Approved by: Chair of Committee, Committee Members, Robert A. Wattenbarger Goong Chen Christine Ehlig-Economides Bryan Maggard Stephen Holditch Head of Department, May 2009 Major Subject: Petroleum Engineering iii ABSTRACT Rate Transient Analysis in Shale Gas Reservoirs with Transient Linear Behavior. (May 2009) Rasheed Olusehun Bello, B. Sc. , University of Lagos, Nigeria; M. Sc. , University of Saskatchewan, Canada Chair of Advisory Committee: Dr. Robert Wattenbarger
Many hydraulically fractured shale gas horizontal wells in the Barnett shale have been observed to exhibit transient linear behavior. This transient linear behavior is characterized by a one-half slope on a log-log plot of rate against time. This transient linear flow regime is believed to be caused by transient drainage of low permeability matrix blocks into adjoining fractures. This transient flow regime is the only flow regime available for analysis in many wells. The hydraulically fractured shale gas reservoir system was described in this work by a linear dual porosity model.
This consisted of a bounded rectangular reservoir with slab matrix blocks draining into adjoining fractures and subsequently to a horizontal well in the centre. The horizontal well fully penetrates the rectangular reservoir. Convergence skin is incorporated into the linear model to account for the presence of the horizontal wellbore. Five flow regions were identified with this model. Region 1 is due to transient flow only in the fractures. Region 2 is bilinear flow and occurs when the matrix drainage begins simultaneously with the transient flow in the fractures.
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Region 3 is the response iv for a homogeneous reservoir. Region 4 is dominated by transient matrix drainage and is the transient flow regime of interest. Region 5 is the boundary dominated transient response. New working equations were developed and presented for analysis of Regions 1 to 4. No equation was presented for Region 5 as it requires a combination of material balance and productivity index equations beyond the scope of this work. It is concluded that the transient linear region observed in field data occurs in Region 4 – drainage of the matrix.
A procedure is presented for analysis. The only parameter that can be determined with available data is the matrix drainage area, Acm. It was also demonstrated in this work that the effect of skin under constant rate and constant bottomhole pressure conditions is not similar for a linear reservoir. The constant rate case is the usual parallel lines with an offset but the constant bottomhole pressure shows a gradual diminishing effect of skin. A new analytical equation was presented to describe the constant bottomhole pressure effect of skin in a linear reservoir.
It was also demonstrated that different shape factor formulations (Warren and Root, Zimmerman and Kazemi) result in similar Region 4 transient linear response provided that the appropriate f(s) modifications consistent with ? Ac calculations are conducted. It was also demonstrated that different matrix geometry exhibit the same Region 4 transient linear response when the area-volume ratios are similar. v DEDICATION I dedicate my work to all those who have lovingly supported me throughout life and all its travails. vi ACKNOWLEDGEMENTS
First, I want to give praises to Almighty Allah for sparing my life, continuously granting me His blessings and allowing me to successfully conclude this phase of my life. I want to acknowledge my supervisor, Dr. Robert. A. Wattenbarger for being a father, mentor, supervisor and friend to me. I am honored to have worked with him. I am also eternally grateful to him. I want to acknowledge the suggestions and contributions of my committee members, Dr. Christine Ehlig-Economides , Dr. Bryan Maggard – whose classes are among my favorites and invaluable- and Dr. Goong Chen.
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Logic Model Development Guide Introduction If you don’t know where you’re going, how are you gonna’ know when you get there? –Yogi Berra In line with its core mission – To help people help themselves through the practical application of knowledge and resources to improve their quality of life and that of future generations – the W.K. Kellogg Foundation has made program evaluation a priority. As ...
The research problem is described and the project objectives are presented. Chapter II presents an extensive literature review. The dual porosity model and its applications to liquids and gas are reviewed. Horizontal well applications are also reviewed. Chapter III describes the linear model to be used in this work. Validation of the linear model is also presented. Chapter IV presents new analysis equations developed using the linear model. Chapter V discusses the transient linear regime in detail and discusses the effects of shape factors and area-volume ratio.
Chapter VI describes the constant bottomhole pressure effect of skin in linear reservoirs Chapter VII presents development of new type curves with application to sample field data. Chapter VIII presents conclusions and recommendations. 9 CHAPTER II LITERATURE REVIEW 2. 1 Introduction Initial studies of fractured reservoirs were concerned with applications to well test analysis of reservoir flow of liquids (constant rate, pressure buildup and drawdown).
