Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Flood front tracking and continuous recording of time lag in immiscible displacement
(USC Thesis Other)
Flood front tracking and continuous recording of time lag in immiscible displacement
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
FLOOD FRONT TRACKING AND CONTINUOUS RECORDING OF TIME LAG IN IMMISCIBLE DISPLACEMENT by Abdollah Orangi A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PETROLEUM ENGINEERING) December 2008 Copyright 2008 Abdollah Orangi ii Dedication To Maryam iii Acknowledgments I would first and foremost like to express my gratitude to God whose blessings and guidance have made the writing of this thesis an enjoyable and rewarding process. May His peace and blessings be upon His beloved Prophet, Muhammad the last one of his messengers who has given light to mankind. I ask God to accept this work out of His generosity and help others to benefit from it. Amen. I wish to express my sincere gratitude to Professor Iraj Ershaghi, advisor and committee chair, who made significant contributions to this work with his guidance, encouragement, and valuable suggestions. Acknowledgment is extended to Professor Kristian Jessen and Professor Stefan Schaal for serving on my dissertation committee and for their helpful and constructive comments. Partial financial support for my graduate studies while working on the IAM project and later as SME on other projects was furnished by Chevron Corporation through the Center of Interactive Smart Oilfield Technology (CiSoft) at University of Southern California. Support of CMG and the use of their code is also acknowledged here. For the love and understanding I received from my wife, Maryam and my daughter, Maideh, during the preparation of this manuscript, I give special thanks that cannot be expressed in words alone. iv Table of Contents Dedication ii Acknowledgments iii List of Tables vi List of Figures vii Abbreviations xii Abstract xiv Chapter 1 Introduction 1 1.1 Scheduled and Repeated Pulse Tests 4 1.2 Methodology 5 1.3 Literature Review 6 Chapter 2 Development of the Mathematical Model for a Radial System 12 2.1 Parameter Sensitivity Based on the First Derivative 15 2.1.1 Rate Changes 17 2.1.2 Storativity/Transmissibility Ratio 22 2.1.3 Interwell Distance 27 2.1.4 Injection Period 30 2.1.5 Discussion 32 2.2 Multiple Rate Model 32 2.3 Parameter Sensitivity Based on the Second Derivative 37 2.3.1 Discussion 43 2.4 Wellbore Storage and Skin Effects 44 2.4.1 The Reciprocity Principle 46 2.4.2 Wellbore Storage and Skin Effects on Time Lag 48 2.5 Composite Reservoir 51 2.6 Reservoir Parameters Distribution 55 Chapter 3 Dynamic Discontinuity Effect on the Time Lag 59 3.1 Linear System 59 3.2 Front Tracking in One Dimensional Immiscible Displacement 62 3.2.1 Buckley-Leverett Frontal Advance Theory 63 v Chapter 4 Analytical Solution vs. Reservoir Simulator 71 4.1 Example 1- Homogeneous System 72 4.2 Example 2- Gradient Heterogeneous System 76 4.3 Example 3- Faulted System 80 4.4 Example 4- Channeled System 84 Chapter 5 Conclusions and Recommendations 89 5.1 Conclusions 89 5.2 Recommendations 93 5.2.1 Continuous Recording of Time Lag by Rate Changes 93 5.2.2 Wellbore Model 95 5.2.3 Vertical Heterogeneity (Multilayer Reservoir) 97 5.2.4 Dykstra-Parsons Approach 97 5.2.4.1 Dykstra-Parsons Model for Two Layers Reservoir 101 5.2.4.2 Dykstra-Parsons Model for Multi-Layer Reservoir 106 Bibliography 109 vi List of Tables Table 2-1 Reservoir and fluid properties 17 Table 4-1 Injection rate schedules and related time lag values at producers for a five spot homogeneous model 73 Table 5-1 The properties of two layer reservoir 101 Table 5-2 Permeability distribution for a multilayer reservoir 106 vii List of Figures Figure 1-1 A schematic of pulse testing terminology and tangent method 10 Figure 2-1 Pressure responses at observation well as results of multi-rate changes at pulsing well 14 Figure 2-2 Single pulse test, pressure and rate profile 16 Figure 2-3 Dimensionless rate effect on time lag in radial system 18 Figure 2-4 Dimensionless rate effect on dimensionless time lag in radial system 19 Figure 2-5 Dimensionless rate effect on pressure difference (amplitude) measurement causes by rate changes 20 Figure 2-6 Dimensionless rate effect on dimensionless pressure amplitude 21 Figure 2-7 Dimensionless (S/T) effect on time lag in radial system 22 Figure 2-8 Dimensionless (S/T) effect on dimensionless time lag in radial system 24 Figure 2-9 First derivative of dimensionless (S/T) vs. dimensionless time lag 25 Figure 2-10 Viscosity effect on time lag 25 Figure 2-11 Total compressibility effect on time lag 26 Figure 2-12 The effect of viscosity and compressibility for gas reservoir on time lag 27 Figure 2-13 The effect of dimensionless rate on time lag for selection of well distance 28 Figure 2-14 the effect of dimensionless “S/T” on dimensionless time lag for selection of well distances 29 Figure 2-15 Time lag vs. interwell distance for selection dimensionless rates 29 Figure 2-16 The effect of dimensionless “S/T” on time lag for various injection rate periods 31 viii Figure 2-17 The effect of injection rate period on dimensionless “S/T” vs. dimensionless time lag in type curve format 31 Figure 2-18 A schematic of unsupervised rate changes at active well and pressure responses at observation well 33 Figure 2-19 Time lag changes vs. the first injection rate (q 1 ) when the second dimensionless rate is variable 35 Figure 2-20 Time lag changes vs. the first injection rate (q 1 ) when the second dimensionless rate is constant 36 Figure 2-21 The effect of the first injection rate period to the following time lags 37 Figure 2-22 The comparison of dimensionless (S/T) effect on time lag between extrema (first derivative) and Inflection point (second derivative) for variable dimensionless rates 39 Figure 2-23 The comparison of dimensionless (S/T) effect on the time lag between extrema and inflection point for variable interwell distance for given dimensionless rate 40 Figure 2-24 The comparison of dimensionless (S/T) effect on time lag between extrema and inflection point for variable injection periods for given dimensionless rate 40 Figure 2-25 the pressure difference measurement causes by rate change based on second derivative 41 Figure 2-26 The dimensionless rate effect on dimensionless pressure difference based on second derivative 42 Figure 2-27 The dimensionless rate effect on dimensionless time lag based on second derivative 42 Figure 2-28 The injection rate effect on pressure difference measurement based on first derivative 43 Figure 2-29 The injection rate effect on pressure difference measurement based on second derivative 44 Figure 2-30 The effect of wellbore storage and skin on pressure at observation well 49 ix Figure 2-31 The effect of wellbore storage on time lag for selected dimensionless rate 50 Figure 2-32 The effect of wellbore storage on time lag for various skin factors (q D =∞) 50 Figure 2-33 The effect of wellbore storage on time lag for various skin factors (q D =4) 51 Figure 2-34 The effect of dimensionless discontinuity interface distance on time lag for selected dimensionless rates 54 Figure 2-35 The effect of dimensionless discontinuity interface distance on dimensionless time lag for selected dimensionless rates 55 Figure 2-36 Log normal distribution for absolute permeability with mean of three and standard deviation of one in logarithmic scale 56 Figure 2-37 The time lag distribution which obtained from log-normal absolute permeability distribution 57 Figure 3-1 Dimensionless rate effect on time lag in linear system 61 Figure 3-2 Dimensionless (S/T) effect on dimensionless time lag in linear system 62 Figure 3-3 the integration of small segments before breakthrough to calculate total transmissibility in a 1-D system 65 Figure 3-4 Total interwell transmissibility at various front locations 66 Figure 3-5 The front location and the computed total transmissibility at various pore volume injections 67 Figure 3-6 Water saturation profile in 1-D system with and without capillary pressure 68 Figure 3-7 The commercial reservoir simulator results for 1D 69 Figure 3-8 Pressure difference at observation well vs. time, after starting pulse 70 Figure 3-9 The comparison of analytical solution and reservoir simulator results 70 Figure 4-1 The front location for a five spot pattern homogeneous model after one year 72 x Figure 4-2 Time lag behavior with front movement for various dimensionless rate in a homogeneous model 74 Figure 4-3 Dimensionless time lag behavior vs. dimensionless front location in a homogeneous model 76 Figure 4-4 The front map for a gradient heterogeneous reservoir (permeability decreases from left to right) 77 Figure 4-5 Dimensionless time lag behavior vs. dimensionless front location for production wells 1 and 2 in a reservoir with gradient heterogeneity 78 Figure 4-6 Dimensionless time lag behavior vs. dimensionless front location for production wells 3 and 4 in a reservoir with gradient heterogeneity 79 Figure 4-7 The comparison of time lag behavior vs. front location for production well 1 in higher permeable zone and production well 4 in lower permeable zone 79 Figure 4-8 A schematic of faulted reservoir 81 Figure 4-9 Water saturation map in faulted reservoir after one year 82 Figure 4-10 Dimensionless time lag vs. dimensionless front location for production well 2 82 Figure 4-11 Dimensionless time lag vs. dimensionless front location for production well 4 83 Figure 4-12 Dimensionless time lag vs. dimensionless front location for production well 1 83 Figure 4-13 A Schematic of Channeled reservoir 84 Figure 4-14 The water saturation map in channeled reservoir 86 Figure 4-15 Dimensionless time lag vs. dimensionless front location for production well 1 which is located in lower permeable zone 86 Figure 4-16 Dimensionless time lag vs. dimensionless front location for production well 2 which is located inside the channel with the highest injector support 87 Figure 4-17 Dimensionless time lag vs. dimensionless front location for production well 3 with interwell lateral discontinuity 87 xi Figure 4-18 Dimensionless time lag vs. front location for production well 4 88 Figure 4-19 The comparison of the rate of front location changes according to time lag 88 Figure 5-1 Dimensionless time lag behavior vs. dimensionless front location in a homogeneous model when bottomhole pressure is constant 95 Figure 5-2 Pressure gradient at the wellbore 97 Figure 5-3 Non-communicating layered reservoir in Dykstra-Parsons model 98 Figure 5-4 Schematic of a linear system in piston-like immiscible displacement 98 Figure 5-5 Schematic of two layer reservoir 101 Figure 5-6 Time lag values vs. pulse sequence number for two layers reservoir (k 1 =100 md, k 2 =300 md) (h 1 =30 ft, and h 2 =30 ft) 102 Figure 5-7 Time lag values vs. pulse sequence number for two layers reservoir (k 1 =100 md, k 2 =300 md) (h 1 =10 ft, and h 2 =50 ft) 103 Figure 5-8 Time lag values vs. pulse sequence number for two layers reservoir (k 1 =100 md, k 2 =300 md) (h 1 =50 ft, and h 2 =10 ft) 103 Figure 5-9 Dimensionless front location vs. dimensionless time lag for the same thickness 104 Figure 5-10 Dimensionless front location vs. dimensionless time lag for high permeable layer for all three cases 105 Figure 5-11 The comparison of time lag obtained from reservoir simulator vs. calculated time lag in layer “i” based on Equation 105 Figure 5-12 Dykstra-Parsons coefficient of Permeability Variation 106 Figure 5-13 Time lag in multilayer reservoir when the injection well in all layers is comingled and each layer has its own perforation for producer 107 Figure 5-14 The comparison of time lag between comingled producer and separate layer perforation 108 xii Abbreviations c t Isothermal Coefficient of Compressibility, psi -1 C D Wellbore storage F i , H i Heterogeneity factor f w Fractional flow h Formation thickness, ft k Permeability, md k ri , (i=o,w) Relative permeability of oil or water l Laplace transform variable M i End point mobility ratio p Pressure, psi P D Defined pulse test pressure, bbl/d = ( ∆pT)/(70.6Bq) q D Dimensionless rate, q i-1 /q i q i Flow rate in i th period , STB/D r Radial Distance, ft r D Dimensionless radius, =r/rw S Storativity = φc t h, ft/psi s Skin s w Water saturation T Transmissibility, md ft/cp = kh/µ t Time, min xiii t D Dimensionless time = (Tt)/(56900Sr w 2 ) t DL Dimension time lag, t L / ∆t t L Time lag, min u i , (i=o,w) Fluid velocity in porous media x Well distance, linear system x D Dimensionless distance in linear system ∆t Injection period, min µ i , (i=o,w) Viscosity, cp ϕ Porosity, Fraction λ i (i=o,w) Mobility, 1/cp xiv Abstract In immiscible displacements processes such as water flooding, an important consideration is how to maximize the areal sweep efficiency. Limited numbers of tools exist to map oil-water interface movement. It is our expectation that the extended and repeated application of interwell pulse tests can provide some insight in tracking the fluid front approaching the producing wells. The improvement of sensor technology and data mining has opened up new opportunities to obtain continuous recording of rate and pressure data at the observation/producing and injection wells. In reality, each injection well can be subjected to pre-scheduled or unsupervised rate changes. These rate changes create pulsation and observation wells detect the pulse after a time lag. The delay is a function of interwell distance and the effective interwell formation and fluid properties. In this study we focus on both flowing wells and observation wells. We propose that the monitoring of the time lags can help in tracking fluid front plus indigenous reservoir properties. To separate the two effects, we postulate that a comparison of time lags with the first available time lag can lead to the estimation of front location. The differences between displacing and displaced fluid properties on the two sides of the interface impacts the observed time lag. We focus also on the sensitivity of this relationship to the strength of pulsation caused by rate changes rather than shutting-in the injection well. Our approach to examine this problem is with using a combination of analytical and numerical solutions. Analytical solutions to calibrate numerical test cases for radial and linear flow geometries are presented for homogeneous and composite reservoirs. We also present the result of this study for some 2-dimensional reservoir cases. From the studies xv of these systems, we have demonstrated the definite potential of monitoring repeated pulses for front tracking in immiscible fluid displacement processes. From the sensitivity studies of the effect of static and dynamic reservoir and fluid properties between observation and the pulsing wells, we note progressive changes in the measured time lags with the movement of interface between the displacing and displaced fluid. 1 Chapter 1 Introduction Increasing oil recovery beyond what is attainable during the primary production phase requires injection water, with or without chemicals, or the use of economically affordable hydrocarbon gases. Fluid injection processes maintain reservoir pressure while they sweep the oil out of the reservoir. An effective flooding process requires uniform sweep of the reservoir. Monitoring of the fluid-fluid interface is necessary to effectively manage water floods. There are limited options available to monitor non-uniform sweep conditions. To assure the success of the process, it is necessary to monitor and detect the fluid displacement behaviors from affordable tests. Dynamic mapping of fluid interfaces before breakthrough is a complex task. Hydrocarbon reservoirs consist of flow units, formed during depositional processes and changed during the post depositional events. The efficiency of displacement processes depends on many factors such as structural setting of the producing formations and their rock and fluid properties. Recovery efficiency for a fluid injection process in an oil reservoir is defined as the ratio of recovered oil to initial volumes of oil in place. Displacement efficiency of reservoir fluid injection process can be expressed in terms of current and original oil saturations. Volumetric efficiency includes areal and vertical sweep efficiencies. Areal sweep efficiency, which is the focus of this research, is defined as swept area compared to total 2 area. Reservoir heterogeneities resulting in non uniform sweep can result in preferential movement of fluid fronts and early breakthroughs. Prudent management of front movement requires dynamic monitoring. In reservoir management of fluid injection processes, one main objective is to keep an eye on fluid-fluid interfaces. Such monitoring can then help in taking corrective measures to improve volumetric sweep efficiencies. These include adjusting number and the location of the wells, completion types and voidage /replacement rates. There are limited options available for dynamic front tracking to monitor areal sweep. Use of single well fall-off tests, for water injection or air injection as a means for obtaining an estimation of approximate location of the front have been discussed in the literature (Kazemi, 1966; Kazemi et al., 1972). Well interference tests (Theis, 1935) and pulse tests (Johnson et al., 1966) have also been generally used to obtain average inter- well properties. In the past, frequent use of such tests for dynamic monitoring interface movement has not been discussed in the literature because of implementation difficulties. Use of 4-D seismic has been proposed to monitor sweep. 