractured reservoirs calls for joint efforts of geoscientists and reservoir engineers, for an integrated and appropriated static/dynamic assessment of fractures. 2.1.1 Classification of Fractured Reservoirs
Fractured reservoirs are heterogeneous reservoirs, where matrix blocks are separated by fractures (Figure 2.1). Geological processes produce different types of fractured reservoirs, which may be classified in several ways.
Figure 2.1 Sketch of a smaller section of a fractured reservoir (HAUGEN, 2010).
In Group 1, the main hydrocarbon volume resides in the matrix, and the fracture pore volume is minimal. In Group 2, the fracture pore volume constitutes between 10-20% of the total pore volume. In Group 3 reservoirs, more than half of the hydrocarbons are stored in the fracture system with insignificant contribution from the matrix.
Later, one common way of classifying naturally fractured reservoirs was also proposed by Nelson (2001), as shown in Figure 2.2.
Figure 2.2 Naturally fractured reservoir classification (NELSON, 2001).
In Type 1 reservoirs, fractures provide both the essential permeability and porosity, exemplified by the Asmari Limestone Field in Iran, where fracture pore volume ranges from 10-20% of the reservoir pore volume(HAUGEN, 2010).In Type 2 reservoirs, fractures provide the essential permeability of the system, whereas the major part of the hydrocarbons resides in the rock matrix. North Sea chalk fields are examples of such reservoirs (HERMANSEN et al. 2000). In Type 3 reservoirs, fractures assist in the permeability of already productive reservoirs. The matrix has low permeability but provides essential hydrocarbon storage capacity with high porosity. In Type 4 reservoirs, fractures provide no additional permeability or porosity, but create significant reservoir anisotropy.
The different types of fractured reservoirs mentioned above highlight the relationship between heterogeneities and the geological features related to hydrocarbon storage (FERNØ, 2012). Classification of a potential fractured reservoir may reveal potential production and reservoir evaluation problems that can be anticipated. At reservoir scale these heterogeneities will have a large impact on the overall productivity and drainage of the field (LIE, 2013).
This dissertation focuses on fractured reservoirs, where oil recovery from the matrix is relevant, and the fracture permeability contributes to the fluid’s transmissibility. That is reservoirs mostly associated with Group 2, according to Firoozabadi (2000) or Type 3, according to Nelson (2001) classification.
2.1.2 Characteristics of Fractured Reservoirs
Fractured reservoirs behave differently from conventional reservoirs during oil production (HAUGEN, 2010). Some typical characteristics of fractured reservoirs, according to Van Golf-Racht (1982), are listed as follows.
Low pressure declining per unit oil produced: The rate of pressure declining per unit of oil produced usually is low in fractured reservoirs compared to conventional reservoirs. This is caused by the ample supply of fluids from the matrix to fracture as a result of gravity and imbibition combined with fluid expansion, segregation and convection. Conventional reservoirs must generally re-inject more than 80% of the produced gas to display similar rate of pressure declining per unit oil produced.
Low producing gas-oil ratio: The gas-oil-ratio is often substantially lower in fractured reservoirs compared to conventional reservoirs. High vertical communication in fractured reservoirs causes liberated gas to segregate towards the top of the reservoir.
Lack of transition zone: Fractured reservoirs often lack transition zones with sharp, horizontal fluid contacts. The fracture permeability is high, and changes in the fluid contacts are rapidly re-equilibrated even during production.
Small pressure drops around producing well: High permeable fractures promote low pressure drops around producing wells even at high producing index.
Constant fluid properties in depth: Compositional gradients are usually absent because fluids have been circulating by convection due to thermal expansion and compression. Unlike conventional reservoirs, which have different bubble point as a function of depth.
2.1.3 Recovery Mechanisms
There are significant differences between the recovery performance of fractured and non-fractured reservoirs. The high contrast of capillarity between the matrix and the fractures is the leading cause of these differences. As mentioned in Section 2.1.2 and pointed out by Firoozabadi (2000), one of the main characteristics of fractured reservoirs is high-rate wells in the early life of the field, due to the high effective single-phase permeability of the combined matrix-fracture porous media. The flow behavior contrast between fracture and matrix media is
emphasized under two- or three-phase flow conditions. For instance, considering gas-oil gravity drainage, the capillary forces hinder positive gravity displacement effects on matrix oil recovery; moreover, the oil drained from the matrix to the fracture can partially or totally re-imbibe neighboring blocks under the effect of capillary forces. Finally, gas-oil displacement turns out to be a complex recovery mechanism that is governed by capillary, gravity and viscous forces and affected by compositional effects including transfer between phases and diffusion (LEMONNIER and BOURBIAUX, 2010).