Subsequent research considered production data analysis (constant bottomhole pressure) and extension of existing models to gas flow.
Most of the literature is devoted to radial reservoir models. In this chapter, review of literature will be conducted in three sections. The first section discusses the dual porosity model and its application to flow of slightly compressible fluids. The second section discusses the application of the dual porosity model to gas flow. The final section discusses the application of the dual porosity model to analysis of naturally fractured reservoirs with horizontal wells. 2. 2 Dual Porosity Model (Slightly Compressible Fluids) Naturally fractured reservoirs (tight gas, shale gas and coal gas) have been described by the dual porosity model.
The dual porosity model was first formulated by Barenblatt et al. 17 and later extended to well test analysis by Warren and Root. 18 The Warren and Root model forms the basis of modern day analysis of naturally fractured reservoirs. In the Warren and Root model, the naturally fractured reservoir is modeled by uniform homogeneous matrix blocks separated by fractures as shown in Fig. 2. 1. The matrix blocks provide storage of the fluid to be produced while the fractures provide the 10 permeability. When a producing well is present, the fluid flows from the matrix to the fractures and to the well.
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There have been two types of approach in applying the dual porosity model based on how flow of the fluid from the matrix to the fractures is modeled – pseudosteady state and transient. Fig. 2. 1 – Dual Porosity Model. 18 2. 2. 1 Pseudosteadystate Matrix-Fracture Transfer Models An equation for interporosity flow from the matrix to the fractures at a mathematical point under pseudosteadystate (quasisteadystate or semisteadystate) conditions was presented by Warren and Root. 18 11 q =? km µ (p m ? pf ) Where q is the drainage rate per unit volume, ? is the Warren and Root shape factor, pm is the matrix pressure at a mathematical point.
Two new parameters which are used to characterize naturally fractured reservoirs were presented by Warren and Root18 – the interporosity flow parameter, ? (a measure of the flow capacity of the system) and the storativity, ? (a measure of the storage capacity of the fractures).
Warren and Root18 were the first to apply Laplace transformation to obtain “f(s)” and solve for the dimensionless pressure distribution. A method of analyzing pressure buildup data for the infinite radial reservoir case was presented. Buildup plots were found to exhibit parallel lines on a semilog plot separated by an Sshaped transition period.
The first line represents flow in the fracture system only while the second line represents flow in the total system (matrix and fractures).
Kazemi et al. 19 investigated the suitability of applying the Warren and Root model to interpret interference results. They presented a model which extends the Warren and Root model to interference testing. They applied the Laplace transformation to obtain “f(s)” and solve for the dimensionless pressure distribution. They also numerically solved the model equations by finite-difference methods and included vertical pressure gradients.
It was concluded that an equivalent homogeneous model was not appropriate at early times but could be used at later times. It was also concluded that the Warren and Root model yielded similar results as their numerical solution and was thus appropriate for analyzing naturally fractured reservoirs. 12 Odeh20 developed an infinite radial reservoir model for the behavior of naturally fractured reservoir. The model incorporates some limiting assumptions. The Laplace transformation is also utilized. Two parallel straight lines were not observed on a semilog plot contrary to Warren and Root’s results.
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It was concluded that buildup and drawdown plots of naturally fractured reservoir transient responses are similar to those of homogeneous reservoirs. Mavor and Cinco-Ley21 present solutions for the constant rate case in an infinite radial reservoir with and without wellbore storage and skin; and a bounded radial reservoir. Solutions are also presented for the first time for a constant pressure inner boundary with skin in an infinite radial reservoir. Da Prat et al. 22 extended the Warren and Root18 solutions to constant pressure inner boundary conditions and bounded outer boundary cases for the radial reservoir.
They also present type curves for analysis. The results do not appear to represent realistic field cases. Bui et al. 23 present type curves for transient pressure analysis of partially penetrating wells in naturally fractured reservoirs by combining the Warren and Root model with the solution for these wells in homogeneous reservoirs. 2. 2. 2 Transient Matrix-Fracture Transfer Models Kazemi24 used a slab matrix model with horizontal fractures and unsteady state matrixfracture flow to represent single-phase flow in the fractured reservoir.
