Rate Decline Analysis for Horizontal Wells with Multiple Sections

The widely used application of horizontal well makes it significant to effectively evaluate rate performance of horizontal well in oil and gas reservoir. However, most models in previous work only focus on rate decline analysis (RDA) of horizontal well with single section (HWSS); they hardly address the problem that production rate distributes nonuniformly along horizontal wellbore in analyzing rate transient behaviors. However, only some horizontal segments contribute to the total production rates, and the production of each section along horizontal wellbore is not the same in fact, which may be caused by reservoir heterogeneity, selective completion, and nonuniform formation damage along horizontal wellbore. Therefore, the effect of these phenomena on rate decline characteristics cannot be ignored. The aim of this paper is to propose an analytical model to investigate transient rate response of a horizontal well with multiple sections (HWMS). The compound type curves, including the normalized production curve, the normalized production integral curve, and the production integral derivative curve, are developed to distinguish the different cases.The influences of some sensitive parameters ondecline curves are further discussed. Results showobviousdifferences on the decline curves between the HWMS and HWSS. The parameters are sensitive on decline curves, which explore the feasible application on production performance evaluation and parameters interpretation through history matching the production data with the compound type curves in this paper.


Introduction
In the past decades, horizontal wells have been widely used in exploration and development of low permeability reservoirs and unconventional reservoirs [1][2][3][4][5][6][7].Rate decline analysis for oil and gas reservoirs is needed for well performance evaluation and reservoir characterization [8].Arps [9] proposed a series of decline charts to make it convenient for decline curves extrapolation.Then, Fetkovich [10] developed theoretical model and log-log curves to demonstrate both infinite and finite reservoir solutions.To analyze rate decline data where pressure drop and flow rate vary significantly, Blasingame [11] established method for transforming variable-rate system into equivalent constant system using material balance time function.
Many researchers investigated transient rate behavior of horizontal wells.Pratikno et al. [12] provided new type curves for rate decline analysis of fractured well.For analyzing tight oil reservoir, Clarkson and Pedersen [13] provided a method which combines traditional techniques (e.g., flow regime analysis, analytical model history matching, and type curve matching).Duong [14,15] developed a novel approach to predict future rate for fracture-dominated well in unconventional reservoirs.With considering slippage and desorption in shale gas reservoir, Nobakht et al. [16] established an analytical methodology to analyze rate transient behavior.Belyadi et al. [17] evaluated the productivity of Marcellus Shale wells based on rate transient analysis.On the basis of the dynamic drainage area concept, Qanbari and Clarkson [18] developed a new rate-transient-analysis technique for shale reservoirs.Kuchuk et al. [19] presented the rate decline response of fractured horizontal wells for both conventional and unconventional formations.
Producing Section In the last decade, numerous research works have been presented on RDA of horizontal wells [20][21][22][23][24][25][26][27].Although most models in previous study focus on rate decline analysis with one whole section, they did not specifically address the problem that production rate distributes nonuniformly along horizontal wellbore.This may be caused by several reasons.On one hand, due to selective completion, total production rate is contributed from less than the entire well interval.On the other hand, even if the entire well interval is open, on account of reservoir heterogeneity and nonuniform formation damage, production rate of each section along horizontal wellbore is not the same.Some of horizontal sections have little or no contribution.Since this phenomenon exists in horizontal wells with selective completion and open-hole completion, when analyzing transient rate behavior, nonuniform distribution of parameters subject to each section (e.g., length, production rate, skin factor, and location) should be taken into consideration [28][29][30].

Malfunctioning section
Pressure transient analysis (PTA) is a reliable method for evaluating well performance and reservoir characteristics of wells with multiple sections.Kamal et al. [31] presented an analytical model which allows for a horizontal well to consist of multiple sections.A new flow regime was found, which behaves as a horizontal line with the value of 0.5 divided by number of horizontal sections on pressure-derivative curve.Other PTA models are proposed in different works [32][33][34][35][36].He et al. [37] proposed a method for estimating malfunctioning horizontal wellbore through bottom hole pressure data.Qin et al. [38] developed novel approach to evaluate nonuniform production distribution of multifractured horizontal wellbore based on PTA.During past years, although plenty of work has been developed for PTA of horizontal wells and multifractured horizontal wells with multiple sections [39][40][41][42][43], rate decline analysis of HWMS is still rarely documented.
The major objective of this paper is to establish an analytical model for analyzing transient rate behavior of HWMS.The effects of some sensitive parameters on decline curves are further investigated (i.e., production rate distribution along wellbore, number of horizontal sections, length of horizontal sections, and spacing between adjacent horizontal sections).