Depending on their matrix block sizes and matrix/fissure permeabilities, fractured reservoirs can be produced using several recovery processes, primary recovery, gas drive (gas cap expansion and solution gas drive combined with possible convection phenomena), waterflooding and miscible or immiscible flooding. The main drive recovery mechanisms involved in these processes are schematized in Figure 2.3. Depending on the nature of the recovery process, one or several mechanisms, among imbibition, water drive, gravity drainage, re-imbibition, and diffusion, can contribute to production.
Figure 2.3 Main drive recovery mechanisms in fractured reservoirs (Lemonnier and Bourbiaux, 2010).
In this work imbibition is considered as a recovery mechanism and waterflooding as a recovery process. First, the principles of this mechanism will be explained, followed by its effects on waterflooding.
2.1.3.1 Imbibition
According to Haugen (2010), imbibition represents an important recovery mechanism in fractured reservoirs, taking place when water displaces oil from the matrix to the fracture by capillary forces.
Zhou et al. (2000) define imbibition as a direct function of capillary and gravity forces, depending on the rock wettability. That is, capillary forces balance defines rock wettability. I Wettability
Wettability can be defined as the tendency one fluid has to spread on or adhere to a solid surface in the presence of another immiscible fluid (CRAIG JR, 1971). The reservoir rock wettability has an essential role in order to determine the success of a waterflood because it has a significant influence on the location, flow, and distribution of the fluids in the reservoir. According to Puntervold (2008), in a system at equilibrium, the wetting fluid is located on the pore walls and occupies the smallest pores while the non-wetting fluid is in the pore bodies. Thus, in a water-wet system, water is found at the pore walls and oil in the pore bodies. Vice-versa, in an oil-wet system the oil is located at the pore walls and water in the pore bodies I.I Wettability classification
A relatively fast way to evaluate the wettability of the system is by measuring the contact angle (θ) between a solid and the two immiscible fluids. The contact angle reflects the equilibrium between the interfacial tensions of the two fluid phases and their own adhesive attraction to the solid. The contact angle is by convention measured through the denser phase, which in Figure 2.4 is the water phase (PUNTERVOLD, 2008).
Figure 2.4 Wettability of a system between oil, water and rock (CRAIG JR, 1971). In a system containing a reservoir rock, oil, and water, as Figure 2.4, the rock is typically preferentially water-wet if water occupies the smallest pores and is the spreading fluid
(θ < 90°), and preferentially oil-wet if oil is the spreading fluid (θ > 90°) occupying the smallest pores. The rock is intermediate (neutral)-wet when the rock has no strong preference for either oil or water, and both fluids can be the spreading fluid (θ = 90°) (CRAIG JR, 1971).
Not all reservoirs have uniform wettability throughout the reservoir, but rather a heterogeneous wettability. Fractional, spotted or dalmatian wettability are terms that are often seen representing a heterogeneously wetted reservoir (ANDERSON, 1986a). Some areas of this type of rock are strongly oil-wet, while the rest is strongly water-wet. A particular type of fractional wettability was introduced by Salathiel (1973) as “mixed wettability”. In mixed wettability, the fine pores and grain contacts are preferentially water-wet and contain no oil, whereas the oil-wet surfaces form continuous paths through the largest pores and contain all the oil. Thus, oil permeability exists down to very low oil saturations during waterflooding, and higher recovery is seen than for either uniformly water-wet or oil-wet conditions (SALATHIEL, 1973).
I.II Wettability measurement methods
There are several methods, qualitative or quantitative, that can be used for wettability measurements (ANDERSON, 1986b). Quantitative methods are the contact angle measurements, Amott Test (imbibition and forced displacement), and the U.S. Bureau of Mines (USBM) method. Qualitative methods are imbibition rates, microscope examination, flotation, glass slide method, relative permeability curves, permeability/saturation relationships, capillary pressure curves, capillarimetric method, displacement capillary pressure, reservoir logs, nuclear magnetic resonance and dye adsorption (PUNTERVOLD, 2008).
