SPE 164638 Determination of Drainage Area and Shape Factor of Vertical Wells in Naturally Fracture Reservoir with Help Well testing and Developed IPR Curve Mohammad Sadeghi, Islamic Azad University, Science and Research Branch; Seyed Reza Shadizadeh, Petroleum University of Technology; Mohammad Ali Ahmadi, Petroleum University of Technology Copyright 2013, Society of Petroleum Engineers This paper was prepared for presentation at the North Africa Technical Conference & Exhibition held in Cairo, Egypt, 15 17 April 2013. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract Shape Factor and drainage area in hydrocarbon reservoir are of the necessary and influencing factors on evaluation and optimization of drilling operation in oil fields. There are numerous methods to calculate the reservoir average pressure in conventional and sandstone reservoirs in that shape and drainage area of the reservoir are of important variables. On the other hand, these relations especially set for sandstone reservoirs and have limited usage in natural fracture reservoirs. This research demonstrates an approach to figure out shape factor and drainage area of wells in carbonated and naturally fractured reservoirs in all pressure and fluidity conditions. To assess this goal, results of well testing are carried out both for development of inflow performance curve and finding the drainage area. To make sure about the validity and preciseness of this method, it was investigated against the data from a well in a natural fracture field and the results were compared with those of other methods. Results dictated that the new approach can provide us with more precise and correct routes for the shape of the well in naturally fractured reservoirs. The suggested approach in this research needs production test data that has been optimized by well testing data for naturally fractured reservoirs. Additionally, this approach implemented to the draw down test and Tiab direct technique which is related to Warren and Root solution. What's more, this approach is reliable for all vertical wells in naturally fractured reservoirs (NFRs) with accessible production test data and well testing. 1. Introduction Drainage area of wells defined as an oil reservoir area met pressure disturbance of production well and depend on production rate of producer well. Drainage area of producer well considered as a cylindrical shape based on imagination and due to this fact, radious of investigation is defined [1-4].
2 SPE 164638 Many authors implemented different expressions to demonstrate the radius of investigation and drainage area for conventional reservoirs [5-6]. Various approaches are proposed to figure out well drainage area and shape factor in conventional reservoirs while low approaches are available for naturally fractured reservoirs [7-9]. Regard to the depletion shape of producer well when the pressure disturbance reach to physical boundaries of the reservoir like as sealing faults (semi-steady state) remains constant, accurate calculations of Shape factor depend on the value of drainage area of well. Miller, Days and Hutchinson proposed approach based on mathematical equation to solving diffusivity equation and build up test analysis to calculate the average pressure in cylindrical reservoir while drainage area of well is an important unknown parameter [4]. Eventually, Matthews - Brown - Hazebroek developed a similar MDH approach while developed approach no longer needed the well testing information about the semi-steady state fluid data and only with information during the initial well-testing (build transient testing) can be obtained average reservoir pressure in term of drainage area of well [10-14]. After MBH and MDH approaches which drainage area of well were provided with respect to the average reservoir pressure, Dietz with aim of continuously superposition equations could determine the shape factor parameter which demonstrate physical shape of boundaries could existed around the wellbore [15]. All the above approaches are based on the equations of fluid flow in the conventional porousmedium like as sandstone reservoirs. Therefore, this study makes effort to develop the approach to calculate the correct well drainage area in Natural Fractured Reservoirs. The current approach implemented production test results which supported by the well testing information while developed approach has been modified for naturally Fractured Reservoirs (NFRs) by means of Tiab approaches which in the analysis of the well test pressure in solution of Warren and Root equation [1] can determine exact routs of drainage area of wells
