ZHANG Lie-hui, GUO Jing-jing, LIU Qi-guo

State Key Laboratory of Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China, E-mail: zhangliehui@vip.163.com

A NEW WELL TEST MODEL FOR A TWO-ZONE LINEAR COMPOSITE RESERVOIR WITH VARIED THICKNESSES*

ZHANG Lie-hui, GUO Jing-jing, LIU Qi-guo

State Key Laboratory of Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu 610500, China, E-mail: zhangliehui@vip.163.com

(Received June 7, 2010, Revised October 1, 2010)

This article presents a new well test model for two-zone linear composite reservoirs, where the rock and fluid properties as well as the formation thicknesses on both sides of the discontinuity are distinctly different. An analytical solution of pressure-transient behavior for a line-source, constant-rate well in this type of reservoir configuration is obtained with Fourier space transformation and Laplace transformation. By applying Duhamel principle, the wellbore storage and skins effects can easily be included. A set of type curves are generated and the sensitivities of the relevant parameters are discussed. A new correlating parameteris proposed to identify the pressure response in the pressure derivative curve. The model as well as the corresponding type curves are quite general that they are useful in predicting the production performance or analyzing the production data from this type of well-reservoir systems.

non-uniform thickness, linear composite, well test model, type curve

1. Introduction

Many prolific reservoirs are located in channelized environment. The well test interpretation is always a challenging issue because of the variations of sand body geometry and reservoir properties. Due to the facies changes in plane, the mobilities change from the center to the edge of the channels, i.e., there are possibly two or even more regions with varied reservoir properties. The geometric model of this type of reservoir configuration can be envisaged as a linear composite strip reservoir with different mobilities, diffusivities and thicknesses.

The mathematical models for radial composite reservoirs are well developed and validated by numerous studies[1-8]. However, to our knowledge, the well test models for strip reservoirs are so far not available in literature. The effect of the formation thickness variation on the transient response of a well in this type of reservoir configuration is not well studied. In 1963, Bixel et al.[9]first proposed a transient well test model for two-zone linear composite strip reservoirs which was then improved by Streltsova and Mckinley[10]. In 1987, Ambastha et al.[11]introduced an additional pressure drawdown at the interface in a revised well test model. However, the effects of skins and wellbore storage were not included. In 1996, in building the Ambastha’s model, Bourgeois et al.[12]neglected the pressure loss at the discontinuity and presented a well test model for three-zone linear composite reservoirs bounded with parallel faults. In 1997, Kuchuk and Habashy[13]extended the model to multi-zone conditions. These models have accounted for the varied porosity, permeability and compressibility, but not so much for the effect of varied thicknesses. It has to be pointed out that the assumption of isopach reservoir is not often valid in a real case and significant changes in reservoir thickness could occur from one region to another.

In this article, based on the theory of fluid flow in porous media, a new well test model for a two-zone linear composite strip reservoir with non-uniformthickness is proposed. The analytical solution is obtained with the finite Fourier cosine space transformation[14]and Laplace transformation. By using Duhamel principle, the corresponding wellbore pressure response is obtained, together with effects of skins and wellbore storage. The type curves are obtained with Stehfest numerical inversion method[15]and the impacts of relevant parameters are discussed. The model and the corresponding type curves in this paper can be applied to the well test analysis of a general channelized reservoir.

2. Mathematical model

The mathematical model proposed in this article is based on oil reservoir conditions, it is also applicable to gas reservoirs by replacing the pressure term with a pseudopressure term. The key assumptions made in developing the mathematical model include:

(1) The strip reservoir can be divided into two semi-infinite regions. The well is located in Region I and is considered as a constant-rate line source, as illustrated in Fig.1. The two reservoir Regions on both sides of the discontinuity may have different rock and fluid properties. But each one of the two zones is homogeneous and isotropic with constant reservoir properties such as permeability and porosity.

(2) Single-phase slightly compressible fluid and isothermal flow.

(3) Reservoir properties change at the interface. Both the width and the flow resistance along the interface are neglected.

(4) Laminar flow in each zone, with negligible gravitational effect and capillary effect.

(5) Uniform initial reservoir pressure (pi).

Based on the assumptions and the coordinate system shown in Fig.1, the diffusivity equations in the dimensionless form for Regions I and II, respectively, are as follows

where p1Dis the dimensionless pressure drop in Region I, p2Dis the dimensionless pressure drop in Region II, xDis the dimensionless distance in x direction, yDis the dimensionless distance iny direction,aDis the dimensionless distance between the well and the interface, bDis the dimensionless y coordinate of the well location, wDis the dimensionless reservoir width, tDis the dimensionless time, ηDis the transmissibility ratio, and δ is the delta function denoting the constant-rate line-source well.

The initial conditions can be described as

The pressure continuity at the interface is expressed as

The flow continuity at the interface means

where M is the mobility ratio, andDh is the thickness ratio.

The dimensionless quantities mentioned above are defined as follows

where the subscripts of 1 and 2 denote the properties for Region I and Region II respectively. k is the permeability, h is the reservoir thickness,μ is the fluid viscosity, φ is the porosity, Ctis the total system compressibility, p is the pressure, piis the initial reservoir pressure, rwis the wellbore radius, t is the production time,x and y are the coordinates of a point, a and b are the distance between the well and the interface inx andy directions, respectively, w is the reservoir width,q is the flow rate at sandface.