Since an experimental approach is not considered on this work, the principles of the wettability measurements will not be further discussed. However, there is a large number of studies in the literature to be consulted about these topics (AMOTT, 1959; KYTE et al. 1961; ANDERSON, 1986b; MORROW, 1990; MA et al. 1999; STRAND, 2005; STRAND et al. 2006; PUNTERVOLD, 2008).
2.1.3.2 Mechanisms of production under Waterflooding
Water injection represents the majority of the projects of secondary recovery compared to gas injection (LEMONNIER and BOURBIAUX, 2010). As the injected water preferentially flows through the fracture network, a high water-saturation boundary condition is established on the matrix blocks. According to Lemonnier and Bourbiaux (2010), the displacement of oil by water in the matrix medium is then due to three mechanisms: (1) spontaneous capillary
imbibition if the matrix rock is water-wet; (2) viscous displacement under the pressure gradient generated by fracture flows; and (3) gravity effects linked to water-oil density difference.
The first mechanism and one of the focus of this work, capillary imbibition, is governed by capillary forces, which occur at the macroscopic level by the existence of interfacial tension and differential wettability. Capillary forces have also the following principal effects: (1) the creation of a "capillary zone" in which the saturation varies progressively; this capillary zone dissipates the saturation discontinuity anticipated in the simplified theory of Buckley and Leverett (1942), that neglects capillary pressure, and (2) the existence of "end effects" (laboratory) that affect the saturation values in the neighborhood of surfaces through which the fluids transfer to and from the porous medium (MARLE, 1981).
In scenarios, where capillary forces are weak, water injection is, intuitively, expected to flow primarily through low-flow-resistance fractures rather than the high-flow resistance matrix. Unless imbibition forces are strong enough to pull water into the matrix and expel nonwetting fluid, there is little mass transfer between matrix and fractures. Thus, capillary forces must be relatively strong if both water injection and oil recovery in fractured systems are to be successful. Without imbibition, water propagates through the fracture network and does not enter the matrix. The injection fails, and oil recovery is low (AKIN et al. 2000). Consequently, it is possible then to conclude the essential impact of imbibition on the oil recovery process by waterflooding.
According to Jerauld and Rathmell (1997), wettability strongly impacts the efficiency of waterflooding projects. In strongly water-wet reservoirs, the smaller pores imbibe water, and the oil is located in larger pores. After the water injection, the residual oil remains in the center of larger pores as an isolated droplet, which can be imprisoned for being disconnected from a continuous mass of oil (RAZA et al. 1968). The oil saturation quickly reaches a constant value after the water breakthrough, and almost all residual oil is irreducible (FAERSTEIN et al. 2011). In strongly oil-wet reservoirs, the advancing water will preferentially move through larger pores pushing the oil forward. Usually, in these cases, the water injection is less efficient than in the cases of strongly water-wet (RAZA et al. 1968).
Some of the first works related to the impact of wettability on secondary recovery by water injection were published by Donalson et al. (1969) and Donalson and Thomas (1971). In these works, micro-models of laboratory (a sand layer confined between two glass slides) were used to observe the effect of wettability on oil recovery. The results of micro-models were then validated with experiments in cores. Donalson et al. (1969) observed a higher recovery of oil in cases of water-wet in comparison to cases of intermediate and oil-wet. For those studies,
intermediate-wet was defined as all cases between slightly water-wet, and slightly oil-wet. Also, small recoveries were attributed to the quick canalization of water and oil production continued for a long time even after the water breakthrough. After stopping the oil production, large oil pockets remained in the porous media not being produced. Otherwise, in water-wet cases, oil pockets stayed behind but in smaller sizes. When subjected to higher rates of water injection, these pockets were displaced.
Jadhunandan and Morrow (1995), in laboratory analyses, found that the maximum recovery of oil is obtained in cases classified by the authors as slightly water-wet. The results showed that the recovery in facies strongly water-wet is independent of the volume of water injected. The recovery in oil-wet facies is weakly dependent on the injected volume, with the increasing dependence as the wettability is approaching to a more intermediate condition.
Later, Li et al. (1997) studied the effect of heterogeneity and wettability on oil recovery through numerical modeling with a two-dimensional model considering three and five facies. The uniform wettability was obtained by the J-function of Leveret (LEVERET, 1940) to calculate the capillary pressure curves to each facie. Under intermediate injection rates, with facies ranging from intermediate to water-wet, the final recovery tended to 1, decreasing as the water wettability became smallest. Oil-wet facies recovered about 30% lower. Under higher injection rates, the capillary effects became less critical, and the effects of wettability on recovery were smallest.