Physical Model of Horizontal Well with Multiple Sections
Figure 1 illustrates the physical model of HWMS (top view and side view).Major assumptions are as follows: (i) Formation is circular-bounded with a constant thickness h, porosity , horizontal permeability k h , vertical permeability k v , initial pressure p i , and total compressibility C t .
(ii) The HWMS consists of n sections.Due to formation damage, each section owns specific properties (e.g., length L k , production rate q k , and skin factor S k ).
(iii) The fluid is assumed to be single-phase oil or gas in this model.Total production rate equals to q.
(iv) The gravity effect and capillarity effect are negligible.

Solution of HWMS Model
In the circular-bounded reservoir, the analytical solution for pressure drop of HWMS at any point can be derived on the basis of the instantaneous source solutions raised by Gringarten and Ramey [44] and Newman product method [45].
Assuming that an HWMS is located in a circularbounded formation, horizontal wellbore can be regarded as continuous-line-sources with length of L. Through the integration of point-source functions, the line-source solutions can be derived.
For a single point source in the infinite-slab reservoir, it can be considered as a result of plane sources in three different directions (x-, y-, and z-).
By using Newman product method, the point-source function can be obtained: where dV represents flow rate per volume. ℎ and  V are diffusivity in horizontal and vertical directions, respectively.Through the integration of point-source function, the line-source solution is given by where ds represents rate per unit length when t equals to .
When there is a continuous flow with the rate of q k from the k-th section located at point (x k , y k , z k ), the pressure drop at point (x, y, z) can be expressed as where ℎ * means the formation thickness considering permeability anisotropy and  is anisotropy coefficient.Through the principle of superposition, pressure drop caused by multiple sections can be obtained: denoting The dimensionless length and production rate of the k-th section, dimensionless length of horizontal well, and dimensionless time are defined as Other dimensionless variables are defined as follows: As a result, dimensionless pressure drop of an HWMS with n sections is given by ) Since formation damage caused by drilling and completion cannot be ignored, dimensionless pressure drops resulted by skin effect can be written as (  ,   ,   ,   ) =   (  ,   ,   ,   ) By converting solution into Laplace domain, dimensionless pressure drop with considering the effect of wellbore storage can be expressed as On the basis of Laplace transformation, analytical solution for unsteady-state pressure drop of HWMS with incorporating skin effect and wellbore-storage effect at any point can be derived: where   and   represent the dimensionless pressure with and without considering wellbore storage, respectively, and u is the Laplace transform variable.Through numerical inversion algorithm developed by Stehfest [46] and Duhamel principle [47], rate solution of HWMS can be obtained.
The circular-boundary effect can be taken into consideration by using the method from Pratikno et al. [12]: By using Duhamel principle [47], dimensionless rate solution of HWMS under constant pressure located in circular-bounded formation in Laplace space can be expressed as can be converted into the real space using the Stehfest inversion algorithm [46].
Furthermore, three types of rate-material balance time curves can be plotted.
(i) The normalized dimensionless decline production curve (  ) (ii) The normalized dimensionless decline production integral curve (  ) (iii) The dimensionless decline production integral derivative curve (  )

Model Comparison
In this part, Blasingame type curves are developed by use of HWMS model proposed in this paper and HWSS model in previous study.
For HWSS model, an HWSS with length of L and constant rate of q is located in circular-bounded reservoir.Since HWSS only consists of one section, in this case, n equals one.For HWMS model, length of each section equals 0.1L and production rate of each section is q/n.Two cases are further discussed (i.e., n=1, n=5).When n equals 1, producing section is located at the heel of horizontal wellbore.When n equals 5, sections are distributed uniformly along wellbore.
By comparing type curves from HWMS model with that from HWSS model, clear distinctions can be observed on Blasingame compound type curves except for the boundarydominated flow period, shown in Figure 2. Three kinds of curves (i.e., q Dd , q Ddi , and q Ddid ) belonging to compound type curves are shown in Figure 3. Compared to Figures 3(a) and 3(b), the use of production integral derivative smooths the derivative curve of measured data and makes it easier to identify different features as shown in Figure 3(c).Therefore, it is necessary and significant to establish the RDA model of HWMS to analyze the rate transient behavior, which enable us to better evaluate production performance.