The assumptions include homogeneous behavior and isotropic matrix and fracture properties. The well is 13 centrally located in a bounded radial reservoir. A numerical reservoir simulator was used. It was concluded that the results were similar to the Warren and Root model when applied to a drawdown test in which the boundaries have not been detected. Two parallel straight lines were obtained on a semilog plot. The first straight line may be obscured by wellbore storage effects and the second straight line may lead to overestimating ? when boundary effects have been detected.
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De Swaan25 presented a model which approximates the matrix blocks by regular solids (slab and spheres) and utilizes heat flow theory to describe the pressure distribution. It was assumed that the pressure in the fractures around the matrix blocks is variable and the source term is described through a convolution term. Approximate linesource solutions for early and late time are presented. The late time solutions are similar to those for early time except that modified hydraulic diffusivity terms dependent on fracture and matrix properties are included. The results are two parallel lines representing the early and late time approximations.
The late time solution matches Kazemi24 for the slab case. De Swaan’s model does not properly represent the transition period. Najurieta26 presented a transient model for analyzing pressure transient data based on De Swaan’s25 theory. Two types of fractured reservoir were studied- stratum (slabs) and blocks (approximated by spheres).
The model predicted results similar to Kazemi. 24 Serra et al. 27 present methods for analyzing pressure transient data. The slab model used is similar to De Swaan25 and Najurieta. 26 The model considers unsteady state 14 matrix fracture transfer and is for an infinite reservoir.
Three flow regimes were identified. Flow Regime 1 and 3 are the Warren and Root18 early and late time semilog lines. A new flow Regime 2 was also identified with half the slope of the late time semilog line. Chen et al. 28 present methods for analyzing drawdown and buildup data for a constant rate producing well centrally located in a closed radial reservoir. The slab model similar to De Swaan25 and Kazemi24 is used. Five flow regimes are presented. Flow regimes 1, 2 and 3 are associated with an infinite reservoir and are described in Serra et al. 27 Flow regime 1 occurs when there is a transient only in the fracture system.
Flow regime 2 occurs when the transient occurs in the matrix and fractures. Flow regime 3 is a combination of transient flow in the fractures and “pseudosteady state” in the matrix. Pseudosteadystate in the matrix occurs when the no-flow boundary represented by the symmetry center line in the matrix affects the response. Two new flow regimes associated with a bounded reservoir are also presented. Flow regime 4 reflects unsteady linear flow in the matrix system and pseudosteadystate in the fractures.. Flow Regime 5 occurs when the response is affected by all the boundaries (pseudosteady-state).
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Streltsova29 applied a “gradient model” (transient matrix-fracture transfer flow) with slab-shaped matrix blocks to an infinite reservoir. The model predicted results which differ from the Warren and Root model in early time but converge to similar values in late time. The model also predicted a linear transitional response on a semi-log plot between the early and late time pressure responses which has a slope equal to half 15 that of the early and late time lines. This linear transitional response was also shown to differ from the S-shaped inflection predicted by the Warren and Root model.
Cinco Ley and Samaniego30 utilize models similar to De Swaan25 and Najurieta26 and present solutions for slab and sphere matrix cases. They utilize new dimensionless variables – dimensionless matrix hydraulic diffusivity, and dimensionless fracture area. They describe three flow regimes observed on a semilog plot – fracture storage dominated flow, “matrix transient linear” dominated flow and a matrix pseudosteadystate flow. The “matrix transient linear” dominated flow period is observed as a line with one-half the slopes of the other two lines. 7,29 It should be noted that the “matrix transient linear” period yields a straight line on a semilog plot indicating radial flow and might be a misnomer. The fracture storage dominated flow is due to fluid expansion in the fractures. The “matrix transient linear” period is due to fluid expansion in the matrix. The matrix pseudosteadystate period occurs when the matrix is under pseudosteadystate flow and the reservoir pressure is dominated by the total storativity of the system (matrix + fractures).
It was concluded that matrix geometry might be identified with their methods provided the pressure data is smooth.