4-D seismic technology depends on the differences between corresponding amplitudes in seismic surveys taken at different times. Successive images from 4-D surveys of the same reservoir may potentially illustrate the fluid flow dynamics. This method has a low resolution, but it can be complementary to other methods. The 4-D seismic monitoring can image the fluid movement, locate bypassed oil, monitor injected fluids, and map the flow paths. The 3 process is, however, very expensive and it may not be practical in all cases especially for small and marginal fields and those fields operating in urban areas. Recent advances in downhole sensors and sensor technologies are making it possible to get continuous recording of real-time injection rates, production rates and pressure data at all active and idle wells in water floods. In this research, a test bed for the proof of concept with injection and observation/production wells equipped with sensors, allowing collection of real time downhole pressure and rate data, is assumed. The focus of this research is on the concept of monitoring the time lag in operationally-caused pulses generated by variations in the injection rates. A pulse is normally generated at the acting/injection well. The surrounding wells do not respond immediately. Well responses will be observed after some time. The delay is called the time lag. Time lags in pulse test are intuitively expected to be affected by static and dynamic reservoir and fluid properties between observation and the pulsing wells. Because reservoir static properties usually do not significantly change during a flood, the dynamics associated with the movement of interface between the displacing and displaced fluid are expected to cause progressive changes in the measured time lags. We have examined the nature of the relationship between the interface movement and recorded time lags and the sensitivity of this relationship to the strength of pulsation caused by rate changes at pulsing wells. 4 1.1 Scheduled and Repeated Pulse Tests The lumping of formation and fluid properties between a producer and an injector well is often done with and representing Transmissivity (T) and Storativity (S) respectively. These lumped parameters include permeability (k), thickness (h), viscosity (µ), porosity ( φ), and total compressibility (c t ). In the case of an immiscible displacement, relative permeability of wetting and non wetting phase will also affect the effective transmissibilities. As shown in the literature (Kamal and Brigham, 1975), by measuring the lag in response time, the composite interwell T and S values can be obtained from pulse tests. Transmissibility embodies both formation and fluid properties and represents the ease with which fluids are transmitted through the formation. Storativity represents the pore volume and the compressibility of the pore volume. Such conventional and infrequent tests can shed light on existence of communication between injection and production wells, detect communication across faults, and identify directions and magnitude of permeability trends caused by the presence of fractures and channels. Repeated pulse tests, because of their interruption effect on operations, are not common in field operations. With permanent downhole recorders, continuous recording of pressures can provide opportunities for obtaining dynamic changes caused by fluid movements. During a fluid injection process, the saturation changes caused by fluid invasion and the corresponding relative permeability effects influence the estimated composite T and S between injectors and producers. These dynamic changes, caused by 5 the movement of the fluid front, are expected to result in variations in the recorded time lag. To avoid the necessity of shutting down injection wells, we consider scheduled or unexpected rate changes for causing the pulsation. These data can then provide a series of time lags at each observation well corresponding to the rate variations in pulsing wells. The purpose of this project is investigating the nature of the relationship between interface movement and time lag values and the sensitivity of this relationship. The goal is to examine and derive correlations, numerically and analytically to show how the time lag is related to interface movement. From that perspective, even for unsupervised operation, the system is able to continuously record the time lag and estimate the interface location. 1.2 Methodology We examine a radial flow system followed by a one dimensional (1D) system to evaluate the predicted changes in time lag during fluid injection. Recorded responses at observation wells can be estimated from the superposition of rate changes affecting fluid dynamics changes in the reservoir. By specifying rate variations at injection wells in a multiple well system, time lags at producing wells can be continuously estimated. Because of the fluid movement effects, different values for repetitive time lag are expected. To have a base case, suppose we conduct a pulse at the start of fluid injection process (t 0 ). 6 At t 0, it is assumed that the displacing fluid has not invaded the reservoir and the time lag represents the formation heterogeneity profile and the flow characteristics of the indigenous fluid. These values of early times are the basis for comparison with time lags at later times. Because the static properties are assumed to be invariant with time, the changes in the observed time lags need to be correlated with the changes in effective and composite transmissibilities caused by saturation changes and interface movements. To study the discontinuity between two zones, by either static condition or dynamic condition, a radial composite system is modeled. Formation and fluid properties such as permeability, porosity, viscosity and total compressibility, which are lumped in transmissibility and storativity for each zone, are expected to affect time lag values. For instance, tight rocks with small permeability result in large time lags. A large value of total compressibility also could damp out the pulse signal. Time lag values are continuously measured for any disturbance in the pulse well. Time lags are normalized with respect to the first value, not necessarily the largest one. Because of this normalization, a family of type curves is generated. These curves show the trend of time lag changes. The current method in pulse testing requires shutting-in the pulsing well to get the largest possible signal to offset wells. We will examine rate changes that could generate measureable pulses to address the front tracking. 1.3 Literature Review Well tests responses characterize dynamic fluid flow and pressure changes in well and 7 through the reservoir. These tests provide a description of the reservoir in dynamic condition during the test. Well log data and geological survey information, however, represent static parameters; well flow rate and pressure. These two variables are related in different ways for each reservoir model. A disturbance or change in one variable (usually flow rate) causes changes in the other variable (usually pressure). The pressure behavior with time, according to changes in flow rate can be used to obtain reservoir properties. This is the basis for pressure transient tests (Kamal, 1976). The objective of well test analysis is to describe the unknown systems, which are the wells and the reservoir, by indirect measurements of pressure responses and flow rate changes. Other direct and indirect methods should also be consulted for validation of well test methods. From pressure analysis, it is possible to estimate horizontal and vertical permeabilities reservoir heterogeneities (layering, natural fracture, etc), boundaries (size and shape, and distance), and reservoir pressure (initial and average). If flow rate changes and pressure responses are measured at the same well, the test is called a "single well" test. These include drawdown, buildup, and fall-off test. On the other hand, if flow rate changes in one well and pressure responses are monitored in another wells, the test is called "multiple well" test. This includes interference and pulse tests. The purpose of multiple well testing is to show communication among wells and to determine the average reservoir properties. This research is focused on pulse tests. Ramey introduced a solution for analysis of the interference tests in anisotropic, 8 reservoirs (Ramey, 1970). Other complexities for pulse testing such as boundary effects (Vela, 1977) and dual permeability (Prats, 1986) have also been studied. Wellbore storage at the pulsing or observation well causes a delay in time lags and reduces the amplitude of pressure responses. Pulse test was introduced in 1966 to describe reservoir flow properties between wells. In this test the flow rate at the acting well is changed into a series of flow and shut-in periods. The changes cause pressure oscillations in the observation well. They introduced the "Tangent Method" to analyze the pressure responses. Figure 1-1 illustrates a schematic of pulse test terminology and the concept of the tangent method. By drawing the tangent between two consecutive peaks or valleys and a parallel tangent at the valley or peak between them, and calculating the distance between these two lines, the pressure amplitude for that specific pulse can be estimated. The difference between the time that a pulse is generated by shut-in or flow and the time of maximum or minimum responses occurring in the observation well indicates the time lag. The measured time lag, pressure response amplitude and theoretical correlation curves can be used to estimate the transmissibility and storativity (Kamal, 1973). The characteristics of these theoretical correlation curves to handle equal and non-equal pulse and shut-in time were introduced in 1975 (Kamal and Brigham, 1975). Their method is based on a few important assumptions. First, their model is derived for single-phase flow. In this research, we are trying to map the fluid front locations for an immiscible displacement which includes two-phase flow system; mainly water and oil. Relative permeability is one of the key 9 parameters in the case of two-phase flow. We will focus on effect of relative permeability and front movement on time lag. They assumed further that the observation wells are inactive and their recorded pressure responses are reflected from the pulse only; that is, the production wells are shut in to record the pressure responses. In reality, the pressure difference at observation well caused by rate disturbance at active well is small. If the observation well is active then the pressure difference caused by production is high. If the observation well is active, depends on mechanistic model and flow regime in the wellbore there will be noises in pressure measurements. For instance the flow regime could be laminar or turbulent and the flow pattern in vertical upward flow could be bubbly, slug, churn, or annular. These criteria could affect pressure measurements and make it non uniform with time. In this research we will follow the same assumption to be able to eliminate the superposition effect of the pressure changes caused by well production. To honor this assumption, the production wells are assumed to be shut in when the rate changes take in place at active (pulsing) well. On the other hand, this solution is similar to conventional pulse test and there will be some interference to field production. Other alternative option is to assume the observation wells with no production/injection are available to record the pressure response. This assumption could be more realistic in water flooding processes. However, analytical solution shows that one may capture the pressure change developed by pulsation while the production wells are active. If production wells are set at constant bottom hole pressure, the result of generated pulses appears in production well’s gross rate. 10 Figure 1-1 A schematic of pulse testing terminology and tangent method In conventional pulse test design, the active/pulsing well has to be shut in to generate the required disturbance in the reservoir. We show in certain conditions, the rate change at active well is enough to generate a detectable pulse at observation wells. In the immiscible displacement processes, the front location can be correlated with time lag only when the pressure changes are detectable. In a two-phase flow process, saturation distributions and corresponding relative permeabilities have significant impact on the entire calculation. The concept of relative permeability effect on front movement will be discussed in the context of this study. Saturation distribution and location of fluid-fluid interface in the reservoir may change very slowly. It is then a reasonable assumption that the interface location remains 11 constant for a short period of time. Then a reservoir with two fluid regions could be simulated as a composite reservoir. As part of this research, the effect of discontinuity location between two zones in a composite reservoir on time lag was will be considered. The composite reservoir model has been used for the fall-off test, in water injection wells and air injection wells in in-situ combustion processes (Abbaszadeh and Kamal, 1989; Kazemi, 1966; Kazemi et al., 1972). The previous studies were based on either single well test or interference test. Analytical solutions of the composite model exist in the literature (Ambastha, 1988; Issaka, 1996; Satman et al., 1980). The radial composite system was introduced in 1960 (Hurst, 1960). Hurst assumed the well is located at the center of a circular zone and a second zone corresponds to the outer part of composite reservoir. For a linear system, two zones in a linear composite reservoir have been modeled where two semi-infinite parts are divided by a vertical plane (Bixel et al., 1963). We apply the pulse test methodology to the composite reservoir model to derive expected responses about inter-well transmissibility and front location. 12 Chapter 2 Development of the Mathematical Model for a Radial System In this Chapter, we consider a horizontal and homogeneous single layer reservoir with two vertical wells in a radial and linear system. First we assume a single phase fluid system. In the absence of gravity and dispersion, potential flow governs the dynamics of the displacement process. For radial systems Theis (Theis, 1935) for the first time introduced a solution for unsteady state pressure changes with time at any point of reservoir, which is caused by change of flow rate at another point. Using the line source approximation, the solution is (Culham, 1969; Odeh and McMillen, 1972): 2-1 Equation (2-1) is a basis for variety of well test analysis, such as interference and pulse test. The mathematics of original pulse test study (Johnson et al., 1966) was based on no boundary effect (infinite reservoir). The solution of an infinite radial system describes pressure as a function of radius and time. Applying the principle of superposition in rate will result the following general form: 2-2 where the variables in field units are: dimensionless radius: 13 defined pulse test pressure (bbl/d): dimensionless time: dimensionless rate: , transmissibility (md.ft/cp): storativity (ft/psi): time (min): t While a series of flow disturbance (supervised or un-supervised) is generated at pulsing well, Equation (2-2) describes the pressure responses at observation well. Figure 2-1 shows a schematic of conventional pulse test which is the simplified version of supervised rate changes and the corresponding pressure responses when the active well is first shut in and then flowed for certain period of time (Johnson et al., 1966; Kamal, 1973). As shown in Figure 2-1, a system of Equations were introduced (Kamal, 1973) for points “A”, “B”, and “C” which indicates the first three pulses (m=4). This Equation can be rewrite in field units as: 2-3 14 Pulse Period Cycle Period (Δt c) Pressure Amplitude (Δp) Pulse Period Shut in Period Flow Rate (bbl/D) Time Pulse Rate Pressure Response P A, t A P B, t B P C, t C t 1 t 2 t 3 t LA=t A-t 1 t LB=t B-t 2 t LC=t C-t 3 q 1 q 2 q 3 q 4 q 5 t 5 t 4 t Figure 2-1 Pressure responses at observation well as results of multi-rate changes at pulsing well Kamal used the tangent method (Johnson et al., 1966), which is the commonly preferred method for pulse test analysis. The first derivatives of pressure responses at points , , , yield the slops of two parallel lines. Solving this system of Equations yields the time lag values and pressure amplitudes for each cycle (Kamal, 1973). His work was based on first derivative on pressure response. This method is applicable if the duration of pulse is long enough for pressure responses to build up to a maximum or minimum. In the next Chapter the application of second derivative will be introduced to indicate the time lag. The second derivative of pressure response at the observation well describes the moment when pressure response, in the case of rate changes at the pulsing well, starts deviating from the case with no changes in rate at the pulsing well. 15 We used general formulation for two-phase flow system which will be demonstrated in the next Chapter. In reality, the transmissibility and storativity in Equation (2-3) are average values and depend on inter-well properties and which can be correlated to front location. 