Unlike the previous authors (DONALSON et al. 1969, DONALSON and THOMAS, 1971, JADHUNANDAN and MORROW, 1995 and LI et al. 1997), who reported low efficiency of oil displacement by water in intermediate and oil-wet rocks, other researchers observed that the recovery in the water and oil-wet rocks have small recoveries compared to rocks of intermediate wettability.
Intermediate to oil-wet field situations are often encountered in practice. In these systems, water will be rapidly displaced in the fracture leaving behind the oil from the matrix (BEHBAHANI et al. 2005). In such scenarios, the waterflooding efficiency will depend on the possible role played by other recovery mechanisms, including gravity forces and also viscous drive due to fracture flow.
For instance, Beliveau et al. (1993) presented an evaluation study of more than 20 years of waterflooding history in the naturally fractured Midale field in Canada, a heavily fractured vuggy limestone overlain by a less fractured chalky dolomite. Sensitivity runs showed that imbibition effects are directly proportional to capillary pressure and matrix permeability, and inversely proportional to the square of fracture spacing. The simulations also suggested that
gravity drainage plays only a minor role in fracture-matrix fluid transfer in this thin reservoir. The different well behaviors were matched by varying only fracture spacing and fracture network conductivity. It can be suggested that an adequate characterization of the fracture system made by a multidisciplinary team was critical to understanding the Midale reservoir behavior. Ultimate waterflooding recovery was predicted to be 24% OOIP. The study also showed that opportunities exist to increase sweep efficiency through infill drilling, especially with horizontal wells to take advantage of the natural fractures. There is also the possibility to alter the wettability of the rock from oil-wet to water-wet, in order to promote imbibition of water. In that case, if it was done, the capillary forces would become positive, water would be imbibed, and oil could be expelled (PUNTERVOLD, 2008).
On the other hand, Firoozabadi (2000) quotes intermediate-wettability field examples, such as the Ekofisk chalk field in the North Sea, where water injection turned out to be very efficient although laboratory spontaneous imbibition measurements would predict minimal oil recovery. Tang and Firoozabadi (2000) conducted laboratory experiments on Kansas outcrop intermediate-wet chalk samples which highlighted the contribution of viscous forces to water injection efficiency.
Simulation of Fractured Reservoirs
Petroleum field development has a very high cost and a substantial degree of uncertainty. It follows from that the need for numerical simulations to estimate reservoir performance under a variety of production schemes. Continuous technology improvements made reservoir simulations easier and more realistic. Today and since about thirty years, results from reservoir simulations are used for significant reservoir development decisions. Reservoir simulation can answer several crucial issues such as the choice of the “best” recovery process (technical and economic) for a given reservoir, the calculation of production profiles and expenses (operating costs and capital expenditure), and the risk evaluation. Numerical simulations have become a reservoir management tool at all stages of reservoir life, and this is particularly true in the case of fractured reservoirs (LEMONNIER and BOURBIAUX, 2010).
The growing awareness of the fracture impact on field production has been the incentive for significant advances in the characterization and modeling of fractured reservoirs in recent years. Integration of geosciences and reservoir engineering resulted in better-constrained flow models of fractured reservoirs. According to Bourbiaux (2010), it unveiled at the same time limitations calling for further progress: (a) at an intermediate scale between small-scale diffuse
fractures and seismic faults, sub-seismic faults and fracture swarms (b) high-performance methodologies for characterizing and up-scaling the flow properties of fracture networks and (c) multi-scale expression of natural fracturing. The rate that a fracture conducts fluid to/from a matrix block at partial saturation is also a significant unknown in numerical reservoir simulation. The difficulty of modeling this behavior comes from the fact that matrix and fracture interact in a specific manner depending on parameters such as fluid viscosity, rock properties, fracture properties, and the ‘combined’ properties. These combined properties include capillary pressure, relative permeability, and fracture continuity. They should be analyzed by coupling the effects of both the matrix and the fracture (RANGEL-GERMÁN, 2012).