SPE 164638 3 independent from the average pressure [15]. Throughout this current study, simple production approach called inflow performance relationship (IPR) evolved well testing approaches to figure out drainage area and relevant shape factor of production well in petroleum reservoir like as fractured reservoirs. Moreover, introduced approach applied to a naturally fracture reservoir (NFR) to illustrate effectiveness of developed approach in determination of drainage area and shape factor of well while various routinely approaches implemented to addressed fractured reservoir. Draw in parallel between developed approach and conventional methods exhibit low uncertainty and high precision of introduced method in estimation well drainage area and shape factor. 2. Drainage Area and Shape Factor of Well Through various attendances to developed relationship between different aspects of petroleum reservoirs like as reservoir pressure and production features like as production rate, radius of investigation is highly appreciated while radius of investigation represent area of reservoir which met pressure disturbance due to production. It is worth bearing in mind that, Radius of investigation assumed as a homogeneous cylindrical shape while expressed as following expression [13]: i = (1) Radius of investigation is considered a circular while the pressure change disturbance reached the physical reservoir boundaries however; pseudo-steady sate conditions found in the well, the radius of investigation will be like as real boundaries of reservoir. It should be noted that In this case the affected area will remain permanently fixed and we will identify it to the drainage area. To simplify the complexity of the physical boundaries, the shape of well drainage with simple mathematical shapes are formulated while these simple shapes are called the shape factor. Equation of fluid flow in porous media which its pressure disturbance met the boundary
4 SPE 164638 is considered as below expression [13]: 162.6 4 0.87 1.781 (2) Various approaches such MBH and MDH, can conducted to figure out the drainage area of wells in conventional reservoirs due to the known average reservoir pressure. 3. Well Testing Analysis The main purpose the well testing is an estimate of the well s potential and reservoir characterization by means of pressure gradient and flow rate. Well testing approaches have various classifications such as transient, stable while Stability testing is implemented to figure out inflow performance relationship (IPR). Various parameters can be obtained from the transient testing which are summarized to permeability, skin factor and average pressure. It is worth bearing in mind that in Transient flow condition, the pressure disturbance not met the boundaries and reservoir acting infinite. Solution of diffusivity Equation for the transient state by entering skin effects at the wellbore of the well is expressed as follows [13]: 162.6 3.23 0.87 (3) Above equation is the Based on drawdown testing in transient state. When the pressure disturbance meets reservoir boundaries, flow regime of Transient state reaches pseudo-steady state. Solution of diffusivity equation for well is placed in the center a cylinder in a term of average reservoir pressure is defined as following expression [13]: 141.2 0.75 (4) Demonstration of pressure drawdown and the pressure buildup tests discussed separately and completely in a lot of open literatures.
SPE 164638 5 3.1 Analysis MBH Method Bottom-hole pressure of producer well at early times of production while producer well produced at constant rate q (stb/day) could be expressed as following expression [11]: P wf = m. Log (t) + p 1hr (5) Where P wf: flowing well pressure; (psi) t: production time; (hr). µ m =.µ Pw = p i.. µ 0.80907 2 (6) (7) Where, m is a slop of linear part of a plot of P w versus log (t) and p 1hr is an intercept of the straight line portion of that plot at t = 1 (hr). When the curve of the given shape presented by MBH method becomes linear then the pressure disturbs behavior follows pseudo steady state for a given shape launches. Dietz tabulates the time of pseudo steady state launching for some given shapes during the pseudo steady state period [15]. P wf = m*.t + p int (8) Where:. m* = p wf = p i. 0.80907 2 (9) (10) A: drainage area ;( ) C A: Dietz shape factor [9] P int is the intercept of the straight line when it is extrapolated to t=0. It should be noted that here, to define the beginning time of pseudo steady state behavior the Dietz dimensionless time implemented as follow expression [15]:
6 SPE 164638 ( t DA ) p.s.s = 0.1833... (11) Where, t p.s.s is the time of start straight line in the plot of p wf versus t. 3.2 Tiab Approach Tiab approach was implemented mathematical calculations, solving the movement of flux equation in dual porosity porous medium with aid of Laplace transfer function then model and draw the two diagrams of P D and p D.t D vs. t D to figure out drainage area [16-18]. Tiab works on the implication of TDS technique to estimate the average reservoir pressure in vertical and horizontal wells as follow expressions [7, 18]: P WD = 2πt DA + P wf = -. µ... 141.2 µ 21. 2 2 A =. +s 2 (12) (13) (14) (15) (16) (17) 3.3 Drainage with Well Testing Through this research inflow performance relationship (IPR) and well testing approach were carried out to figure drainage area and shape factor out while This new approach open new perspective to determine physical indexes of petroleum reservoir by implementing two kinds of reservoir information [9]. It should be noted this crucial point that here, this work provides the approach which generate inflow performance relationship (IPR) equation implementing the fluid flow movement in reservoir. Moreover, due to its intrinsic ability, this expression can be