2.1 Parameter Sensitivity Based on the First Derivative Conventional boundary conditions, for multiple well interference or pulse test analysis (Johnson et al., 1966; Kamal, 1975), require closing the observation wells and generating disturbances at pulsing well by sequences of shut in and following conditions. Because of the interference with field productivity, such tests are not conducted on a routine basis. In addition, episodic pulse testing obtains the transmissibility and storativity for a specific time and gives instantaneous values. Our hypothesis is that continuous recording of time lags can be correlated with front location. To demonstrate the pressure responses behavior, we now rewrite the Equation (2-3) for the first or single pulse (m=2) as follows: 2-4 Equation (2-5) represents the relationship between time lag and fluid and rock properties; lumped into transmissivity and storativity terms for the first pulse for condition where the rate variation is the cause for pulsation; ( ). 16 2-5 This Equation is the first derivative of Equation (2-4) made equivalent to zero which has been simplified. Figure 2-2 shows a schematic of a single pulse test and the related terminologies. 8 10 12 14 16 Injection period (Δt) Flow Rate (q) Time (t) Pulse Rate Pressure Response (Δp) Time Lag ___ q D = q 1 q 2 q 2 q 1 t L Figure 2-2 Single pulse test, pressure and rate profile We now examine the sensitivity of estimated parameters’ from the observed pressure responses at observation wells. Dependency of time lag to S/T was presented (Daltaban and Wall, 1998). As the composite T and S vary with the movement of the fluid-fluid interface, using the expression relating the time lag to the T and S, one can examine the expected changes in the time lag. In our approach, we focus on the rate changes to generate the pulse and to derive expected responses at observation wells. To test the effectiveness of weaker pulses, that can be produced by variations in injection rates, we represent this succession of rate changes by a series of dimensionless rates within a range from infinity (shut-in the pulsing well) to close to one (no rate changes). 17 We now compare this relationship to that representing the conditions before the start of the injection process: 2-6 The influences of those parameters on time lag have been demonstrated by comparison of time lag at given time to the t 0 for a family of curves for different dimensionless rate. Because we assume a lower mobility fluid (such as oil) is being displaced by a higher mobility fluid (such as water), during this process; the higher time lag in each case is representative of first conducted pulse (t 0 ). Table 2-1 shows the range of parameters used in this study to examine the effect of composite “S/T” on estimated time lag. Table 2-1 Reservoir and Fluid properties Property Range Porosity 0.05< φ < 0.25 Permeability (md) 1<k<625 correlated with porosity (Balan et al., 1995) Thickness (ft) 50-100 Viscosity (cp) 1<µ<5 Compressibility (1/psi) 10 -6 <c t <50×10 -6 Injection period (min) 10 3 -10 5 Well distance (ft) 100-500 2.1.1 Rate Changes In our hypothesis, we predict the rate changes, in some cases, generate pulses that can be sensed at observation wells. It is obvious the variation in injection rates create weaker pulse. The root of the first derivative of pressure response indicates the maximum and 18 minimum values and it is applicable in case of injection rate’s reduction. We will show how increasing the injection rate can be detected by second derivative in the next section. Figure 2-3 demonstrates how reservoir and fluid properties affect time lag values. This Figure illustrates the same behavior for variable dimensionless rates. The sensitivity of dimensionless rate changes for large values of is depicted in this Figure. Figure 2-3 Dimensionless rate effect on time lag in radial system 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 0.0 0.2 0.4 0.6 0.8 1.0 t L (min) 1/q D (r^2/∆t)(S/T)=1.39E-08 8.33E-08 1.85E-07 3.89E-07 1.25E-06 3.06E-06 19 Figure 2-4 Dimensionless rate effect on dimensionless time lag in radial system Figure 2-4 shows the relationship between dimensionless time lag and dimensionless rate for a various values of in the Equation (2-5). The “y” axis in this graph is the time lag at each rate variations divided by the time lag at maximum rate change ( ). This lumped parameter is dimensionless with a multiplier of in the field unit system. This lumped parameter includes all of the effective parameters on the time lag. When the q D is infinity, corresponding to when the injection well has been shut in to generate the pulse, the time lag has the least value, because of the strongest pulsation energy. We expect, for variation in the rate, we may have longer time lag. For lower value of the effect of rate variations on the time lag is small. Increasing this lumped parameter increases the sensitivity of time lag with rate variations. The time lag evaluations are based on maximum or minimum of the pressure response amplitude measurement at the observation well. In this 1.0 1.5 2.0 2.5 0 0.2 0.4 0.6 0.8 1 t L / t L (at q D =∞) 1/q D (r^2/∆t)(S/T)=1.39E-08 8.33E-08 1.85E-07 3.89E-07 1.25E-06 3.06E-06 20 study, the ability to measure the pressure is essential. Figure 2-5 Dimensionless rate effect on pressure difference (amplitude) measurement causes by rate changes Figure 2-5 illustrates the estimated pressure changes at observation well, causes by rate changes at active well. For instance, similar to the time lag, for the small value of , a minor pressure change may be recorded. For larger value of , a significant pressure change will be recorded. It is obvious the time lags and pressure changes are directly related and the sooner that time lag is observed, there will be less time to build-up the pressure change. The practical implication of this concept is the availability of high resolution and accurate pressure sensors. Figure 2-5 shows for high the pressure difference is in the range of 5 psi while there are downhole sensors with accuracy of 0.01 psi. 1.E-4 1.E-3 1.E-2 1.E-1 1.E+0 0.0 0.2 0.4 0.6 0.8 1.0 ∆p (psi) 1/q D (r^2/∆t)(S/T)=1.39E-08 8.33E-08 1.85E-07 3.89E-07 1.25E-06 3.06E-06 21 For practical purposes, we will provide a set of type curves that can be used for analysis. Fundamental use of the type curves are discussed in detail in the literatures (Kamal, 1973; Ramey, 1970). One can generate similar type curves for different reservoir conditions to cover a wider range of parameters. A particular type curve is defined for given parameters that can be set and retrieved through these properties. Figure 2-6 illustrates a set of type curves which show the relations between dimensionless rates and dimensionless pressure amplitudes for variety of . Figure 2-6 Dimensionless rate effect on dimensionless pressure amplitude Here, pressure amplitude is defined the difference between the pressure at time lag and the pressure when the rate change started. The pressure amplitude is the largest in the case of shut in the active/pulsing well ( . 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0.0 0.2 0.4 0.6 0.8 1.0 ∆p / ∆p(at q D =∞) 1/q D (r^2/∆t)(S/T)=1.39E-08 8.33E-08 1.85E-07 3.89E-07 3.89E-07 1.25E-06 22 2.1.2 Storativity/Transmissibility Ratio The definition of storativity and transmissibility simplify the “S/T” ratio to “ . This ratio is the inverse of hydraulic diffusivity coefficient which is the fundamental grouping of parameters that plays a major role in this study. As mentioned, the time lag values depend on T and S between pulsing and observation wells. Our goal in this study is to approximate the fluid-fluid interface location embedded in average transmissibility and storativity between two wells with continuous time lag monitoring. Figure 2-7 shows the relationship between time lag and the ratio of “S/T” at each time to “S/T” at t 0. Figure 2-7 Dimensionless (S/T) effect on time lag in radial system We assumed the displacing fluid with higher “S/T” replaces the displaced fluid with lower “S/T” ratio. At t 0 the original fluid between wells indicates a specific time lag which represents “(S/T) 0 ”. The flooding process causes decreasing of average “S/T” 0 20 40 60 80 100 120 140 160 180 200 0.0 0.2 0.4 0.6 0.8 1.0 t L (min) (S/T)/(S/T) 0 qD=∞ qD=10 qD=3.33 qD=2 qD=1.43 qD=1.25 23 between wells. The Dimensionless “S/T” compares “S/T” at each time to “S/T” at time zero. Then the “x” axis starts from one that indicates the beginning of flooding process to the small value that represents the ratio of the interwell “S/T” at the end of the flooding process to initial “(S/T) 0 ” which has non zero value. For a given dimensionless rate, the time lag decreases with decreasing the dimensionless “S/T”. This trend shows the displacement of a fluid with lower transmissibility and/or higher storativity with a fluid with higher transmissibility and/or lower storativity. This concept will be discussed in two-phase flow system. The dimensionless rate is defined and varies from infinity to one. When the pulsing well is shut in, the dimensionless rate is infinity and when there is no change in the pulsing well rate the dimensionless rate is one. The partial rate change at pulsing well varies the dimensionless rate between infinity to one. Figure 2-7 shows the effect of dimensionless “S/T” on time lag for variant values of dimensionless rate. As we expected the infinite dimensionless rate causes the strongest pulse which makes smaller time lag. On the other hand, in the case of lower dimensionless rate the time lag is more sensitive to dimensionless “S/T”. Our strategy is to normalize these curves and develop the type curves for each parameter. Figure 2-8 shows the same information as Figure 2-7 for dimensionless time lag. In this Figure “y” axis is also normalized and the time lag values have been compared with the first one at t 0 . The first advantage of this approach is to provide a series of type curves which have similar trends. 24 These type curves fit on the second order polynomial with linear derivatives. Figure 2-9 shows the linear trend of the first derivative of dimensionless “S/T” with respect to dimensionless time lag. Figure 2-8 and Figure 2-9 both can be used to obtain the interwell average “S/T” with measuring time lag. Figure 2-8 Dimensionless (S/T) effect on dimensionless time lag in radial system 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t L /t Lmax (S/T)/(S/T) 0 qD=∞ qD=10 qD=3.33 qD=2 qD=1.43 qD=1.25 25 Figure 2-9 First derivative of dimensionless (S/T) vs. dimensionless time lag Figure 2-10 Viscosity effect on time lag Figure 2-10 and Figure 2-11 show the effect of viscosity and total compressibility 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 d[(S/T)/(S/T) 0 ]/d[t L /t Lmax ] (S/T)/(S/T) 0 qD=∞ qD=10 qD=3.33 qD=2 qD=1.43 qD=1.25 0 10 20 30 40 50 60 1 2 3 4 5 6 7 8 9 10 t L (min) µ (cp) qD=∞ qD=10 qD=3.33 qD=2 qD=1.43 qD=1.25 26 respectively when the other parameters are constant. These Figures illustrate the cases which one phase flow presence in the reservoir and relative permeabilities don’t have any effect. The advantage of these illustrations is to emphasize the effect of total compressibility and viscosity individually on the time lag values. Figure 2-10 shows that with increasing of viscosity, the time lag increases for a given dimensionless rate. Increasing the viscosity causes lower transmissibility which affect on time lag. As we expect the higher dimensionless rate dilute the effect of viscosity. Higher compressibility damps the pulse energy and increases the time lag. High compressibility could be a representative of gas injection. Figure 2-11 represents the effect of total compressibility on time lag for variant dimensionless rate. It is obvious the compressibility has significant effect on the time lag. Figure 2-11 Total compressibility effect on time lag In the case of gas flooding, gas properties such as viscosity and compressibility depend 0 200 400 600 800 1,000 1,200 1,400 1,600 1,800 2,000 0 10 20 30 40 50 60 70 80 90 100 t L (min) c t ×10 6 (1/psi) qD=∞ qD=10 qD=3.33 qD=2 qD=1.43 27 on pressure, temperature and gas compositions. Figure 2-12 The effect of viscosity and compressibility for gas reservoir on time lag Figure 2-12 shows the same effect on time lag when the gas compressibility and viscosity changing accordingly. It should also be noted that this graph is for single gas phase and the effect of relative permeability has been ignored. The time lag values have been calculated for a gas reservoir in radial coordinate. 2.1.3 Interwell Distance Figure 2-13 shows the effect of dimensionless rate on dimensionless time lag for given interwell distances. For larger interwell distance, the changes of the time lag with dimensionless rate are more significant than smaller distance. For instance for the same dimensionless rate value, the associated time lag to the larger interwell distance is greater than the time lag associated with the smaller interwell distances. That means it takes 0 200 400 600 800 1,000 1,200 1,400 1,600 0.020 0.025 0.030 0.035 0.040 0.045 0.050 t L (min) µ (cp) qD=∞ qD=10 qD=3.33 qD=2 qD=1.43 qD=1.25 28 more time for the pulses caused by the rate changes to travel to the observation well and be captured. As shown earlier in Figure 2-7, the lower values of “S/T”, pulses caused by the rate changes may provide almost the same time lag as shutting in this well. Obviously these values are small and very sensitive sensors are required to detect the pulses. The same concept also applies for well distance. Figure 2-13 The effect of dimensionless rate on time lag for selection of well distance Figure 2-14 shows the relationship between dimensionless “S/T” and dimensionless time lag for different interwell distances for a given dimensionless rate. This graph also provides a series of type curves that can be used for prediction of dimensionless “S/T” with having time lag in a given condition. Figure 2-15 demonstrates the relation between time lag values with interwell distance, as a function of dimensionless rate. This Figure shows when the active well is shut in to 1.0 1.2 1.4 1.6 1.8 2.0 0.00 0.20 0.40 0.60 0.80 1.00 t L / t L (at q D =∞) 1/q D r= 100 (ft) r= 200 r= 300 r= 400 r= 500 29 Figure 2-14 the effect of dimensionless “S/T” on dimensionless time lag for selection of well distances Figure 2-15 Time lag vs. interwell distance for selection dimensionless rates 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t L /t Lmax (S/T)/(S/T) 0 r= 100 (ft) r= 200 r= 300 r= 400 r= 500 qD=4 0 100 200 300 400 500 600 700 50 100 150 200 250 300 350 400 450 500 t L (min) Distance (ft) qD=∞ qD=10 qD=3.33 qD=2 qD=1.43 qD=1.25 30 generate a pulse, the interwell distance has less effect on time lag. On the other hand, for the lower dimensionless rates, the time lag is more sensitive to the interwell distance. Figure 2-15 also indicates for very close well distance, all of the curves are overlap and even with small changes in rate, the observation well detects the same response. 2.1.4 Injection Period Figure 2-16 illustrates the effect of injection period on time lag which is based on Equation (2-5). This Figure shows the effect of dimensionless “S/T” on time lag at various injection periods for a given dimensionless rate. Longer injection periods cause smaller time lag value at the same given condition of dimensionless rate and “S/T”. The injection duration before pulse indicates the stabilization of reservoir. Equation (2-4) simply shows longer injection or flowing periods damp the first term in the bracket and have less effect in the first derivative. Figure 2-16 illustrates how the variation of injection rate changes the relationship between dimensionless “S/T” and the time lag. Figure 2-17 shows the same behavior in dimensionless format which organized in a series of type curves. 31 Figure 2-16 The effect of dimensionless “S/T” on time lag for various injection rate periods Figure 2-17 The effect of injection rate period on dimensionless “S/T” vs. dimensionless time lag in type curve format. 