Effect of Sensitive Parameters on Decline Curves
To understand the effects of main parameters on rate transient behaviors of a HWMS, we develop the Blasingame compound type curves to perform the sensitivity analysis through the RDA model proposed in this paper (i.e., production rate distribution along wellbore, number of horizontal sections, length of horizontal sections, and spacing between adjacent horizontal sections).

Production Rate Distribution along
Wellbore.Dimensionless production rate is defined as the production rate of each  sections are shown in Figure 5.As the number of horizontal sections decreases, all curves move downward.There exist big differences between the results for n=1 and n=5 during the transient flow period.Although the distinctions between the normalized production curves (  ) and the normalized production integral curve (  ) are not obvious enough, the production integral derivative curves (  ) show clear differences among them.Therefore, the number of horizontal sections is sensitive and can be determined through matching the production data with the compound type curves in this paper.

Length of Horizontal
Sections.Dimensionless length of horizontal sections equals length of each section divided by length of horizontal wellbore (  =   /L).Two sections with the same dimensionless length are located at the heel and toe of horizontal well.Both sections have the same production rate, skin factor, etc. Figure 6 shows the effect of length of horizontal sections on type curves.The normalized production curves and the normalized production integral curve are quite different especially when dimensionless length of horizontal sections become smaller.The production integral derivative curves differ from other cases for different   , which is benefit of estimating effective producing length of the horizontal well.

Spacing between Adjacent Horizontal Sections.
Dimensionless spacing between adjacent horizontal sections is defined as the spacing between adjacent horizontal sections divided by length of horizontal wellbore (ûx D = ûx/L).Five cases are designed to discuss the influences of dimensionless spacing on decline curves.As the distance from the second producing section to the heel becomes farther, decline types move up, shown in Figure 7. Though the influences of spacing between horizontal sections on type curves are not obvious compared to other parameters, it is possible for us to distinguish them based on the production integral derivative curves during the transient flow regime.

Conclusions
This paper establishes the RDA model of a HWMS and the compound rate decline type curves to analyze the effect of production distribution, number and length of producing horizontal sections, and spacing between sections on rate transient behaviors.
(1) Since different horizontal segments make unequal contribution to total production rates in fact, the effect of these phenomena on rate decline behaviors should be considered.
(2) The effects of production rate distribution along wellbore on rate characteristic are obvious especially for the production integral derivative curve during the transient flow regime, which makes it possible to estimate the production distribution through the history matching of production data.
(3) The normalized production curves and the normalized production integral curve are quite different especially when dimensionless length of horizontal sections become smaller.The production integral derivative curves differ from other cases for different   , which is benefit of estimating effective producing length of the horizontal well.
(4) Although the distinctions between the normalized production curves and the normalized production integral curve are not obvious, the production integral derivative curves show clear differences among them.Therefore, the number of horizontal sections can be determined through matching the production data.
In conclusion, the proposed RDA model enables engineers to evaluate the rate performance of horizontal wells and interpret reservoir and well parameters more effectively and accurately (e.g., estimating the production distribution of different locations along horizontal wellbore and effective producing length, permeability, well controlled area and reserves, and skin factor) through the history matching of long-term production data.If the field production data can match well with these three curves at the same time, the results can be reliable.

Figure 1 :
Figure 1: Physical model of HWMS (top view and side view).

Figure 4 :Figure 5 :
Figure 4: The effect of rate distribution on type curves.

Figure 6 :Figure 7 :
Figure 6: The effect of length of two sections on type curves.
Length of horizontal well, cm   : Dimensionless length of horizontal well Cartesian coordinates   ,   ,   : Dimensionless Cartesian coordinate   ,   ,   : Coordinates of the center of the k-th section   ,   ,   : Dimensionless coordinates of the center of the k-th section ℎ : Horizontal permeability, D  V : Vertical permeability, D :