SPE 164638 7 implemented for other naturally fractured reservoirs. Through this research, general fluid flow equation through porous media for below bubble point pressure condition is arranged based on average pressure. Then, with aid of well testing and laboratory, the required relative parameters and average pressure are figured out and substituted in addressed rearranged equation. In next step, the transition or pseudo steady state (P.S.S.) condition for the fluid flow is checked with a test points for the interested well which revealed that fluid flow satisfied pseudo steady state (P.S.S.) condition. Moreover, inflow performance relationship (IPR) is drawn for various r e while by means of implemented test points correct value of r e which satisfy transition condition could be determined [15]:. q o =.,. 3.23 0.87. (18) The previous studies on the reservoir equation ended in the introduction of A/C A fraction to the field. Based on these studies, in all the resulted equations the reservoir was assumed as a cylinder with the well inside by a simplified assumption. Unfortunately, this process means replacement of the actual shape of the reservoir with a circle in mathematic equations which caused high uncertainty in obtained results. Therefore, the actual shape of the reservoir based on the reverse engineering for imagination of the well should be figured out and be implemented in the original form of the equation with A/C A. This approach has lower errors and obtained results from that is closed to the real answers. As can be seen from equation 2, well drainage area and shape factor are in one fraction which unfortunately are related together however, it is a reasonable relation because we defined shape factor from well drainage area. Therefore, any approach which obtained the shape factor without identifying drainage area, at least in naturally fractured reservoirs (NFRs), will be incorrect [10]. As previously mentioned in this article, the methods which calculate average pressure are
8 SPE 164638 demonstrated, because after developing the Dietz theory [15], some of addressed approaches implemented Dietz table to estimate the average reservoir pressure either with correlation or calculations. But in this research enormous efforts have been made to put forth a novel approach based IPR and well testing to estimate well drainage area and related shape factor with low uncertainty and high precision. This section exhibits the best approach to identify both drainage area and shape factors which satisfy all the complexities of naturally fractured reservoirs and shape factor while we implement the newest approach by means of Inflow Performance Relationship (IPR) and Well Testing for P.S.S condition as following expression [15]:.., (19) To figure out the addressed variables two equations are needed which this research proposed that one equation must satisfy the fluid flow through porous media in naturally fractured reservoirs while flow in both of matrix and fissure regions and the second equation demonstrate the relationship between pressure and time. As mentioned previously, first equation should satisfy the fluid flow behavior while this article suggested inflow performance relationship (IPR) equation which is developed for naturally fracture reservoirs, is a better option to be implemented. It should be noted that the inflow performance relationship (IPR) curves are correlation between flow rate and bottom-hole pressure therefore, the second equation which related to pressure and time/rate should be implemented to calculate the second variable. Thus, well testing provides more selective options to be implemented however; well test approach selection is a challenge to get low uncertainty and high precise answers. It is worth bearing in mind that, this research implements the TDS approach along with Tiab s Direct Synthesis Technique.
SPE 164638 9 5. Results and Discussion It is worth bearing in mind that implemented data of this work such as drawdown test data taken from lgbokoyi and Tiab research in (2006) [16] while circular geometry is assumed. Implemented and figured out Reservoir characterizations along with used well data are reported in Table 2. Figure 1 exhibits draw dawn test for long time of the considered well that reached semi steady state condition. Figure 2 demonstrate semi-log plot of draw dawn test in consider case study. Results of inflow performance relationship (IPR) illustrated in Figure 3. The log-log plot of pressure derivative data was smoothened in order to have a good definition of the system (See Figure 4). Figure 2 is the horner plot of considered draw down test [16] which confirmed implemented reservoir is naturally fractured reservoir and reached to semisteady state flow regime. Figure 3 was implemented to estimate the drainage area while real value is 3,008,340ft 2. To figure out permeability the value of on the stabilized line at 0.5 should be considered and replaced in following expression: k =. (20) To calculation ω in TDS technique, should be identify the minimum coordinate and t min on the log derivative plot. Below equation was carried out to indicate storability ratio (ω): ω=0.15866( 0.54653 (21) To determine the inter-porosity flow ratio following equation was implemented: λ =. (22) To determine the drainage area, TDS approach for semi-steady state period was implemented. It should be noted that, drainage area can compute that pressure changes met the boundaries, in the other hand; we need all equation which satisfying semi-steady state flow in fractured reservoirs. Therefore, following expression was conducted to figure out drainage area in TDS