0 50 100 150 200 250 300 350 400 0.0 0.2 0.4 0.6 0.8 1.0 t L (min) (S/T)/(S/T) 0 ∆t=1E3 (min) ∆t=1E4 ∆t=1E5 ∆t=1E6 q D =4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t L /t Lmax (S/T)/(S/T) 0 ∆t=1E3 (min) ∆t=1E4 ∆t=1E5 ∆t=1E6 q D =4 32 2.1.5 Discussion In this section, the effective parameters on the time lag are studied. The time lag is sensitive to injection period, interwell distance, rate variation and “S/T”. The variations of these parameters cause significant changes on the time lag values. On the other hand, normalized curves provide series of type curves. One can generate all possible type curves and expand them for wider range of effective parameters. As mentioned, these type curves have been generated based on the first pulse and don’t cover the following pulses in a system. The variation of rate causes unsupervised multiple pulsations during the injection process that will be discussed in the next section. 2.2 Multiple Rate Model In reality, unsupervised rate changes results in unstructured pulse tests. We now examine for a general case, the multiple rate changes and derive the predicted pulse responses caused by such variations in rate. Figure 2-18 shows a schematic of this case with related terminologies. The Equation (2-2) represents the general form of pressure response at observation wells, together with its first and second derivatives. 2-2 2-7 33 2-8 In this section we focus on the first three points which indicate first three time lags caused by first three rate changes at pulsing well. These points are indicated in Figure 2-18 with “A”, “B”, “C” accordingly. Figure 2-18 A schematic of unsupervised rate changes at active well and pressure responses at observation well The root of the first derivative for the points “A”, “B”, and “C” in Figure 2-18 are as follow: Flow Rate (bbl/D) Time Injection Rate Pressure Response t 1 t 2 t 3 t L q 1 q 2 q 3 q 4 q 5 t 5 t 4 q Di q i q i-1 =---- A B C 34 2-9 2-10 2-11 In the previous section as shown in Equation (2-5), we indicated that the time lag is independent of the rate. This argument is valid only for the first pulse. Equation (2-9) shows the first time lag at point “A” depends on second dimensionless rate only which is defined as: . The first time lag doesn’t change as long as the associated dimensionless rate is constant. Figure 2-19 show the effect of first injection rate (q 1 ) on the time lag when the second dimensionless rate (q D2 ) is not constant. All other variables such as rates and the duration of each rate change are given. The time lags’ trends are subjected to other rates and duration of each rate. Figure 2-19 has been constructed based on equal pulse and injection period. 35 On the other hand, Figure 2-20 illustrates the same behavior when the second dimensionless rate is constant. Dotted lines for the second and third time lag are copied from Figure 2-19. This Figure supports the hypothesis of dependency of the first time lag only on first dimensionless rate. Figure 2-19 Time lag changes vs. the first injection rate (q 1 ) when the second dimensionless rate is variable 100 150 200 250 300 350 400 450 500 550 0 100 200 300 400 500 600 700 800 900 1000 Time Lag (min) 1st Injection Rate, q 1 (bbls/d) 1st time lag 2nd time lag 3rd time lag 36 Figure 2-20 Time lag changes vs. the first injection rate (q 1 ) when the second dimensionless rate is constant However, to solve the Equations (2-10) and (2-11), the dimensionless rates and actual rate needed to be considered. This argument emphasizes that the injection rates affect the time lag from the beginning of flooding process. However, above Equations also show with increasing time, the coefficients to the early rates are vanished and have less effect. Then only later rates would affect on the time lags. For instance in Equation (2-11) the multiplied bracket to q 1 has two terms. The nominator term increases with increasing the dimensionless time. On the other hand, the nature of exponential stabilizes this term and its asymptote would be one. The denominator term increases with increasing the dimensionless time as well. The result of increasing the dimensionless time lag vanishes this bracket and dilutes the effect of rates. Figure 2-21 shows the second and third time lag values flatten with increasing the first injection rate period (t 1 ). The result of above 100 150 200 250 300 350 400 450 500 550 50 100 150 200 250 300 350 400 450 500 Time Lag (min) 1st Injection Rate, q 1 (bbls/d) 1st time lag 2nd time lag 3rd time lag 2nd time lag (q2=cte) 3rd time lag (q2=cte) 37 discussion clarifies in the Equation (2-7) the most recent rate and rate changes have more influence on the latest time lag. Figure 2-21 The effect of the first injection rate period to the following time lags 2.3 Parameter Sensitivity Based on the Second Derivative By definition, the pressure response function at observation well (called f) has a local maximum at a point t=t Lmax if for all of t in a range of R where f has been defined. In this case R is defined as the range between the pressure that are associated when pulse starts and the pressure at the next rate change. Similarly, f has a minimum at t Lmin if for all of t in R. If f is a differentiable function and has maximum and/or minimum at point t L in R, then the first derivative must be zero. If the first derivative provides information about the rate of change of a pressure, the second 150 200 250 300 350 400 0 5,000 10,000 15,000 20,000 Time Lag (min) 1st Injection Period, t 1 (min) 2nd time lag 3rd time lag 38 derivative provides information about the rate of change of the first derivative. An inflection point is a point on a pressure response curve at which the curvature changes. In this study, we are interested to have the starting point of pressure response deviation when the rate changes, compared to the case of no changes in rate. One could use this deviation point as new definition for time lag. Technically, the deviation point and inflection point are different. However, we examined that these two points overlap in the case of high transmissibility and low storativity. In this section we assume the deviation point to be inflection point and use for time lag identification. As mentioned, to find the extrema and inflection point, the first and second derivatives are set to be zero respectively. The simplified forms of the first and the root of the second derivative of single pulse are shown in Equation (2-12) and (2-13) respectively. 2-12 2-13 Because the parameter sensitivities based on first derivative have been discussed in previous section in detail, the difference between these two characteristics are discussed in this section. In general the second derivative’s root is smaller than the first derivative’s root in pressure response. 39 Figure 2-22 The comparison of dimensionless (S/T) effect on time lag between extrema (first derivative) and Inflection point (second derivative) for variable dimensionless rates Figure 2-22 emphasizes that the time of inflection point is much smaller than the time of maximum or minimum for the same pulse. It also clarifies that the inflection points’ trends are similar and if these points are detectable, that would be a very good time lag indicator. Figure 2-23 and Figure 2-24 point out the same concept which is applicable for various interwell distance and injection duration respectively. 0 20 40 60 80 100 120 140 160 0.0 0.2 0.4 0.6 0.8 1.0 t L (min) (S/T)/(S/T) 0 qD=∞ (2nd) qD=10 (2nd) qD=3.33 (2nd) qD=2 (2nd) qD=1.43 (2nd) qD=∞ (1st) qD=10 (1st) qD=3.33 (1st) qD=2 (1st) qD=1.43 (1st) 40 Figure 2-23 The comparison of dimensionless (S/T) effect on the time lag between extrema and inflection point for variable interwell distance for given dimensionless rate Figure 2-24 The comparison of dimensionless (S/T) effect on time lag between extrema and inflection point for variable injection periods for given dimensionless rate 0 50 100 150 200 250 300 350 400 450 0.0 0.2 0.4 0.6 0.8 1.0 t L (min) (S/T)/(S/T) 0 r= 300 (2nd) (ft) r= 400 (2nd) r= 500 (2nd) r= 300 (1st) r= 400 (1st) r= 500 (1st) qD=4 0 50 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 t L (min) (S/T)/(S/T) 0 ∆t=1E3 (2nd) (min) ∆t=1E4 (2nd) ∆t=1E5 (2nd) ∆t=1E3 (1st) ∆t=1E4 (1st) ∆t=1E5 (1st) qD=4 41 Figure 2-25 the pressure difference measurement causes by rate change based on second derivative The main assumption in the above Figures is measuring the pressure difference at the starting point of rate change and inflection point. Figure 2-25 demonstrates the order of magnitude of pressure changes for variety of reservoir conditions. The range of pressure differences in some cases varies between 10 -4 to 10 -2 psi. Most of pressure sensors in current technology don’t have the accuracy to capture this range of pressure. However, one could implement this technique for applicable reservoir condition. Figure 2-26 and Figure 2-27 demonstrate the type curves of dimensionless rate effect on dimensionless pressure changes and associated time lag respectively. 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 1.0E+01 0.0 0.2 0.4 0.6 0.8 1.0 ∆p (psi) 1/q D (r^2/∆t)(S/T)=1.39E-08 8.33E-08 3.89E-07 3.89E-07 1.25E-06 3.06E-06 42 Figure 2-26 The dimensionless rate effect on dimensionless pressure difference based on second derivative Figure 2-27 The dimensionless rate effect on dimensionless time lag based on second derivative 1.0 1.1 1.2 1.3 1.4 0 0.2 0.4 0.6 0.8 1 ∆p/∆p (at q D =∞) 1/q D (r^2/∆t)(S/T)=1.39E-08 8.33E-08 1.85E-07 3.89E-07 1.25E-06 3.06E-06 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 0 0.2 0.4 0.6 0.8 1 t L /t L (at q D =∞) 1/q D (r^2/∆t)(S/T)=1.39E-08 8.33E-08 1.85E-07 3.89E-07 1.25E-06 3.06E-06 43 2.3.1 Discussion A main assumption in this research is that high resolution continuous pressure measurements at observation wells are available. In the section 2.1 we focused on single pulse in injection well. Equation (2-5) emphasized that in addition to transmissibility, storativity, interwell distance, and injection period, only dimensionless rate affected the time lag values. On the other hand, Equation (2-4) point out the pressure change is affected by actual injection rate in addition to other parameters. Figure 2-28 and Figure 2-29 illustrate the calculated pressure change at observation well with dimensionless rate for variable initial injection rate. Furthermore, the interwell distance has significant influence on pressure changes and related time lags. The time lag decreases for smaller interwell distance between observation well and active well. Figure 2-28 The injection rate effect on pressure difference measurement based on first derivative 0.0 0.3 0.6 0.9 1.2 1.5 0.0 0.2 0.4 0.6 0.8 1.0 ∆p (psi) 1/q D q1=100 (1st)(bbl/d) q1=500 (1st) q1=1000 (1st) q1=1500 (1st) q1=2000 (1st) 44 Figure 2-29 The injection rate effect on pressure difference measurement based on second derivative The argument in the previous section about available pressure measurement based on inflection point should be reconsidered case by case. In some reservoirs, the expected pressure change caused by rate change is significant enough, that current permanent downhole pressure sensors are able to detect these values. 2.4 Wellbore Storage and Skin Effects Wellbore storage effect causes based on the compressibility of the fluid in the wellbore. The effects of wellbore storage and skin on the pressure transient data was introduced by Van Everdingen (Van Everdingen, 1953) and Hurst (Hurst, 1953). They introduced the presence of an “infinitesimally” thin; steady state skin at the wellbore that reduces the productivity. The analysis of the influences of skin and wellbore storage effects on short time pressure data was extensively studied in 1970 (Agarwal et al., 1970; Ramey, 1970; 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.20 0.40 0.60 0.80 1.00 ∆p (psi) 1/q D q1=100 (2nd) q1=500 (2nd) q1=1000 (2nd) q1=1500 (2nd) q1=2000 (2nd) 45 Wattenbarger and Ramey, 1970). They introduced the type curve matching to analyze the effect of wellbore storage. They made correction and improved the interpretations of early data of interference well test influenced by wellbore storage and skin. However, these studies investigated the pressure response at production and/or observation well in presence of wellbore storage and skin at production well. Effect of skin and wellbore storage under well interference and pulse tests were reported by Prats et al. in 1975 (Prats and Scott, 1975). They showed that wellbore storage at the responding well increase the time lag and decrease the pressure response amplitude. That means, ignoring the wellbore storage in the pulse test would overestimate the storativity and underestimate the transmissibility. Jargon (Jargon, 1976) concluded the same results considering the effect of skin and wellbore storage at active well and evaluated the pressure response at observation well. He showed that the transmissibility between wells is low and storativity is high if the wellbore storage and skin are neglected. In 1979 Mondragon (Mondragon and Menzie, 1979) introduced an empirical method to make the correction for pulse test result with wellbore storage at pulsing well based on previous work (Jargon, 1976). The pressure responses at the observation well for interference test considering wellbore storage and skin in both wells, active and observation well, was introduced in 1977 (Fenske, 1977). He showed the dependency of pressure response on the magnitude of wellbore storage and interwell distance and illustrated that the effect of wellbore storage is more significant. 46 The reciprocity principle for interference test had been considered later on (Tongpenyai and Raghavan, 1981). They showed if the skin and wellbore storage exist on only one well, either at active well or observing well, same pressure responses will be recorded. The effect of wellbore storage and skin on the pulse test, when the wellbore storage exists at acting well or both at acting and observation wells, quantitatively studied (Ogbe and Brigham, 1987; Ogbe and Brigham, 1989). In this research, we used their mathematical formulation to demonstrate the effect of wellbore storage and skin on the time lag. 2.4.1 The Reciprocity Principle The reciprocity principle in well testing may be stated as follows: “The pressure drop at time t at an arbitrary point A in a heterogeneous and anisotropic porous medium as a result of production/injection at point B will be identical to the pressure drop that would result at point B at time t owing to production/injection at point A.” (Raghavan, 1993) The reciprocity principle was introduced by McKinley for the first time on interference test analysis (McKinley et al., 1968). Tongpenyai at al. (Tongpenyai and Raghavan, 1981) showed that in presence of wellbore storage and skin at one of the wells, either active or responding well, the pressure response at observation well is the same for the same values of wellbore storage and skin. In 1987, Ogbe et al. showed that that the pressure response at observation well depends on five dimensionless lumped numbers: , and ; where the refer to active and observation well. With respect to reciprocity principle, they concluded the same pressure response 47 would be obtained at observation well for the same values of and only either for active well or observation well (Ogbe and Brigham, 1987; Ogbe and Brigham, 1989). They developed their formulation for single phase fluid in an infinite, homogenous, uniform, and isotropic porous medium. The fluid has small and constant compressibility and constant viscosity. Small pressure gradient was assumed and gravity was neglected. The dimensionless pressure drop (Equation(2-2)) in Laplace domain at any dimensionless distance (r D ) and dimensionless time (t D ) is(Ogbe and Brigham, 1987): 2-14 Where “l” is Laplace transform variable for t D and s 1 , C D1 , s 2 , C D2 are skin factor and dimensionless wellbore storage constant for observation and active well respectively. The pressure behavior at responding well in Laplace domain is (Ogbe and Brigham, 1987; Ogbe and Brigham, 1989): 2-15 It is obvious the dependency of pressure responses in Equation (2-15) to wellbore storages and skin factors represent in the denominator. The exchange of the corresponding parameters of acting and observation well will yield the same result. This behavior is the basis for reciprocity principle. 48 2.4.2 Wellbore Storage and Skin Effects on Time Lag We expect that wellbore storage at the responding/production or active/injection well increase the time lag and decrease the pressure response amplitude. Furthermore, the reciprocity principle clarifies that skin and wellbore storage exist either at observation or activate well have the same effects on time lag and pressure amplitude. Equation (2-15) shows the pressure response at the observation well in Laplace domain when a series of rate changes take in place at active well. The superposition principle applies to Equation (2-15) and the result in general form at observation well would be as follows: 2-16 The derivative of Equation (2-16) is: 2-17 In this section, the pressure behavior and its derivative for the first pulse are studied. The following Equations are simplified format of Equation (2-16) and (2-17) for a single pulse. 