Lai et al. 31 utilize a one-sixth of a cube matrix geometry transient model to develop well test equations for finite and infinite cases including wellbore storage and skin. Their model was verified with a numerical simulator employing the Multiple Interacting Continua (MINC) method. Ozkan et al. 12 present analysis of flow regimes associated with flow of a well at constant pressure in a closed radial reservoir. The rectangular slab model similar to De 16 Swaan25 and Kazemi24 is used. Five flow regimes are presented. Flow regimes 1, 2 and 3 are described in Serra et al. 7 Two new regimes are presented- Flow regime 4 reflects unsteady linear flow in the matrix system and occurs when the outer boundary influences the well response and the matrix boundary has no influence. Flow Regime 5 occurs when the response is affected by all the boundaries. Houze et al. 32 present type curves for analysis of pressure transient response in an infinite naturally fractured reservoir with an infinite conductivity vertical fracture. Stewart and Ascharsobbi33 present an equation for interporosity skin which can be introduced into the pseudosteadystate and transient models.
The effect of interporosity skin is to delay flow from the matrix to the fractures. This equation is given by s ma = 2k mi h s hm k s where kmi is the intrinsic matrix permeability, hs is the thickness of the interporosity skin layer, hm is the matrix block dimension and ks is the permeability of the interporosity skin layer. It should be noted that all the transient models previously described were developed for the radial reservoir cases (infinite or bounded).
El-Banbi16 was the first to present transient dual porosity solutions for the linear reservoir case.
New solutions were presented for a naturally fractured reservoir using a dual porosity, linear reservoir model. Solutions are presented for a combination of different inner boundary (constant pressure, constant rate, with or without skin and 17 wellbore storage) and outer boundary conditions (infinite, closed, constant pressure).
This model will be used in this work. 2. 3 Dual Porosity Model (Gas) Kucuk and Sawyer34,35presented a model for transient matrix-fracture transfer for the gas case. Previous work had been concerned mainly with modeling slightly compressible (liquid) flow. They considered cylindrical and spherical matrix blocks cases.
They also incorporate the pseudopressure definitions for gases. Techniques for analyzing buildup data are also presented for shale gas reservoirs. Their model results plotted on a dimensionless basis matched Warren and Root18 and Kazemi24 for very large matrix blocks at early time but differ at later times. They also conclude from their tests that naturally fractured reservoirs do not always exhibit the Warren and Root behavior (two parallel lines).
Carlson and Mercer15 coupled Fick’s law for diffusion within the matrix and desorption in their transient radial reservoir model for shale gas.
Modifications include use of the pressure-squared forms valid for gas at low pressures to linearize the diffusivity equation. They provide a Laplace space equation for the gas cumulative production from their model and use it to history match a sample well. They also show that semi-infinite behavior (portions of the matrix remain at initial pressure and is unaffected by production from the fractures) occurs in shale gas reservoirs regardless of matrix geometry. They present an equation for predicting the end of this semi-infinite behavior. 18 Gatens et al. 6 analyzed production data from about 898 Devonian shale wells in four areas. They present three methods of analyzing production data – type curves, analytical model and empirical equations. The empirical equation correlates cumulative production data at a certain time with cumulative production at other times. This avoids the need to determine reservoir properties. Reasonable matches with actual data were presented. The analytical model is used along with an automatic history matching algorithm and a model selection procedure to determine statistically the best fit with actual data.
Watson et al. 37 present a procedure that involves selection of the most appropriate production model from a list of models including the dual porosity model using statistics. The analytical slab matrix model presented by Serra et al. 27 is utilized. Reservoir parameters are estimated through a history matching procedure that involves minimizing an objective function comparing measured and estimated cumulative production. They incorporate the use of a normalized time in the analytical model to account for changing gas properties with pressure.
Reasonable history matches were obtained with sample field cases but forecast was slightly underestimated. Spivey and Semmelbeck38 present an iterative method for predicting production from dewatered coal and fractured gas shale reservoirs. The model used is a well producing at constant bottomhole pressure centered in a closed radial reservoir. A slab matrix is incorporated into these solutions. These solutions are extended to the gas case by using an adjusted time and adjusted pressure. Their method also uses a total compressibility term accounting for desorption. 19 2. Horizontal Wells in Naturally Fractured Reservoirs There have been different traditional approaches to modeling horizontal wells in homogeneous reservoirs. Horizontal wells are normally modeled as infinite conductivity (pressure is uniform along the wellbore).