10 SPE 164638 technique: A =., (23) As previously mentioned here the real drainage area is 3,008,340 ft 2 and figured out value by TDS technique is 3,020,664 2 with 0.00004% error which could be ignored. Due to this reliable estimated drainage area value, high precise with low uncertainty shape factor could be obtained by implementation inflow performance relationship (IPR) which developed for naturally fracture reservoir (NFR) with equation (19). It should be noted that here, Defining productivity index of a system as: P.I= J = (24) In which; q: flow rate P ave P wf = is the pressure drop. Also productivity index can be written in the following form: J = =. µ./ (25) By productivity index we mean how much production we have by a special pressure drop. Eq. 25 could be written in more general form as: J=. µ /γ (26) ɣ :Euler constant=1.781 some believe that in Eq. 26 both shape factors introduced by Dietz (for constant rate production) and for constant bottom-hole pressure could be implemented but Helmy [18] illustrated that constant rate shape factors produce an error of 10% in ultimate recovery of shape factor if implemented instead of constant bottom-hole pressure. By implementation two points of the pressure and rate dataset the value of 0.87 is assigned to J while this approach is satisfied the fluid flow in oil reservoir that in under-saturated condition with one phase, because
SPE 164638 11 IPR curve in this period have a linear shape and two point enough to calculated J. determined shape factor by above mentioned procedure is C A =10.8. 5.1. Comparison of the Proposed Approach for Determination of Shape Factor with Well Testing Approach In this section, popular approach for shape factor determination by means of well testing approach was carried out. To achieve addressed issue of this section, following expression was implemented to indicate shape factor: C A = 5.456.. ) (27) Where m = slop of straight line in log-log drawdown test m * = slop of straight line in limit test for semi-steady state p 1hr = pressure at t=1(hr) p int = intercept of straight line in limit test Determined shape factor by means of above equation is C A =4.3 while there isn t any value in Dietz Table by means of dimensionless time which defined by Dietz [15] as below. Obtained shape factor by means of well testing approach has high uncertainty due to comparison with real value C A =10.8.. t DA =( ) (28) 4. Conclusions New approach to figure out shape factor and drainage area of naturally fractured reservoirs with implementing integration of inflow performance relationship (IPR) and well testing approaches was illustrated. Based on routes were figured out through this research following main conclusions can be drawn:
12 SPE 164638 1. Implementing inflow performance relationship (IPR) and well testing provided more precise, reliable and robust information to figure out fundamental equation of petroleum reservoir. 2. Implementing equation with lower error levels lead to better predictions of the future production scheme while the shape and environment of reservoir are crucial information to categorize and recognize the petroleum reservoir. 3. Routes of this research which relevant to the naturally fractured reservoir (NFR) have high precision and fewer mathematical functions. Due to this addressed advantage, introduced approach could be applied in various conditions like as homogenous reservoirs. Nomenclature NFR: natural fracture reservoir A: reservoir area C A : shape factor parameter P D: dimensionless pressure P i: initial reservoir pressure,psi P wf : wellbore flowing pressure,psi P avr : average reservoir pressure,psi K f : fracture permeability,md K m : matrix permeability,md H: formation thickness,ft Q: flow rate,stb/d B: formation volume factor,rb/stb µ: oil viscosity,cp Porosity fraction C t: total formation compressibility,psi -1
SPE 164638 13 R w : wellbore radius,ft T d: dimensionless time Storability ratio fraction λ: inter porosity transfer coefficient s: skin factor SI Metric Conversion Factors bbl 1.589873 E-01 = m 3 cp 1.0 E-03 = Pa.s ft 3.048 E-01 = m ft 3 2.831685 E-02 = m 3 psi -1 1.450377 E-04 = Pa -1 psia 6.894757 E+00 = kpa md 9.869233 E-04 = μm 2 References [1] Warren, J. E., Root, P. J., The Behavior of Naturally Fractured Reservoirs, SPE Journal, (1963), Vol.15, 9,, 51-63. [2] Fetkovich, M. J., Decline Curve Analysis Using Type Curves, JPT, Vol.32, issue 7, 1980, 1065-1077. [3] van Evardingen A. F., Hurst, W. The Application Of The Laplace Transformation To Flow Problem In Reservoirs, Trans, AIME (1949),186, 305-24. [4] Ahmed, T., Reservoir Engineering Handbook, Second Edition, Gulf Professional Publishing Company, Houston, Texas, U.S.A, (2001),1-86.