3. Model solution

The finite Fourier cosine transformation and Lapalce transformation are employed to solve the dimensionless well testing model.

With Eqs.(3) and (7), the Laplace transformation with respect toDt and the finite Fourier cosine transformation with respect toDy of Eq.(2) yields

u is the Laplace variable with respect to time, m is the Fourier variable,is the dimensionless pressure of Region II in Fourier-Laplace space.

The solution to Eq.(10) is

With Eq.(5) and Eq.(11), one obtains

With Eqs.(3) and (6), the Laplace transformation with respect toDt and the finite Fourier cosine transformation with respect toDy of Eq.(1) give

The solution of Eq.(13) is not readily obtained because of the delta function at the right hand side. The Laplace transformation of Eq.(13) with respect to xDyields

where s is the Laplace variable with respect to xD, and W1 refers toin Laplace space with respect to xD.

W1can be obtained by solving Eq.(14). A term-by-term Laplace inversion transformation and using conditions at the interface, one obtains

Thus, the dimensionless pressure drops in the Laplace-Fourier space at any location in Region I and II at any time are given by Eqs.(17) and (18), respectively

4. Type curves and discussions

The steps to generate the type curves are as follows:

(1) The finite Fourier cosine transformation is inverted numerically.

(2) The wellbore pressure with effects of wellbore storage and skins is calculated by using Duhamel principle.

(3) Inverse numerical transformation of Laplace equation using Stehfest algorithm.

(4) Calculate the pressure expression in the real space considering the effects of skins and wellbore storage.

(5) The dimensionless wellbore pressure drawdown then can be calculated by setting xD=aD?1 and yD=bD.

(6) Plot type curves based on the results obtained in above steps.

It is shown that a number of variables may affect the characteristics of type curves, such as thickness ratioDh, mobility ratio M, transmissibility ratio Dη, well location in the reservoir (Da andDb) and reservoir widthDw. The effects of all these parameters on type curves were extensively discussed in literature except for thickness ratioDh. Therefore, the effect of thickness ratioDh is discussed in detailin this article. In addition, the effect of a parameter combination ofis also discussed.

Figures 2, 3 and 4 show the pressure responses as a result of varied thickness ratio and different well locations within the strip reservoir. The wellbore storage period and the homogeneous radial flow period can be seen from all the three figures irrespective of well locations. The effect of thickness ratio on type curves becomes significant after the pressure wave reaches the interface.

For the case aD＜bD, as shown in Fig.2, the pressure wave first reaches the interface and the effect of interface begins to appear. A second horizontal line can be observed in the pressure derivative curve which is the reflection of the radial flow in the equivalent homogeneous reservoir. The elevation of the second horizontal line depends on the average reservoir properties of the two regions. It is smaller than 0.5 if hD＞1 and greater than 0.5 ifhD＜1. The line remains horizontal until the pressure wave reaches the parallel impermeable boundaries. After that, the flow in reservoir becomes a linear flow in the equivalent homogeneous reservoir characterized as a half-slope line on the log-log graph of the pressure-derivative responses.

For the case aD＞bD, as shown in Fig.3, the effect of one impermeable boundary appears first. As a result, the pressure derivative rises to 1.0 and continues rising until the pressure wave reaches the interface. Then, the linear flow behavior is detected as a half-slope line. In general, the greater the thickness ratio is, the lower the pressure and the pressure derivative curves become.

For the case aD=bD, as shown in Fig.4, the effects of parallel impermeable boundaries and interface appear simultaneously, and the linear flow in the strip reservoir is observed as a half-slope line. The effect of thickness ratio on type curves is similar to that of Figs.2 and 3.

Figure 5 shows a correlating parameter,, for wellbore pressure behavior for a well in a strip reservoir. Identical pressure responses are observed with the samevalue. Therefore, it can be established as a correlating parameter for the pressure responses in a non-isopach, linear composite strip reservoir.

5. Model validation

6. Conclusions

(1) A new well test model for a two-zone linear composite strip reservoir is proposed, which takes into account the changes in reservoir thickness, wellbore storage effect and skin factor. The analytical solutionis obtained in Fourier-Laplace space but it can be easily inverted to the real space. The type curves are obtained and the effects of correlating parameters are analyzed. This new well test model is very useful in predicting the pressure performance or analyzing the test data for this type of reservoir configurations.

(2) The effect of thickness ratio can be identified after the pressure wave reaches the interface. The greater the thickness ratio is, the lower the pressure and the pressure derivative curves are.

(4) The mathematical model proposed in this article is applicable to both oil and gas reservoirs. Just by replacing the pressure term with a pseudopressure term, the model can be applied to gas reservoirs.

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10.1016/S1001-6058(09)60119-0

* Project supported by the National Key Basic Research Program of China (973 Program, Grant No.2006CB705808), the State Major Science and Technology Special Project during the 11th Five-year Plan (Grant No.2008ZX05054).

Biography: Zhang Lie-hui (1967-), Male, Ph. D., Professor