2-18 2-19 Figure 2-30 illustrates the pressure responses with and without wellbore storage and skin. 49 These parameters delay to well responses and increase the time lag and decrease the pressure amplitude. To study these effects we refer back to transmissibility and storativity definitions. Transmissibility incorporates both formation and fluid properties and represents the ease with which fluids are transmitted through the formation. The positive skin acts in the way that a low transmissible zone around the well, either active or observation well, decreases the total transmissibility. The wellbore storage at active or observation well increases the pore volume and its compressibility which is the same as fluid compressibility. Figure 2-30 The effect of wellbore storage and skin on pressure at observation well Figure 2-31 demonstrates the effect of wellbore storage on time lag for constant skin factors for both wells. This Figure clarifies that even with presence of wellbore storage and skin, rate changes have the same behavior as before. On the other hand, for given rate changes for the first pulse, increasing the wellbore storage increases the time lag. 0 0.2 0.4 0.6 0.8 1 1.2 20 25 30 35 0.95 0.97 0.99 1.01 1.03 1.05 1.07 1.09 1.11 1.13 1.15 Dimensionless Rate ∆p (psi) t /∆t Without CD & S With CD & S Dimensionless Injection Rtae 50 Figure 2-31 The effect of wellbore storage on time lag for selected dimensionless rate Figure 2-32 The effect of wellbore storage on time lag for various skin factors (q D = ∞) 0 50 100 150 200 0 5,000 10,000 15,000 20,000 Tiime Lag (min) CD qD=∞ qD=4 qD=2 qD=1.33 S 1 =S 2 =2 0 50 100 150 0 5,000 10,000 15,000 20,000 Time Lag (min) CD S1=S2=1 S1=S2=2 S1=S2=3 q D = ∞ 51 Figure 2-33 The effect of wellbore storage on time lag for various skin factors (q D =4) Figure 2-32 and Figure 2-33 show the effect of wellbore damage on time lag for two given dimensionless rate. As we expect higher skin factors, cause larger time lags. These Figures are generated based on the solution for Equation (2-15) for single pulse which is simplified in Equation (2-19). The wellbore storage and skin factors for both well are substantiated in the denominator of Equation (2-15) and the exchange of these two sets of parameters don’t affect on the time lag result. 2.5 Composite Reservoir The main goal of this research is to estimate the front location by tracking the time lag. In immiscible displacement, if the transition zone between displacing and displaced fluids is assumed to be small, the system becomes analogous to a composite reservoir with two 0 50 100 150 0 5,000 10,000 15,000 20,000 Time Lag (min) CD S1=S2=1 S1=S2=2 S1=S2=3 q D = 4 52 zones with different fluid properties. A composite reservoir includes at least two zones with different formation and fluid properties. For instance, the influence of skin around the well, and an injection process with a bank of fluid in the reservoir could be represented with composite system. The main difference between these two above examples is that, the interface boundary between damaged or stimulated zone and original formation is almost stationary and the other one is dynamic. The only concern is that the fluid-fluid interface in injection process is dynamic and moves with time. In this section, we investigate and study the front location by representing an injection process in a homogeneous reservoir as a dynamically changing composite system. Profiling temperature distribution in a composite reservoir with implementation of Laplace Transformation technique was introduced in 1941 (Jaeger, 1941). In the petroleum engineering literatures, composite reservoir were examined early 1960s (Carter, 1966; Hurst, 1960). The mathematical solutions of composite reservoir were summarized for different geometry by Ambastha (Ambastha, 1988) and Issaka (Issaka, 1996) . They assumed the effect wellbore storage and skin at discontinuity, around the well, are negligible. A radial composite system will be used to simulate a homogeneous system under flooding process and study the discontinuity interface movement. The diffusivity Equations for radial system in composite reservoir are discussed in literature (Ambastha, 1988; Issaka, 1996) and the solutions in Laplace space are: 53 2-20 2-21 Where “l” is the transformed time variable, in Laplace transform. The Stehfest algorithm is used to obtain pressure responses at each zone. The superposition on rate applied to Equation (2-21) which represents the pressure behavior at the observation well. Equation (2-22) and (2-23) are the pressure behavior for the first pulse and the first derivative equivalent to zero, respectively. 2-22 2-23 The results of Laplace inversion would be pressure response at observation well. Next few Figures are based on a radial composite system to illustrate the importance of effective parameters on time lag when the discontinuity boundary (fluid-fluid interface) moves from active well toward observation well. Figure 2-34 demonstrates the effect of discontinuity movement on the time lag. Based on discontinuity boundary movement the average transmissibility between two wells increases and the result is decreasing the time lag values. In comparison between various dimensionless rates, Figure 2-34 shows higher value makes lower time lag. 54 Figure 2-34 The effect of dimensionless discontinuity interface distance on time lag for selected dimensionless rates Dimensionless time lag has been constructed the way to carry on the reservoir static properties and interpret the dynamic part. The time lag, that intentionally being measured, when the displacing fluid hasn’t been invaded to the reservoir is called initial time lag. In the composite system when the discontinuity boundary is close to the injection well the first time lag is measured. We use this time lag as a base to compare other values to initial time lag. This normalization generates a set of type curve which usually collapse to the narrower region. 0 50 100 150 200 250 300 350 0.0 0.2 0.4 0.6 0.8 1.0 Time Lag (min) Dimensionless Discontinuity Interface Distance qD=∞ qD=4 qD=2 qD=1.34 T 1 /T 2 =2.0 S 1 /S 2 =0.5 55 Figure 2-35 The effect of dimensionless discontinuity interface distance on dimensionless time lag for selected dimensionless rates Figure 2-35 reconstructed from Figure 2-34 to show the variation of dimensionless time lag with dimensionless discontinuity movement. This Figure shows the trends of various dimensionless rates are very similar. 2.6 Reservoir Parameters Distribution In this research we will examine few general 2D cases to illustrate the effect of reservoir heterogeneity on the time lag. As shown in methodology, the time lag value depends on the average interwell properties. The average transmissibility directly related to the absolute permeability distribution in the reservoir and eventually related to the time lag. However, in general we used a single value for absolute permeability to represent the interwell permeability distribution as shown below. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless Time Lag Dimensionless Discontinuity Interface Distance qD=∞ qD=4 qD=2 qD=1.34 56 Log-normal distribution is one the most common distribution for absolute permeability. If we assume a log-normal permeability distribution based on a mean and standard deviation values, we may expect different time lags. A log-normal distribution with mean value of three and standard deviation of one, both in logarithmic scale has been assumed for Figure 2-36. Figure 2-36 Log normal distribution for absolute permeability with mean of three and standard deviation of one in logarithmic scale It is obvious that an algorithm needed to average out the permeability distribution to a single value to obtain the time lag. On the other hand the time lag values have been calculated for the above permeability 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 110 120 130 Frequency Absolute permeability (md) 57 distribution and Figure 2-37 shows the distribution of the time lags. The histogram very similar to a log-normal distribution, but it is not necessarily true. Figure 2-37 The time lag distribution which obtained from log-normal absolute permeability distribution We introduce the relationship between time lag and absolute permeability distribution in this section. With the assumption of single phase and constant thickness the transmissibility also represents the absolute permeability. We wish to determine the time lag as a function of transmissibility distribution. Determination of the density of a function in term of the density of another variable discussed (Papoulis and Pillai, 2002). The result for our case illustrated as below: 2-24 In the above Equation is calculated from Equation (2-5) as shown below: 0 10 20 30 40 50 60 70 80 0 50 100 150 200 250 300 350 400 450 500 Frequency Time lag (min) 58 2-25 On the other hand the transmissibility follows the log-normal distribution: 2-26 Equation (2-25) and (2-26) are substituted to the Equation (2-24) and result in Equation (2-27). 2-27 Equation (2-27) shows the behavior of time lag based on log-normal distribution of transmissibility. This Equation is similar to log-normal distribution, but this argument is not necessarily true. 59 Chapter 3 Dynamic Discontinuity Effect on the Time Lag In Chapter 2, we reported the results on the influences of average interwell parameters such as transmissibility and storativity. However, these values change with fluid movement. In this Chapter, we investigate the effect of fluid movement in a linear system. 3.1 Linear System The theoretical behavior of a linear system for an oil reservoir adjoining a high pressure aquifer has been described by Miller (Miller, 1962). His solution is modified for linear oil reservoir including wells as shown by Ehlig-Economides, et al (Ehlig-Economides and Economides, 1984). The solution for an infinite reservoir and assuming that half of the production comes from observation side toward the production well, can be shown as follows (Ehlig-Economides and Economides, 1984; Miller, 1962): 3-1 Therefore the pressure behavior for superposition of rates can be expressed as: 60 3-2 Where the variables in field units are: Dimensionless radius: Defined pulse test pressure (bbl/d): Dimensionless time: Dimensionless rate: Transmissibility (md.ft/cp): Storativity (ft/psi): Time (min): t Similar to the radial system, from the derivative of Equation (3-2), we obtain the relationship between the first time lag with rock and fluid properties as follows: 3-3 Equitation (3-3) represents a homogeneous reservoir with average transmissibility and storativity. The average transmissibility is harmonic for lateral discontinuity and average storativity is arithmetic. One could study the parameters effectiveness such as “S/T”, dimensionless rate and injection period time on time lag for linear system as same as 61 radial system. In this Chapter we focus on two-phase flow for linear systems. Figure 3-1 shows the relationship between dimensionless time lag and dimensionless rate for various values of in the Equation (3-3). Similar to the radial system, for lower value of the effect of rate on the time lag is small. With increasing this value the sensitivity of time lag with dimensionless rate increases. Figure 3-2 shows the relationship between dimensionless time lag and the dimensionless “S/T” for a series of dimensionless rate. These type curves are similar to the type of curves in radial system. Figure 3-1 Dimensionless rate effect on time lag in linear system 1.0 1.5 2.0 2.5 3.0 0 0.2 0.4 0.6 0.8 1 t L / t L (at q D =∞) 1/q D (L^2/∆t)(S/T)=1.39E-08 8.33E-08 1.85E-07 3.89E-07 62 Figure 3-2 Dimensionless (S/T) effect on dimensionless time lag in linear system The fractional flow theory (Buckley and Leverett, 1942) can be applied for immiscible displacement. This theory illustrates a saturation profile with a sharp front along the flow direction. Capillary pressure and gravity effects are ignored. In a case of uniform initial saturation, the breakthrough (BT) saturation can be obtained (Welge, 1952). With having saturation profile for both phases, the average lumped parameters such as transmissibility and storativity can be obtained. 3.2 Front Tracking in One Dimensional Immiscible Displacement For the estimation of fluid-fluid interface during an immiscible flooding process, we need to define the concept of the interface. The Buckley and Leverett frontal advance theory can be used to find the breakthrough saturation for a given fractional flow curve (Welge, 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t L /t Lmax (S/T)/(S/T) 0 qD=∞ qD=10 qD=3.33 qD=2 63 1952). In this study, we define the front to be the location of breakthrough saturation. Given the assumed relative permeabilities and the fractional flow curve, we can obtain breakthrough saturation from the frontal advance formula and will use this saturation to identify front location before breakthrough. 3.2.1 Buckley-Leverett Frontal Advance Theory Multiphase flow in porous media has been modeled by the Darcy’s law. This empirical relation modified for multi phase systems relates the pressure gradient with the velocity of each phase in porous media. To simplify this case we assumed a homogeneous system. The gravity, capillary pressure, compressibility and diffusion forces have been neglected. The viscosities of two phases are assumed to be constant. The Darcy’s law with these assumptions can be described as: 3-4 3-5 The proportionality factors between the velocities and corresponding pressure gradients in Darcy’s law include absolute permeability (k), viscosity (µ), and relative permeability (k r ) which are rock, fluid and rock/fluid properties respectively. Conservation of mass for 1-D with incompressible water and oil gives: 3-6 3-7 64 In this research we use Corey’s correlation for the relative permeability of oil and water as follows: 3-8 3-9 The fractional flow of water denoting the fraction of the total flow allocated to water, when the gravity and capillary pressure are neglected, is: 3-10 where and . The total mobility for each point in time and space is defined by and pressure distribution for the linear system is: 3-11 Equation (3-11) shows that pressure behavior depends on the integration of within the well distance in this case. For the numerical approach to obtain average transmissibility for given front location in this system, we break the saturation profile into small segments as shown in Figure 3-3. For infinite small increments, the assumption of linear change in saturation is valid and the relative permeability correlations provide these values for each increment. The total transmissibility of the segment after the breakthrough point can be represented by one value. The harmonic average of total transmissibility of each increment before 65 breakthrough added to mobility after breakthrough provides the total average transmissibility of the entire system. Figure 3-3 the integration of small segments before breakthrough to calculate total transmissibility in a 1-D system Figure 3-4 shows estimated total transmissibility with front location. The question is how we can predict average transmissibility from time lag value and related to front location for variable viscosity. This Figure illustrates the total transmissibility changes with increasing the oil viscosity. As expected, higher oil viscosity has lower total transmissibility for the same front location. These graphs have been generated based on given relative permeability correlations and could be different with respect to these correlations. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 S w Dimesionless Front Location 66 Figure 3-4 Total interwell transmissibility at various front locations The following example was generated to study the analytical solution for linear system and to compare the results with commercial reservoir simulation results. A similar linear case in reservoir simulator has been modeled with frequent pulses in the first year of reservoir life. In the analytical solution we consider similar values for pore volume injection (PVI) as shown in Figure 3-5. This Figure shows the estimated total transmissibility at various pore volume injected. 