14 SPE 164638 [5] Daungkaew, S., Hollaender, F., and Gringarten, A. C. (2000). Frequently asked questions in well test analysis. SPE paper No. 63077, SPE Annual Technical Conference and Exhibition, October 1 4, Dallas, Texas. [6] Ewens, S., and Pooladi-Darvish, M. (2006). When does a longer shut-in lead to larger radius of investigation? SPE paper No. 103608, SPE Annual Technical Conference and Exhibition, September 24 27, San Antonio, Texas. [7] Taheri, A., and Shadizadeh, S. R. (2005). Investigation of well drainage geometries in one of the Iranian south oil fields. SPE paper No. 2005-028, Canadian International Petroleum Conference, June 7 9, Calgary, Alberta, Canada. [8] Karimi, S., Shadizadeh, S. R.,: A New Equation for the Determination of a Well Drainage Area in Naturally Fractured Reservoirs, Petroleum Science and Technology, (2012), 30,18, 1953-1964 [9] Jahanbani, A., Shadizadeh, S. R., Determination of Inflow Performance Relationship (IPR) By Well Testing, Paper 2009-086, Canadian International Petroleum Conference, Canada, 2008. [10] Matthews, C. S. and Lefkovits, H.C.: Studies On Pressure Distribution in Bounded Reservoir at Steady State, Trans., AIME (1955) 204, 182-189 [12] Matthews, C. S., Born, F. and Hazebroek, p.: A Method For Determination Of Average Pressure In A Bounded Reservoir, Trans., AIME (1954) 201, 182-191. [13] Matthews, C. S., and Russell, D. G., Pressure Buildup and Flow Tests in Wells, Monograph Vol. 1, Society of Petroleum Engineers of AIME, Dallas, U.S.A, TX, Millet the Printer, 1967. 1-111 [14] Ahmad, T., McKinney, P.D., Advance Reservoir Engineering, First Edition, Gulf Professional Publishing Company, Houston, Texas, U.S.A, 2005, 1-164. [15] Dietz, D. N.: Determination of Average Reservoir from Buildup Survey, J. Pet. Tech.
SPE 164638 15 (1961), Vol.13, 8, 803-805. [16] lgbokoyi, A. O., Tiab, D., Estimation of Average Reservoir Pressure and Drainage Area in Naturally Fractured Reservoir-Tiab s Direct Synthesis, SPE paper No. 104060, First International Oil Conference and Exhibition in Mexico, August 31 September 3, Cancun, Mexico. [17] Restrepo, D., Tiab, D. Fracture Porosity of Naturally Fractured Reservoir SPE 104056, Proceedings, International Oil Conference and Examination, Cacun, Mexico, 31-Aug-2006 to 2-Sep-2006. [18] Tiab, D., Donaldson, Erle C,; Petro physics: Theory And Practice Of Measuring Reservoir Rock and Fluid Transport Properties 2 nd Ed. Elsevier, pages 541-542 [19] Helmy, M. W.,Wattenbarger, R. A.: New shape factors for wells produced at constant pressure, 1998 SPE Gas Technology Symposium held in Calgary, Canada, (March,1998),Canada. SPE 39970
16 SPE 164638 12000 10000 flowing pressure (psi) 8000 6000 4000 p int y = 4.7459x + 9794.4 2000 0 0 20 40 60 80 100 120 140 160 180 200 Time(hour) Figure1. Draw down test for long time to reach semi-steady state time 11500 11000 10500 Pwf=10686-594.5logt Pwf, (Psia) 10000 9500 Pwf=10376-609.7logt 9000 8500 8000 0.1 1 10 100 1000 Time (Hour) Figure2. Draw down conventional test to obtain slop of reservoir in logarithmic time [11]
SPE 164638 17 12000 10000 Flowing Pressure (Psia) 8000 6000 4000 2000 0 0 500 1000 1500 2000 2500 3000 Producing Flow Rate (STB/D) Figure 3. Inflow performance relation (IPR) Curve 10000 1000 T* P', P' 100 10 T*DP/DT P 1 0.1 1 10 100 1000 Time (Hour) Figure 4: Log-log drawdown TDS plot [11]
18 SPE 164638 Table 1. implemented and figured out reservoir property along with carried out well characterstics c t 24.5 ( psi -1 ) 0.048 r w 0.2917 (ft) p i 11347 (psi) H 65 (ft) Q 2700 (STB/D) Β 1.8235 (RB/STB) µ 0.362 (cp) K from plotting 7 (MD) S from plotting 4.7 ω 0.13 λ 1.02 10-6 Table 2. Inflow performance relation (IPR) data 11347 0 8981 2700