Once a field-representative flow model has been set up and parameterized, the reliability of a fractured reservoir flow simulation still depends on the adequate transcription of the physical flow mechanisms taking place in and between the constitutive media, matrix-fractures, especially for porous fractured reservoirs where a dominant share of the field oil reserves are held in the matrix medium (BOURBIAUX, 2010). In this respect, the following section is dedicated to fractured reservoir simulation with specific attention to these matrix-fracture interactions.
2.1.4 Fractured Porous Media
Fractured porous media are divided into matrix and fracture systems, usually modeled using the dual-continuum formulations: dual-porosity and dual-permeability. The accuracy of these formulations lies in the accurate evaluation of the parameters involved. Under these circumstances, the relative permeability, capillary pressure, and the matrix-fracture transfer term are, therefore, the main parameters for numerical simulation work to be successful.
The objective of this section is to describe the two main formulations of the dual-continuum type briefly, as well as explain the importance of the parameters involved in these formulations.
2.1.4.1 Formulations for Fractured Porous Media
Currently, conventional fine-grid simulation is not feasible for modeling fractured porous media. It is costly to achieve enough refinement of the computational grid to simulate matrix-fracture flow rigorously. Modeling of naturally fractured reservoirs is conducted by differing formulations depending on the heterogeneity degree.
In general, mathematical formulations for fractured porous media can be divided into three groups. The first uses dual-continuum models based on sugar-cube type systems, where matrix and fracture communicate through the matrix-fracture interface. In these models, the reservoir is discretized into two collocated continua (two sets of grid blocks located in the same space), one called the matrix, and the other called the fracture. The matrix continuum is assumed to be comprised of matrix blocks which are separated spatially by fractures. The dimensions of these matrix blocks can be variable throughout the reservoir and are a function of the fracture spacing, orientation, and width. The fluids flow through the reservoir occurs through the fracture system, which is the primary conduit for fluid flow. This type of approach was described by Barenblatt and Zheltov (1960) and modified by Warren and Root (1963) and Kazemi et al. (1989). This approach is time-consuming as it requires solving equations both for fluids in the fracture and the matrix. The second approach type is based on averaged representative elementary volumes (REV) and was introduced by De Swaan (1978) in his pioneering integro-differential formulation of the Buckley-Leverett displacement in a fracture surrounded by matrix rock undergoing imbibition. However, this type of approach requires a matrix-fracture transfer function to express the fluid exchange between matrix and fracture. The third approach is the laboratory measurement or experimental model. While yielding results that closely match real data, it has been difficult, to date, to generalize from experimental models because they are tailored to specific conditions and scales. Each of these approaches has advantages and limitations, and their usefulness varies depending on the problem and scale of interest.
I Dual-Porosity
The dual-continuum model was introduced by Barenblatt and Zheltov (1960) who described the double porosity concept for single-phase flow in naturally fractured reservoirs as flow through two superimposed media: a continuous fracture system and a discontinuous system of matrix blocks. The fracture system has a high transmissivity and low storability, while the matrix system has a low transmissivity and contains most of the fluid storage. A porosity formulation does not allow direct flow between matrix blocks; whereas, a dual-permeability formulation allows the transfer of fluids from one matrix block to another as well as transfer from matrix blocks to the fracture system. Figure 2.5 shows a schematic diagram of connectivity for the dual-porosity model.
Figure 2.5 Schematic diagram of connectivity for the dual-porosity model (Adapted from Correia, 2017).
The transfer of fluids between these two systems, as described by Barenblatt and Zheltov (1960), assumes pseudo-steady state flow between the matrix blocks and the fracture system. The matrix to fracture transfer is expressed as Darcy’s law with an appropriate geometrical factor that describes the characteristic length and flow area between the matrix blocks and the fracture system. The flow equation in finite-difference form is for the fracture shown in Equation 2.1.
∆ 𝑇 (∆𝑝
− 𝜌 ∆𝐷 ) − 𝜏
+ 𝑞
=
∆
∆
Equation 2.1where 𝑇 is the transmissibility of the phase p in the fracture system, ∆𝑝 is the change in the phase pressure between grid blocks in the fracture system for phase p, 𝜌 is the density of phase p in the fracture system, ∆𝐷 is the change in the depth between grid blocks in the fracture system, 𝜏 is the matrix-fracture transfer function for phase pin a grid-block volume V, 𝑞 is the source term for phase p in the fracture system, ∆t is the time step size for the simulation, and ∆ is the accumulation term expressed as the porosity change over the time step, phase saturation, and formation volume factor of phase p for the fracture system.