0.0 1.0 2.0 3.0 4.0 5.0 0.0 0.2 0.4 0.6 0.8 1.0 T otal transmissibility Dimensionless front location visc=2 cp visc=4 visc=6 visc=8 67 Figure 3-5 The front location and the computed total transmissibility at various pore volume injections The velocity of front movement vs. total transmissibility is a straight line (Haugse, 1996). It means with increasing the pore volume injection the saturation profile shape stretches and the integration of total transmissibility with respect to velocity doesn’t change. We have assumed that capillary pressure is neglected. This assumption is valid for not very large saturation gradients. On the other hand for large saturation gradients and in the presence of capillary pressure, the shock front won’t be sharp. Figure 3-6 shows the nature of transition zone when capillary pressures are important. On the other hand when the flow rates are high enough, the assumption of discontinuity in the saturation profile is valid. There are techniques to find the breakthrough saturation for given fractional flow data. The shock saturation at tangency point using Welge’s technique includes drawing a tangent on the fractional flow curve from initial saturation. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 S w Dimensionless front location PVI=0.0096, X_front=0.0225, T_total=0.2385 PVI=0.0642, X_front=0.1501, T_total=0.2731 PVI=0.1625, X_front=0.3803, T_total=0.3699 PVI=0.2609, X_front=0.6105, T_total=0.5727 PVI=0.3582, X_front=0.8382, T_total=1.2514 S wf =0.5649 68 Figure 3-6 Water saturation profile in 1-D system with and without capillary pressure As such, the inclusion of capillary pressure can have impact on the definition of the flood front. This effect may change the time lag behavior. In this study, we ignore the capillary pressure effect and our front location is based on the breakthrough saturation. A commercial reservoir simulator (Computer Modeling Group, CMG) has been used to simulate a similar case with analytical solution. In this model, the reservoir and fluid properties forced are similar to the properties for analytical solution. Figure 3-7 shows a schematic of a 1-D reservoir with one injector and one producer. The time lags are recorded five times in one year. The first one is after few days and other four time lags are recorded later. Because a fluid with higher mobility pushes a fluid with lower mobility, time lag decreases. The location of breakthrough saturation and corresponding 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 S w Dimensionless Front Location With Capillary Pressure Without Capillary Pressure Transition Zone 1-S or S wc S wf 69 time lag are shown in Figure 3-7. Figure 3-7 The commercial reservoir simulator results for 1D The well performance data provide the rate and pressure at injection and observation/ production well. The starting point of each significant rate changes, which indicates starting of the pulse, would be the basis for measuring the pressure changes at observation well. Figure 3-8 shows the obtained information from reservoir simulator for five pulses in various times in a linear model. This Figure clarifies that time lag values decrease with front movement. We verify our analytical solution with numerical approach. Figure 3-9 includes two sets of data. The lines in this graph are the outcome of the analytical solution for linear system. First using the Buckley-Leverett frontal advance theory for linear system, we obtain the average values of “S/T” for variable front location. These calculations are based on the corresponding rock and fluid properties. Then we obtain the time lag values 70 for variable “S/T” based on their relationship in linear system. Figure 3-8 Pressure difference at observation well vs. time, after starting pulse Figure 3-9 The comparison of analytical solution and reservoir simulator results -2.0 -1.0 0.0 1.0 2.0 3.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ∆p (psi) ∆t (day) Pulse 1, tL=0.705 d Pulse 2, tL=0.711 d Pulse 3, tL=0.618 d Pulse 4, tL=0.524 d Pulse 5, tL=0.433 d 0.018, 1045 0.135, 1023 0.355, 889 0.576, 754 0.798, 623 0 200 400 600 800 1,000 1,200 1,400 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time Lag (min) Dimensionless Front Location (Linear System) ∆t=9 d ∆t=60 d ∆t=152 d ∆t=244 d ∆t=335 d Reservoir Simulator Results Error= -8.4 % -8.4 % -2.7 % -4.8 % -9.1 % 71 Chapter 4 Analytical Solution vs. Reservoir Simulator In this section, we consider two-dimensional reservoir models with multiple wells. Implementing water injection with rate changes to cause pulsation provides the opportunity to test the proposed formulations and to compare the estimated values of the front location as obtained from the numerical simulation. In the simulation runs using the CMG models, an intentional pulse is included in injection well rate plan each three month. The first pulse happens after few days of injection and we assume that water injection has not significantly altered the “S/T” between the pulsing well and the responding well. In this Chapter we provide Figures that show the dimensionless time lag and front location for different rate changes. It is seen for large q D ’s, the estimated time lag is close to the time lag expected when q D is infinity (shut in well). However, for lower q D ’s and for the same front location, the estimated time lag takes longer and we may not be able to match the estimated front location. For the purpose of discussion here we refer q D as a measure of pulse energy. Higher q D ’s correspond to higher pulse energies. To simplify multiple well systems, a five spot pattern including an injection well at the center and four production wells at the corners of a reservoir is modeled. This five wells system is designed to be without interference from boundaries or other wells. We assume injection and production wells are recording the pressure and the rate continuously. 72 4.1 Example 1- Homogeneous System In this example we consider a 2-D homogeneous reservoir as depicted in Figure 4-1. Because the injection well is surrounded with four equally distanced producers and in a single horizontal layer with no gravity effects, we expect to have symmetric fluid movement towards all the production wells. We assumed the location of water breakthrough saturation represents the front. For a given relative permeability for water and oil the breakthrough saturation can be obtained ( =0.565). We generated pulses during the reservoir life. The first time lag is the bases to normalize later time lags and is used as a reference for comparison. Figure 4-1 shows the front location after one year. All grids with breakthrough saturation are specified with red color. Figure 4-1 The front location for a five spot pattern homogeneous model after one year The first column in Table 4-1 indicates the sequences of rate changes at the injection 73 well. These same schedules are applied for all the models under consideration here. Other three columns in Table 4-1 illustrate the associated time lag values for given dimensionless rates for homogeneous model. Dimensionless rates are selected to be infinity, five, and two when the injection rates went to zero, 20% and 50% of original rate respectively. Time lag values are in minutes. The red arrow shows the time lag increases with decreasing the dimensionless rate. The blues arrow shows that with fluid front movement toward producers, the time lag decreases as expected. It is obvious that these time lag values completely depend on the reservoir and fluid properties and could have various values for different system. Table 4-1 Injection rate schedules and related time lag values at producers for a five spot homogeneous model Time (day) Time lag (min) Case 1, q D = ∞ Case 2, q D =5 Case 3, q D =2 9 637 688 790 19 496 518 556 29 447 459 481 40 419 426 443 50 414 420 433 60 407 412 422 152 274 288 295 244 252 245 266 355 223 230 238 As Table 4-1 shows, for the early time of reservoir life, we have intentionally generated more pulses. Because he injection rate is constant, the velocity of front in the early time of flooding process is higher. The velocity of the front in a radial system is proportional with the inverse of radial distance. This fact results in faster change in average 74 transmissibility and storativity and eventually the time lag value. Figure 4-2 shows the time lag values vs. front location for three given dimensionless rates. By using reservoir simulator, we have the opportunity to examine the location of the front at any time. Because the fluid front moves very slowly in the reservoir, the assumption of freezing the front location for a short period of time is reasonable. Each model ran for three cases of different dimensionless rates. Because the injection rates have been changed for a short period of time for each pulse, the pore volume injection for these cases is almost the same; therefore, the front locations are expected to be the same. Figure 4-2 Time lag behavior with front movement for various dimensionless rate in a homogeneous model As shown, with the movement of the front and the original fluid replacement with higher 0 10 20 30 40 50 60 0 100 200 300 400 500 600 700 800 900 Front location (ft) Time lag (min) qD=∞ qD=5 qD=2 75 transmissibility fluid, the time lag decreases. Also as a result of front movement, the time lags are more controlled and dominated by higher transmissibility fluid and then the time lag values for various dimensionless time lags get closer to each other. Figure 4-3 shows the dimensionless time lag vs. dimensionless front location for one of the producers which represents all of the producers. Dimensionless time lag is defined as the time lag values at each time divided by first time lag. It means that the first values are one. The relation of dimensionless and corresponding front location provides useful information. For instance if we examine a piston-like displacement, the dimensionless time lag is one at the beginning of flooding process. There will another time lag value at the end of displacement. This value divided by initial time lag will be other limit for dimensionless time lag axis. On the other hand the dimensionless front location starts from zero to one respectively. In reality this relationship indicates rough information about interwell properties. We will discuss this subject in the next examples. 76 Figure 4-3 Dimensionless time lag behavior vs. dimensionless front location in a homogeneous model 4.2 Example 2- Gradient Heterogeneous System In the second example, a reservoir with gradient heterogeneity is modeled. The permeability decreases from left to right. We expect to have lower time lags in higher permeability area in addition to faster front movement in that direction. This fact indicates the trends of time lag changes are also different. Figure 4-4 shows the water saturation profile after a year in the reservoir and the location of breakthrough water saturation which is an indication of fluid front location specified with red color. This Figure shows the front moves much faster toward the left side than the right side. If one can predict this behavior before breakthrough happens, it could be a great tool for better reservoir management. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 77 Figure 4-4 The front map for a gradient heterogeneous reservoir (permeability decreases from left to right) This model is symmetric in the “y” direction and the same time lag values and behavior are expected in this direction. Figure 4-5 and Figure 4-6 show the dimensionless time lag vs. dimensionless front location for the producers 1 and 2 and the producers 3 and 4 respectively. These wells have the same behavior as expected and the time lag is decreasing with the movement of the front. But the time lag scale of these two Figures is important and illustrates that in high transmissivity zones, one observes the time lag much faster than that in lower transmissivity zones. The comparison of these two Figures illustrates that for higher transmissible zone almost three dimensionless rates have the same behavior and almost collapsed on top of each other. Furthermore, the front locations almost reach the producers in higher permeable zone but at the same time the location of the front in lower permeable zones reaches to 78 almost 50%. The third observation is the rate of time lag changes. For instance in the higher permeable zone the dimensionless time lag reaches to 30% and at the same time this value in the lower permeable zone is 50%. As it is expected the displacing fluid invasion in the lower permeable zone is much slower than the higher permeable zone. Figure 4-5 Dimensionless time lag behavior vs. dimensionless front location for production wells 1 and 2 in a reservoir with gradient heterogeneity 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 79 Figure 4-6 Dimensionless time lag behavior vs. dimensionless front location for production wells 3 and 4 in a reservoir with gradient heterogeneity Figure 4-7 The comparison of time lag behavior vs. front location for production well 1 in higher permeable zone and production well 4 in lower permeable zone 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 0 20 40 60 80 100 120 140 160 180 0 50 100 150 200 250 300 350 Front location (ft) Time lag (min) Production # 4, qD=∞ qD=5 qD=2 Production # 1qD=∞ qD=5 qD=2 Higher permeable zone Lower permeable zone 80 Finally Figure 4-7 shows the behavior of dimensionless time lag with fluid movement for two wells, one from higher and one from lower permeable zones. This Figure illustrates that the time lag changes with respect to front location for higher permeable zone is much faster than the lower permeable zone. Furthermore, it shows the similarity of time lag behavior at the first producer. 4.3 Example 3- Faulted System The third example is a homogeneous system with two faults. These faults are semi- sealing and almost sealing where limited communications are expected. These faults divide this reservoir into three compartmentalized zones. Producers 2 and 4 have reasonable communications with the injection well while limited communication exist for producers 1 and 3. According to reservoir heterogeneity, it is expected to observe a non- uniform front. Because of existing of the fault the front movements toward faults are expected to be limited. Figure 4-8 shows a schematic of this faulted reservoir and Figure 4-9 illustrates the front map for this model after almost one year. 81 Figure 4-8 A schematic of faulted reservoir Because of the front’s configuration, with different average interwell transmissibility and storativity, we expect that the time lag values and trends to be different for different wells. Figure 4-10 and Figure 4-11 illustrate the relation between dimensionless time lag and dimensionless front location for wells between the faults and behave exactly the same. 82 Figure 4-9 Water saturation map in faulted reservoir after one year Figure 4-10 Dimensionless time lag vs. dimensionless front location of production well 2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 83 Figure 4-11 Dimensionless time lag vs. dimensionless front location of production well 4 Figure 4-12 Dimensionless time lag vs. dimensionless front location of production well 1 Figure 4-12 illustrates that producer 1 has seen limited effect by injection well rate 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 84 changes. Because there is limited interaction between pulsing and observation well, in this case producer 1, there is a very small change in time lag values even with front movement. Because producer 3 is behind a sealing fault and there is no support and communication between the injector and producer 3, no time lag is detected. 4.4 Example 4- Channeled System The next case is the examination of the role of a high permeability channel in a reservoir as shown in Figure 4-13. Front location because of higher permeability is exhibited in Figure 4-14. Figure 4-15 to Figure 4-18 show this relationship for each well. Figure 4-13 A Schematic of Channeled reservoir 85 The shape and the location of channel are modeled in such a way that all wells have different behavior. Producer 2 is one the most influenced by the injection well and is expected to have a small time lag and fastest front movement. The Interwell properties between injector and producers 4 include the combination of higher and lower permeable zone. The interwell permeability distribution for producers 1 and 3 also show these wells are affected by high and low permeable zones. In sequence, the producers 2, 4, 3 and 1 respectively have the support from injection well. Figure 4-15 shows the producer 1 which has less support by injection well, has been less influenced by injection rate changes. The actual time lag values in some cases show significant differences for various dimensionless rates, but in dimensionless format we may see more similar behavior and trend. Figure 4-15 through Figure 4-18 illustrate the dimensionless time lag with front movement for the wells. 86 Figure 4-14 The water saturation map in channeled reservoir Figure 4-15 Dimensionless time lag vs. dimensionless front location for production well 1 which is located in lower permeable zone 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 87 Figure 4-16 Dimensionless time lag vs. dimensionless front location for production well 2 which is located inside the channel with the highest injector support Figure 4-17 Dimensionless time lag vs. dimensionless front location for production well 3 with interwell lateral discontinuity 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 88 Figure 4-18 Dimensionless time lag vs. front location for production well 4 Figure 4-19 The comparison of the rate of front location changes according to time lag Figure 4-19 shows for higher transmissible zones; the font movement is much faster with respect to time lag changes. That means front is moving faster with less time lag changes. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 3 4 5 6 Dimensionless front location / Dimensionless time lag Pulse Number Prod. #1 Prod. #2 Prod. #3 Prod. #4 Decreasing the interwell transmisibility 89 Chapter 5 Conclusions and Recommendations In this Chapter we will summarize the conclusions and provide few suggestions for future research. 5.1 Conclusions In an effort to monitor the location of flood front before breakthrough, we focused on tracking the dynamic position of the fluid-fluid interface during immiscible displacement of oil by water. In particular we developed analytical relationships between the position of the interface and the time lag of pressure or rate change response between injectors and producers. Given the possibility of continuous recording of rate and pressure data at the producing and injection wells, pressure pulses in pre-scheduled or unsupervised rate changes at injection wells can help in tracking and monitoring of fluid fronts. Intuitively one would expect that the measured time lag as a parameter in detecting the pulsation at an injection well relates to the transmissivity characteristics between the wells. We also focused on the sensitivity of this relationship to the strength of pulsation caused by rate changes rather than shut-in the injection well. We developed new analytical solutions for radial and linear flow systems. Our solution requires availability of a reference point for a pulse conducted at the start point of an injection process. The reference point captures the in-situ characteristics of the reservoir space between the two wells. With the assumption that static reservoir properties are constant with time, the time lag changes with time are correlated with the changes of 90 effective and composite transmissibility (T) and storativity (S) caused by the movement of the interface. In terms of detecting the time lag from pressure data measured at an observation well, we considered two separate definitions. The first is the difference between the time that a pulse is generated by rate changes and the time when identifiable responses occur in the observation well. Another definition is based on detecting the inflection point using the second derivative of the pressure response at the observation well. To examine a correlation between time lag and front location, for a one dimensional system, the Buckley and Leverett frontal advance theory was used in this study to find the breakthrough saturation for a given set of fractional flow data. We defined the moving fluid interface as the location of breakthrough saturation. We also showed how the capillary pressure impacts the tracking of flood front and accordingly affects the estimated time lag. We demonstrated that the time lag values increased when we incorporated the capillary pressure effects. We also examined interface movement in a composite one dimensional reservoir system. An important observation in this study is that the pulsation detected at the observation wells caused by rate changes in injection wells can indeed be used to track interface movements. Pressure changes in the observation must be detectable by sensitive sensors. We showed that limitations in pressure measurement accuracy negatively impact the time 91 lag detection. We showed that a lumped parameter, ( ), which is directly related to the initial storability/ transmissivity ratio controls the design of field injection rate change experiments. We showed for systems with higher static values of we can do the monitoring with minimal rate changes. One of our contributions is the development of type curves based on three dimensionless parameters. These parameters are dimensionless rate, dimensionless time lag and dimensionless lumped parameter of . The effects of wellbore storage and skin factor at the pulsing and/or observation well on time lag values were studied in this research. We showed that wellbore storage and skin at either responding or pulsing well increased the time lag and decreased the pressure response amplitude. In general case we examined unsupervised injection rate changes. In the case for a single pulse we showed the time lag depends on dimensionless rate only. On the other hand, for multiple rates, after first pulse, the time lags depend on dimensionless and actual rate. However, the effect of early rates values vanishes with time. That analysis demonstrated for time lag evolution in multiple rate case, only recent rate values were affected on time lag. Extrema and inflection point in pressure behavior at observation well, caused by injection 92 rate changes were examined. We showed that the inflection point also could be a potential to be used as an improved time lag indicator. We demonstrated that if the inflection point is detectable, use of this point generates better results than by using extrema. The inflection point happens much sooner than the extrema. Furthermore, we found the normalized type curves, with respect to initial time lag based on inflection points, collapses on top of each other and follow the same trend in different parametric studies. Following our methodology, we looked into a linear system. The intention was to reconstruct the time lag behavior based on fluid front movement. The main influence on the time lag has been caused by relative permeability effects in the immiscible displacement. In this study we used the Corey’s correlations. As indicated, we have undertaken the breakthrough saturation as an indication for front lactation. The capillary pressure has been ignored and the assumption of sharp front was valid. We divided the saturation profile into small segments from injection well to the front location. The total transmissibility was calculated by the harmonic average of mobilities at each segment. For a given set of relative permeabilities and other required information, we correlated the dimensionless front location with total transmissibility and eventually indirectly with the time lag. The above results were based on the assumptions of immiscible displacement, with primary focus on water flooding. The fluid interface is expected to be uniform in one 93 direction and other cases such as viscous fingering or thief zone effects on front location haven not been studied. In this Chapter, we will discuss some of possible extensions around this research topic. 5.2 Recommendations 5.2.1 Continuous Recording of Time Lag by Rate Changes When a production well produces at a constant rate, it may exhibit a varying bottomhole flowing pressure. Similarly, a well produced at a constant bottomhole pressure can exhibit a varying rate decline. The mathematics behind this study was based on constant rate boundary condition at the well. This assumption delivered the changes in pressure at observation well. As we discussed, we define time lag as the delay between starting time of rate disturbance at pulsing well and sensing the pressure at observation well. In this section, the assumption of constant bottomhole pressure will be reviewed very briefly. The concept is simply to monitor the rate changes at production wells according to any rate disturbance at an observation well. To do this we used a linear productivity model. The productivity index of a well is a parameter related to the rate and the difference between the wellbore pressure and the average reservoir pressure (Raghavan, 1993). The productivity index is often used as a measure of the capacity of a well. Once the well production is “stabilized”, the ratio of production rate to some pressure difference between the reservoir and the well must depend on the geometry of the reservoir/well system only (Raghavan, 1993). After the productivity index is stabilized, 94 the definition of the productivity index can be restated to the Equation that shows the relation between rate and pressure difference at bottomhole and reservoir as shown below: 5-1 The communication between wells in a reservoir based on the supervised or unsupervised rate changes in production and injection rates has been extensively studied in the literature (Lee et al., 2008; Yousef et al., 2005). Lee at al. used data mining techniques to characterize the flow units between injection and production wells in water-flood. Their method calculated the weight factors representing the influences of any injectors surrounding a given producer. Their algorithm can be used to obtain the weight factors to show the influences of one injector to surrounding producers. Yousef et al. among many other researches in this area used the concept of capacitance model to infer interwell connectivity (Yousef et al., 2005). They proposed a procedure where a nonlinear signal processing model was used to provide information about preferential transmissibility trends and the presence of flow barriers. Above studies are based on long history of production and injection data. In our study we consider short changes in injection rate. In other word we consider the pulse testing approach instead of interference. In this section we ran the same homogeneous case as an example 1 with the assumption of constant bottomhole pressure. The time lags were measured based on the maximum influences on the rate changes at production well with injection rate disturbance. Figure 5-1 shows the time lag values vs. 95 dimensionless front location when the bottomhole pressure at producing well is constant. In these cases the time lag values are measured based on bottomhole gross rate at production wells. Figure 5-1 illustrates the same behavior in dimensionless format. In conclusion, continuous recording of time lag based on rate measurement also can be used for front flood tracking. However, downhole rate measurement is needed; otherwise flow regimes and wellbore modeling need to be considered. In the next section we will consider wellbore modeling. Figure 5-1 Dimensionless time lag behavior vs. dimensionless front location in a homogeneous model when bottomhole pressure is constant 5.2.2 Wellbore Model Fluid flow in the wellbore has variety of forms and complexities in the production of hydrocarbons. For instance near the bottomhole, there might be only single liquid phase 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless front location Dimensionless time lag qD=∞ qD=5 qD=2 96 or two liquid phase consisting water and oil. Because oil phase contains a significant amount of dissolved gases, with changing pressure and temperature in the wellbore, there might form two phase flow. The mechanical energy balance Equation is related to pressure drop and it governs static, kinetic and friction in the wellbore. There are many empirical correlations to calculate pressure drop for multiphase flow which are complicated task. In this study we chose a commonly used empirical correlation which is based on an experimental study of pressure gradient for two phase flow (Hagedorn and Brown, 1965). For given wellbore information and fluid properties one can calculate the pressure distribution in depth. Figure 5-2 shows the pressure data vs. depth in the wellbore. In the right side of this graph there is a schematic view of a wellbore and different flow regime could be formed. Obviously Hagedorn and Brown method is simplified version of wellbore model. 97 Figure 5-2 Pressure gradient at the wellbore 5.2.3 Vertical Heterogeneity (Multilayer Reservoir) Predicting the fluid displacement behavior during secondary recovery in layered systems is one of the first problems that engineers had to examine for accurate forecasting. Permeability distribution usually varies between different strata and poses the main problem. Describing the degree of vertical heterogeneity and defining the proper permeability stratification is essential. In this section we review a method called the Dykstra-Parsons model (Dykstra and Parsons, 1950). 5.2.4 Dykstra-Parsons Approach The Dykstra-Parsons model describes 1-D oil displacement by water in multilayered reservoirs. The assumptions of this model are: piston-like displacement, no cross flows between the layers, individual homogeneous layers, constant total injection rate, and the 0 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 0 500 1000 1500 2000 2500 3000 Depth (ft) Pressure (psi) 98 same pressure drop between injector-producer for all layers. Figure 5-3 a schematic of non-communicating layered reservoir in this method. Figure 5-3 Non-communicating layered reservoir in Dykstra-Parsons model In this section we will review the mathematical modeling that they used and relate them to time lag behavior in multilayer reservoir. With the assumption of piston like displacement for a linear flow, Darcy’s law can be described as: 5-2 5-3 The following Figure shows the schematic of piston-like displacement in a layer in the Dykstra-Parsons model. Figure 5-4 Schematic of a linear system in piston-like immiscible displacement Layer 1: h 1 , ϕ 1 , κ 1 , ∆s 1 , M 1 Layer 2: h 2 , ϕ 2 , κ 2 , ∆s 2 , M 2 Layer N-1: h N-1 , ϕ Ν−1 , κ Ν−1 , ∆s N-1 , M N-1 Layer N: h N , ϕ Ν , κ Ν , ∆s N , M N Layer i: h i ϕ ι , κ ι , ∆s i , M i Water Injection Water and Oil Production L P 1 P 2 L ∆p 1 ∆p 2 x 1 99 Because the pressure drop for all layers is the same, then: 5-4 Incompressible flow indicates u w =u o . Then the actual velocity may be expressed as: 5-5 Rearrange the combination of Equations of (5-2), (5-3), and (5-6) the frontal velocity in layer “i” would be: 5-6 This result has been used to compare the front location of each layer to reference layer which is the highest permeable zone. The relative front position for layer “i” to reference layer (“R”) as follows (Dykstra and Parsons, 1950): 5-7 where the variables are: End point mobility ratio: Heterogeneity factor: Dimensionless distance: Equation (5-7) indicates the relative effect of the different variables. This Equation can find the position of the front in layer “i” when breakthrough occurs in the layer “R”. 100 We take another approach starting from Equation (5-6) to relate the time lag of each layer to reference layer. The denominator of Equation (5-6) can be reshaped to the following: 5-8 This above Equation denotes the average transmissibility in a single linear layer system. The relationship between the location of the front “x i ” and the average transmissibility “T i ” for the layer “i” is essential. To study the effect of time lag with average transmissibility change the variable in Equation (5-6) needs to be revised as follows: The Equation (5-6) with new variable will be result in: 5-9 Equation (5-9) shows the rate of the average transmissibility changes with time for layer “i”. The comparison of Equation (5-9) for layer “i” to layer “R” which is the reference layer will be as follows: 5-10 Where 5-11 On the other hand the relation between time lag for the first pulse and average transmissibility was introduced in Equation (3-3) 101 Because we assumed that there is no communication between layers, Equation (5-11) could be used to estimate the time lag of layer “i” with other know variable in this Equation. 