It is not practical, as Gringarten et al. 39 demonstrated with infinite conductivity fractures, to compute the wellbore pressure from the infinite-conductivity model because of the computational work involved. Gringarten et al. 39 suggested computing the pressure drop from the uniform flux model (flowrate is the same for each individual segment along a wellbore) at a value of xD = 0. 32. This value was the point at which the uniform flux model yields the same results as the infinite conductivity model. This computation has also been incorporated into horizontal well models. 40-46 The mathematical problem to be solved for the anisotropic case is usually given by kx ? 2 p ? 2 p ? p ? 2 p + k y 2 + k z 2 = ? µct 2 ? t ? x ? y ? z Several authors have used a model of a line source well in a semi-infinite45,47 or infinite reservoir. 40-44,48-50 Others41-44,48,51 have used a line source well in a closed rectangular reservoir.
The infinite model has no-flow boundaries at the top and bottom. The semi-infinite reservoir model has three no-flow boundaries (top, bottom and left).
The closed reservoir model has all four no-flow boundaries. 20 It should be noted that in these models, the well is usually not completely penetrating but the models by Ozkan 41-44 and Odeh and Babu51 provide this possibility once the appropriate well and reservoir dimensions are specified. The differential equation and boundary conditions have been mostly solved by the Newman product method and source functions. 0-45 These concepts for the homogeneous reservoir case have been extended to model horizontal wells in naturally fractured reservoirs. Ozkan41-44 presents Laplace space solutions for horizontal wells in a reservoir for infinite and closed rectangular boundary cases in terms of f(s).
The line source approach previously described is utilized. As demonstrated by Ozkan, there is a possibility of applying this to the naturally fractured reservoir by substituting the appropriate f(s) for a selected matrix geometry. Carvalho and Rosa52 present solutions for an infinite conductivity horizontal well in a semi-infinite reservoir.
The reservoir is homogeneous and isotropic. The horizontal well is modeled as a line source. The solutions for the homogeneous case were then extended to the dual porosity case by substituting s*f(s) for s in Laplace space for the pressure derivative (homogeneous).
Wellbore storage and skin are incorporated into their model using Laplace space. Aguilera and Ng53 present analytical equations for pressure transient analysis. Their model is a horizontal well in a semi-infinite, anisotropic, naturally fractured reservoir. Transient and pseudosteadystate interporosity flow is considered.
Six flow periods are identified –First radial flow (at early times, from fractures), Transition 21 period, Second radial flow in vertical plane, First linear flow, Pseudoradial flow and Late linear – with expressions for determining skin provided. Ng and Aguilera54 present analytical solutions using a line source and then compute pressure drop on a point away from the well axis to account for the radius of the actual well. A method for determining the numerical Laplace transform is presented. This method was then used to compute the dual porosity response (pseudosteady state).
Their solutions were compared to other solutions. Thompson et al. 55 present an algorithm for computing horizontal well response in a bounded dual porosity reservoir. Their model is a horizontal well in a closed rectangular reservoir. Their procedure involves converting a known analytic solution to Laplace space numerically point by point and then inverting using the Stehfest algorithm. 56 This is similar to the procedure presented by Ohaeri and Vo46 who use a numerical Laplace space algorithm57 but also present alternative equations determined by parameter ranges which result in computational efficiency.
Du and Stewart58 describe situations which can yield linear flow behavior – a multi-layered reservoir (one layer has a very high permeability relative to the other); naturally fractured reservoir (flow from matrix into horizontal well intersecting fractures); and areal anisotropy (vertical fractures aligned predominantly in one direction).
Their model is that of a horizontal well in a homogeneous, infinite acting reservoir. Three flow regimes are identified – radial vertical flow, linear flow opposite completed section and pseudoradial flow at late time.
A bilinear flow behavior was also identified. 22 The model presented in this work has the advantage of being simpler than the horizontal well models. The model will be presented in Chapter III. It also allows the direct use of Laplace space techniques not easily seen with these horizontal well models. Review of literature also shows that the transient linear flow regime has not been investigated in the manner presented in this work. 23 CHAPTER III MATHEMATICAL MODEL 3. 1 Introduction A schematic of the model to be used in this work is shown in Fig. . 1. A rectangular grid is imposed on the microseismic results as shown in Fig. 3. 1. The model is shown in detail with representative cube matrix blocks in Fig. 3. 2. The features of the model to be used in this work are described below. • A closed rectangular geometry reservoir containing a network of natural and hydraulic fractures (as in Mayerhofer et al. 14).
The fractures do not drain beyond the boundaries of this rectangular geometry. • • • • The perforated length of the well , xe is the same as the width of the reservoir.