On the other hand, the flow equation in finite-difference for the matrix is defined by Equation 2.2.
∆ 𝑇
(∆𝑝
− 𝜌
∆𝐷 ) + 𝜏
+ 𝑞
=
∆
∆
Equation 2.2where 𝑇 is the transmissibility of the phase p in the matrix system, ∆𝑝 is the change in the phase pressure between grid blocks in the matrix system for phase p, 𝜌 is the density of phase p in the matrix system, ∆𝐷 is the change in the depth between grid blocks in the matrix
system, 𝑞 is the source term for phase p in the matrix system and ∆ is the accumulation term for the matrix system.
According to Equation 2.3 and Equation 2.4, the transmissibility terms for fracture and matrix, respectively, are,
𝑇
=
𝑘
Equation 2.3𝑇
=
𝑘
Equation 2.4
where A is the grid-block area in the direction of flow, L is the grid-block length in the direction of flow, 𝑘 is the grid-block permeability of the fracture, 𝑘 is the grid-block permeability of the matrix system, 𝑘 is the grid-block relative permeability of phase p and 𝜇 is the viscosity of phase p.
The matrix-fracture transfer function, 𝜏 , is modeled as a source function given by the Equation 2.5.
𝜏
= 𝜎V𝑘
𝑝
− 𝑝
− 𝜌 (𝐷 − 𝐷 )
Equation 2.5
where 𝜎 is the shape factor and hasdimensions of reciprocal area (1/L²), reflecting the geometry of the matrix elements in pseudo-steady state, single-phase flow at all times.𝑝 is the pressure of grid blocks in the fracture system for phase p,𝑝 is the pressure of grid blocks in the matrix system for phase p,𝜌 is the density of phase p,𝐷 and𝐷 are the fracture and matrix depths.
The shape factor reflects the geometry of matrix elements and controls the matrix-fracture interaction. It is, in turn a function of the surface to volume ratio of the matrix block, typically, with an implicit assumption of single-phase pseudo-steady state flow. One of the main shortcomings of Equation 2.5 and the entire dual-continuum formulations is that these are not the conditions to which these models are applied. These formulations were developed for pseudo-steady state flow between the fracture and the matrix and uniformly distributed fractures that are totally immersed instantaneously. It means, fractures being totally covered by water instantaneously during waterflooding with an uniform pressure distribution between fractures blocks.
The dual-continuum model was applied later by Warren and Root (1963). They idealized naturally fractured reservoirs as a stack of identical, rectangular parallelepipeds (sugar cubes), and fractures of constant width oriented parallel to one or more of the principal axes of
permeability. These authors introduced the concept of dual-porosity into petroleum engineering for single-phase flow applied to Well Testing.
Kazemi et al. (1969) extended the concept to multiphase flow, visualizing reservoirs as layers separated by horizontal fractures. When using the dual-porosity formulation, the 𝑇 term in Equation 2.2 is set to 0 and, for grid blocks with no source terms, the matrix-fracture transfer function in the matrix equation simplifies to the Equation 2.6.
𝜏
=
∆
∆
Equation 2.6II Dual-Permeability
The dual-permeability formulation allows the transfer of fluids from one matrix block to another as well as transfer from matrix blocks to the fracture system. Figure 2.6 shows a schematic diagram of connectivity for the dual-permeability model.
Figure 2.6 Schematic diagram of connectivity for the dual-permeability model (Adapted from Correia, 2017).
2.1.4.2 Capillary Pressure and Relative Permeability
The shape of both fracture relative permeability and fracture capillary pressure curves is still a topic of great debate. Several authors have stated that fracture capillary pressures are negligible (KAZEMI and MERRILL, 1979; BECKNER, 1990; GILMAN et al. 1994). Others have shown experimentally that capillary continuity becomes important when gravity provides a driving force (HORIE et al. 1988; FIROOZABADI and HAUGE, 1990; LABASTIE, 1990; FIROOZABADI and MARKESET, 1992; HUGHES, 1995; RANGEL-GERMÁN, 1998).
The relative permeability functions resulting from multiphase flow in fractures have received considerably less attention than those in porous media. Regarding flow in isolated fractures, Romm (1966) performed two-phase flow experiments into smooth, vertical parallel plates divided in wetting and nonwetting strips in the flow direction. He presented the X-curves for fractures commonly used in numerical reservoir flow simulation, where the limiting values
of the relative permeabilities 𝑘 and 𝑘 were both 1.0, and the residual saturations were both 0.0. It means that the sum of the relative permeabilities is always one. Romm (1966) also stated that these results could not be applied to flow in fractured media, where a system of interconnected fractures is present.