5.2.4.1 Dykstra-Parsons Model for Two Layers Reservoir In this section the analytical solution will be examined with numerical solution with using of commercial simulator. Figure 5-5 Schematic of two layer reservoir Table 5-1 The properties of two layer reservoir Case # Layer No. Permeability (md) Layer Thickness (ft) Porosity 1 1 100 30 0.3 2 300 30 0.3 2 1 100 10 0.3 2 300 50 0.3 3 1 100 50 0.3 2 300 10 0.3 Figure 5-6 to Figure 5-8 show the time lag for three cases of double layers reservoir. The pulse sequence schedule is similar to Table 4-1. There are three pressure measurements 102 and production well. Two continuous pressure measurements at each layer and one pressure measurement represents comingled pressure changes at production well. We expect to have lower time lag values for more permeable layer and higher time lag values for less permeable layer. Obviously we expect to have comingled time lag values to be in between. Figure 5-6 shows these expectations. This Figure also shows the comingled responses are close to higher permeable layer’s responses. To evaluate the effect of thickness, we examined cases 2 and 3 where the ratios of more permeable to less permeable layer are 5 and 1/5 respectively. This is illustrated in Figure 5-6 and Figure 5-7. In these two Figures, we note that the comingled responses are close to high permeable layer responses. In conclusion, the presence of high permeable layer in a multilayer reservoir carries the pulse energy to observation well. Figure 5-6 Time lag values vs. pulse sequence number for two layers reservoir (k 1 =100 md, k 2 =300 md) (h 1 =30 ft, and h 2 =30 ft) 400 500 600 700 800 900 1,000 1 2 3 4 5 6 7 8 9 Time lag (min) Pulse Number Layer_1 Layer_2 Comingled 103 Figure 5-7 Time lag values vs. pulse sequence number for two layers reservoir (k 1 =100 md, k 2 =300 md) (h 1 =10 ft, and h 2 =50 ft) Figure 5-8 Time lag values vs. pulse sequence number for two layers reservoir (k 1 =100 md, k 2 =300 md) (h 1 =50 ft, and h 2 =10 ft) The goal of this research is to show the relation of front location and time lag. Figure 5-9 500 600 700 800 900 1 2 3 4 5 6 7 8 9 Time lag (min) Pulse Number Layer_1 Layer_2 Comingled 200 400 600 800 1,000 1,200 1,400 1 2 3 4 5 6 7 8 9 Time lag (min) Pulse Number Layer_1 Layer_2 Comingled 104 illustrates the dimensionless locations and related time lags. This Figure shows in the higher permeable layer the rate of the front movement vs. time lag is much faster than the lower permeable zone as is expected. Figure 5-10 illustrates the comparison of higher permeable layer for all three cases of different thicknesses. This Figure shows the front movements vs. time lag for three cases are the same. On the other hand the case with smaller reaches the breakthrough faster. One can use Equation (5-11) to calculate the time lag for layer “i” with having the time lags for reference layer at each time in addition to initial values for reference and layer “i”. Figure 5-11 shows this Equation obtains a good result if we have the time lag in reference layer. Figure 5-9 Dimensionless front location vs. dimensionless time lag for the same thickness 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless front location Dimensionless time lag Low permeable Layer (1) High permeable Layer (2) 105 Figure 5-10 Dimensionless front location vs. dimensionless time lag for high permeable layer for all three cases Figure 5-11 The comparison of time lag obtained from reservoir simulator vs. calculated time lag in layer “i” based on Equation 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless front location Dimensionless time lag Thcikness ratio =1 Thcikness ratio =5 Thcikness ratio =1/5 550 600 650 700 750 800 850 900 2 3 4 5 6 7 8 9 Time lag (min) Pulse number Reservoir simulator Analytical estimation 106 5.2.4.2 Dykstra-Parsons Model for Multi-Layer Reservoir Dykstra-Parsons also introduced the concept of the permeability variation coefficient “V” which is a statistical measure of non-uniformity of a set of data. We examined the effect of vertical heterogeneity on time lag for a simple case. A reservoir with five layers which has two high permeable layers is modeled. Table 5-2 shows the permeability distribution and thicknesses. Table 5-2 Permeability distribution for a multilayer reservoir Layer No. Permeability (md) Layer Thickness (ft) 1 60 60 2 500 2 3 50 70 4 300 5 5 40 60 Figure 5-12 Dykstra-Parsons coefficient of Permeability Variation Using the Dykstra-Parsons model, for the given layers’ information the variation 1 10 100 1000 10000 2 5 10 15 20 30 40 50 60 70 80 85 90 95 98 Permeability (md) Percent K50 44 K84,1 25 V: 0.4318 H: 197 107 coefficient is 0.4973. In this study we check to see if this model confirms our approach. For this reason we ran the same case in the reservoir simulator to see the behavior of time lag in that regards. We modeled a five layers reservoir with no communication between layers. The same pulsing schedule as Table 4-1 specified for this model. We ran three models, in the first case, all layers are comingled at injection well and production well. In case two only injection well is comingled and we have separate controlled perforation at production well. In case three, the injection and production wells have smart perforation. The total injection rate is constant and water injection rate for each layer in the beginning is calculated as: .It is obvious the effect of relative permeability can change this relationship. Figure 5-13 Time lag in multilayer reservoir when the injection well in all layers is comingled and each layer has its own perforation for producer 0 100 200 300 400 500 600 700 800 1 2 3 4 5 6 7 8 9 Time lag (min) Pulse sequence Comingled Injector_ Layer#2 Comingled Injector_ Layer#4 Smart wells_ Layer#2 Smart wells_ Layer#4 108 Figure 5-13 shows having sensors at each layer can predict the time lag either with separate injection perforation or comingled. The time lag values for these cases are close. Figure 5-14 compares the case that producer is also comingled. The dashed lines that illustrate the single perforation or singles sensor provide values for time lag between the real values of two high permeable zones. In summary, the higher permeable layer carries the strongest pulse energy and the pressure response can be sensed in the production well if proper sensors are provided. In conclusion continuous measurement at each layer is highly recommended. With using this information, one could provide a reasonable estimation of front location. Figure 5-14 The comparison of time lag between comingled producer and separate layer perforation 0 100 200 300 400 500 600 700 800 1 2 3 4 5 6 7 8 9 Time lag (min) Pulse sequence number Comingled Injector_ Layer#2 Comingled Injector_ Layer#4 Comingled Injector and producer 109 Bibliography Abbaszadeh, M. and Kamal, M.M., 1989. Pressure-Transient Testing of Water-Injection Wells. Society of Petroleum Engineering(16744): 115-124. Agarwal, R.G., Al-Hussainy, R. and Ramey, H.J., 1970. An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment Society of Petroleum Engineering 10(2466-PA): 279-290. Ambastha, A.K., 1988. Pressure transient analysis for composite systems Balan, B., Mohaghegh, S. and Ameri, S., 1995. State-Of-The-Art in Permeability Determination From Well Log Data: Part 1- A Comparative Study, Model Development, Society of Petroleum Engineering Bixel, H.C., Larkin, B.K. and Poollen, H.K.V., 1963. Effect of Linear Discontinuities on Pressure Build-Up and Drawdown Behavior Society of Petroleum Engineering(611-PA): 885-895. Buckley, S.E. and Leverett, M.C., 1942. Mechanism of Fluid Displacement in Sands. Petroleum Transactions, AIME, 146: 107-116. Carter, R.D., 1966. Pressure Behavior of a Limited Circular Composite Reservoir. Society of Petroleum Engineering(1621): 328-334. Culham, W.E., 1969. Amplification of Pulse-Testing Theory. Society of Petroleum Engineering(2509): 1245-1247. Daltaban, T.S. and Wall, C.G., 1998. Fundamental and Applied Pressure Analysis. Imperial College Press. Dykstra, H. and Parsons, R.L., 1950. The Prediction of Oil Recovery by Waterflood. Secondary Recovery of Oil in the United States, Principles and Practice: 160-174. Ehlig-Economides, C.A. and Economides, M.J., 1984. Pressure Transient Analysis in an Elongated Linear Flow System. Society of Petroleum Engineering(12742-MS): 1-10. Fenske, P.R., 1977. Radial flow with Discharging - Well and Observation Well Storage. Journal of Hydrology (32): 87-96. Hagedorn, A.R. and Brown, K.E., 1965. Experimental Study of Pressure Gradients Occurring During Continuous Two-Phase Flow in Small-Diameter Vertical Conduits. Journal of Petroleum Technology, 17(940-PA): 475-484. 110 Haugse, V., 1996. Identification of Mobilities for the Buckley-Leverett Equation by Front Tracking. Conference of the European Consortium for Mathematics in Industry(038): 1- 8. Hurst, W., 1953. Establishment of the skin effect and its impediment to fluid flow into a well bore. The Petroleum Engineers: 6-16. Hurst, W., 1960. Interference Between Oil Fields. Society of Petroleum Engineering(1335-G): 175-192. Issaka, M.B., 1996. Well Test Analysis for Comopsite Reservoir in Various Flow Geometries. Jaeger, J.C., 1941. Heat Conduction in Composite Circular Cylinders. Philosophical Magazine, 32(7): 324-335. Jargon, J.R., 1976. Effect of Wellbore Storage and Wellbore Damage at the Active Well on Interference Test Analysis. Society of Petroleum Engineering(5795-PA): 851-858. Johnson, C.R., Greenkorn, R.A. and Woods, E.G., 1966. Pulse-Testing: A New Method for Describing Reservoir Flow Properties Between Wells. Society of Petroleum Engineering(1517): 1599-1604. Kamal, M.M., 1973. Pulse testing Response for Unequal Pulse and Shut-in Periods. Kamal, M.M., 1975. The Effect of Linear Pressure Trends on Interference Tests. Society of Petroleum Engineering(4967): 1383-1384. Kamal, M.M., 1976. Design and Analysis of Pulse Tests With Unequal Pulse and Shut-In Periods. Society of Petroleum Engineering(4889-PA): 205-212. Kamal, M.M. and Brigham, W.E., 1975. Pulse-Testing Response for Unequal Pulse and Shut-In Periods. Society of Petroleum Engineering(5053): 399-410. Kazemi, H., 1966. Locating a Burning Front by Pressure Transient Measurements. Society of Petroleum Engineering(1271): 227-232. Kazemi, H., Merrill, L.S. and Jargon, J.R., 1972. Problems in Interpretation of Pressure Fall-Off Tests in Reservoirs With And Without Fluid Banks. Society of Petroleum Engineering(3936): 1147-1156. Lee, K.H., Ortega, A., Mohammad Nejad, A. and Ershaghi, A., 2008. A Method for Characterization of Flow Units Between Injection-Production Wells Using Performance Data. Society of Petroleum Engineering(114222). 111 McKinley, R.M., Vela, S. and Carlton, L.A., 1968. A Field Application of Pulse-Testing for Detailed Reservoir Description. Society of Petroleum Engineering(1822). Miller, F.G., 1962. Theory of Unsteady-State Influx of Water in Linear Reservoirs. Journal of the Institute of Petroleum, 48(467): 365-379. Mondragon, J.J. and Menzie, D.E., 1979. An emperical method to correct pulse test results for wellbore storage at the pulsing well. Society of Petroleum Engineering(8207- MS). Odeh, A.S. and McMillen, J.M., 1972. Pulse Testing: Mathematical Analysis and Experimental Verification. Society of Petroleum Engineering(3536): 403-409. Ogbe, D. and Brigham, W., 1987. Pulse Testing With Wellbore Storage and Skin Effects. Society of Petroleum Engineering(12780): 29-42. Ogbe, D. and Brigham, W., 1989. A Correlation for Interference Testing With Wellbore- Storage and Skin Effects Society of Petroleum Engineering(13253): 391-396. Papoulis, A. and Pillai, S.U., 2002. Probability, Random Variables and Stochastic processes. Mc Graw Hill. Prats, M., 1986. Interpretation of Pulse Tests in Reservoirs With Crossflow Between Contiguous Layers. Society of Petroleum Engineering(11963-PA): 511-520. Prats, M. and Scott, J.B., 1975. Effect of wellbore storage on pulse-test pressure response. Society of Petroleum Engineering(5322-PA): 707-709. Raghavan, R., 1993. Well Test Analysis. Printice Hall. Ramey, H.J., 1970. Short-Time Well Test Data Interpretation in the Presence of Skin Effect and Wellbore Storage. Society of Petroleum Engineering(2336-PA): 97-104. Satman, A., Eggenschwiler, M., Tang, R. and Jr., H.J.R., 1980. An Analytical Study of Transient Flow in Systems with Radial Discontinuity Society of Petroleum Engineering(9399-MS): 21-24. Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage. Trans Am Geophys Union(16): 519-524. Tongpenyai, Y. and Raghavan, R., 1981. The Effect of Wellbore Storage and Skin on Interference Test Data Society of Petroleum Engineering(8863-PA): 151-160. Van Everdingen, A.F., 1953. The Skin Effect and Its Influence on the Productive Capacity of a Well. Society of Petroleum Engineering(203-G): 171-176. 112 Vela, S., 1977. Effect of a Linear Boundary on Interference and Pulse Tests - The Elliptical Influence Area Society of Petroleum Engineering(5886-PA): 947-950. Wattenbarger, R.A. and Ramey, H.J., 1970. An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: II. Finite Difference Treatment. Society of Petroleum Engineering(2467-PA): 291-297. Welge, H.J., 1952. A Simplified Method for Computing Oil Recovery by Gas or Water Drive. Petroleum Transactions, AIME, 146: 91-98. Yousef, A.A., Gentil, P., Jensen, J.L. and Lake, L.W., 2005. A Capacitance Model To Infer Interwell Connectivity From Production and Injection Rate Fluctuations Society of Petroleum Engineering(95322).
Abstract (if available)
Abstract
In immiscible displacements processes such as water flooding, an important consideration is how to maximize the areal sweep efficiency. Limited numbers of tools exist to map oil-water interface movement. It is our expectation that the extended and repeated application of interwell pulse tests can provide some insight in tracking the fluid front approaching the producing wells. The improvement of sensor technology and data mining has opened up new opportunities to obtain continuous recording of rate and pressure data at the observation/producing and injection wells. In reality, each injection well can be subjected to pre-scheduled or unsupervised rate changes. These rate changes create pulsation and observation wells detect the pulse after a time lag. The delay is a function of interwell distance and the effective interwell formation and fluid properties. In this study we focus on both flowing wells and observation wells. We propose that the monitoring of the time lags can help in tracking fluid front plus indigenous reservoir properties. To separate the two effects, we postulate that a comparison of time lags with the first available time lag can lead to the estimation of front location. The differences between displacing and displaced fluid properties on the two sides of the interface impacts the observed time lag. We focus also on the sensitivity of this relationship to the strength of pulsation caused by rate changes rather than shutting-in the injection well. Our approach to examine this problem is with using a combination of analytical and numerical solutions. Analytical solutions to calibrate numerical test cases for radial and linear flow geometries are presented for homogeneous and composite reservoirs. We also present the result of this study for some 2-dimensional reservoir cases. From the studies of these systems, we have demonstrated the definite potential of monitoring repeated pulses for front tracking in immiscible fluid displacement processes.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Real-time reservoir characterization and optimization during immiscible displacement processes
PDF
Modeling and simulation of complex recovery processes
PDF
Control of displacement fronts in porous media by flow rate partitioning
PDF
Mass transfer during enhanced hydrocarbon recovery by gas injection processes
PDF
Coarse-scale simulation of heterogeneous reservoirs and multi-fractured horizontal wells
PDF
The role of counter-current flow in the modeling and simulation of multi-phase flow in porous media
PDF
A study of diffusive mass transfer in tight dual-porosity systems (unconventional)
PDF
Reactivation of multiple faults in oilfields with injection and production
PDF
Waterflood induced formation particle transport and evolution of thief zones in unconsolidated geologic layers
PDF
Steam drive: Its extension to thin oil sands and reservoirs containing residual saturation of high gravity crude
PDF
Application of data-driven modeling in basin-wide analysis of unconventional resources, including domain expertise
PDF
Continuum modeling of reservoir permeability enhancement and rock degradation during pressurized injection
PDF
Uncertainty quantification and data assimilation via transform process for strongly nonlinear problems
PDF
The study of CO₂ mass transfer in brine and in brine-saturated Mt. Simon sandstone and the CO₂/brine induced evolution of its transport and mechanical properties
PDF
Investigation of gas transport and sorption in shales
PDF
Efficient simulation of flow and transport in complex images of porous materials and media using curvelet transformation
PDF
Optimization of CO2 storage efficiency under geomechanical risks using coupled flow-geomechanics-fracturing model
PDF
Modeling and simulation of multicomponent mass transfer in tight dual-porosity systems (unconventional)
PDF
Multiscale and multiresolution approach to characterization and modeling of porous media: From pore to field scale
PDF
Defect control in vacuum bag only processing of composite prepregs
Asset Metadata
Creator
Orangi, Abdollah
(author)
Core Title
Flood front tracking and continuous recording of time lag in immiscible displacement
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Petroleum Engineering
Publication Date
12/09/2008
Defense Date
10/28/2008
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
flood front,front tracking,immiscible displacement,OAI-PMH Harvest,pulse test,rate change,smart filed,time lag
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Ershaghi, Iraj (
committee chair
), Jessen, Kristian (
committee member
), Schaal, Stefan (
committee member
)
Creator Email
orangi@usc.edu,orangiir@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m1899
Unique identifier
UC166033
Identifier
etd-Orangi-2482 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-142319 (legacy record id),usctheses-m1899 (legacy record id)
Legacy Identifier
etd-Orangi-2482.pdf
Dmrecord
142319
Document Type
Dissertation
Rights
Orangi, Abdollah
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
flood front
front tracking
immiscible displacement
pulse test
rate change
smart filed
time lag