Pruess and Tsang (1990) presented a conceptual and numerical model of multiphase flow in fractures, showed that relative permeabilities depend strongly on the nature and spatial correlation between the apertures, and suggested that the sum of wetting and nonwetting phase relative permeabilities in fractures may be considerably less than one at intermediate saturations. The theory expressly excludes the possibility of "blobs" of one phase being conveyed by the other. The authors also stated that relative permeability behavior becomes more complicated for highest capillary numbers, where enough pressure drive may be present to permit a phase to invade pore spaces that would not be allowed based on a static criterion of local capillary pressure. Their study also predicted ranges of saturation values at which neither phase can flow at all. The model of Pruess and Tsang assumes that fracture properties can be locally approximated by those of smooth-walled plates.
Rossen and Kumar (1992) performed air-water flow experiments in horizontal smooth and artificially roughened fractures. Their findings suggest a range of possibilities between the "sub-Corey" results of Pruess and Tsang (1990) at the lower end and the X-curves at the upper end.
Fouraret al. (1993) also conducted air-water experiments in both smooth- and rough-walled fractures and proposed that the relative permeability concept was not useful to describe multiphase flow in fractures since the apparent relative permeability values appeared to be functions of velocity. The apparent relative permeability curves shown in Fouraret al. (1993) do not appear to follow either X-curve or Corey behavior at any velocity.
Persoff and Pruess (1995) dealt with rough-walled fractures using epoxy replicas. They obtained the best possible matches for an isolated fracture. Their results suggest that fracture relative permeability should not be considered as a straight line with a value equal to the phase saturation or as a Corey-type behavior, showing lower values than either of those two models. Diomampo (2001) performed nitrogen-water experiments on both smooth and rough parallel plates to determine the governing flow mechanisms for fractures and a methodology for data analysis. The author found that the nitrogen-water flow through fractures is better described by using the relative permeability model. The sum of the relative permeabilities was not equal to one, indicating phase interference.
Using a similar apparatus to that used by Diomampo (2001), Chenet al. (2002) performed experiments on steam-water flow, finding that the average steam-water relative permeabilities show less phase interference and behave closer to the X-type relative permeability functions.
On the other hand, Pan et al. (1996) dealing with oil-water experiments in smooth fractures showed that straight-line fracture relative permeabilities could be used, but they also stated that the values are not necessarily equal to the phase saturation. These observations were verified by Rangel-Germán (1998), and Rangel-Germán et al. (1999).
Rangel-Germán (1998), and Rangel-Germán et al. (1999) built an apparatus to obtain detailed measurements of pressure, rate, and in-situ saturation distribution in fractured, sandstone laboratory cores. The combined experimental and simulation study resulted in a much better understanding of the physical processes that occur when there is multiphase flow in fractured systems. This work demonstrated that capillary continuity could occur in any direction, depending on the relative strengths of the capillary and Darcy terms in the flow equations. Thin fracture systems were found to have a more stable front in the direction typical to the displacement and slower breakthrough compared to wide fracture systems. Capillary pressure was shown to have more effect when fractures were narrow. Neither the capillary pressure nor the relative permeability curves in the fracture affected the results for the narrow fracture system.
Rangel-Germán (1998) concluded that the thin fracture was so thin and/or each half mated so well that there was excellent capillary continuity, and the blocks acted very similarly to a solid block. However, the fronts become less stable as the fracture aperture increased. The assumption of fracture relative permeabilities equal to the phase saturation was tested using a commercial numerical simulator to history match their experimental results. This work showed that straight-line fracture relative permeability functions could be used, but the relative permeability of a phase is not necessarily equal to the phase saturation. So, called X-type relative permeability curves can be used for fractured systems using more significant flow resistances in the fractures, such as straight lines with slopes of 0.75 or 0.6 instead. They noted the more substantial resistance of phases in multiphase flow as compared to single-phase flow when walls were rough. Their results also showed the effect of high flow rate on these parameters.
In real rocks, fluids will flow simultaneously in both fractures and the porous matrix. The combination of these two flow processes may result in an effective relative permeability
