Abstract

A new well-test model is presented for unsteady flow in multizone with crossflow layers in non-Newtonian polymer flooding reservoir by utilizing the supposition of semipermeable wall and combining it with the first approximation of layered stable flow rates, and the effects of wellbore storage and skin were considered in this model and proposed the analytical solutions in Laplace space for the cases of infinite-acting and bounded systems. Finally, the stable layer flow rates are provided for commingled system and crossflow system in late-time radial flow periods.

1. Introduction

Many reservoirs are formed from layers of different physical properties because of the different geological deposition rotary loops. Among them, if these layers do not communicate in terms of fluid flow through the formation but may be produced by the same wellbore, these types of reservoir are called commingled systems; if there exits fluid that connects between these layers, they are referred to as crossflow systems. The pressure-transient behavior depends on the comprehensive properties of these multilayers.

For the unstable flow of Newtonian fluids in a multilayered reservoir, Russell et al. [1, 2] studied pressure behavior of single-phase fluid in two layers with formation crossflow and derived the conclusion that it is similar to the flow behavior of the two layers without formation crossflow in early time. Raghavan et al. [3] studied the problem of well test in a multilayered reservoir. Bourdet [4] established using steady state approximation to the presentation interlayer flow model. Gao and Deans [5] studied the behavior of multilayered reservoir with formation crossflow. Ehlig-Economides [6] has systematically established the combination of commingled system and crossflow system unsteady flow models and provided the rule of the pressure and flow for each layer. Bidaux et al. [7], using layered pressure and flow data, conducted a study on the theory and practical application of multilayered reservoir.

For the unsteady flow of non-Newtonian fluids in a multilayered reservoir, van Poollen and Jargon [8] studied non-Newtonian power-law fluid unsteady flow in porous media and showed that the transient pressure response characteristics are different from that of Newtonian fluid. Ikoku and Ramey [9] studied non-Newtonian power-law fluid unsteady flow characteristics in porous medium, and the consideration of wellbore storage and skin effect is obtained in homogeneous infinite reservoir model in Laplace space solution. Lund and Ikoku [10, 11] proposed non-Newtonian power-law fluid (polymer solution) and Newtonian fluid (oil) composite model of transient well-test analysis method. Xu et al. [12] proposed the infinite reservoir Laplace space spherical transient pressure solution and discussed the characteristics of wellbore pressure at early times and later times. The above research result is to solve the problem of single layer. Escobar et al. [13] presented equations to estimate permeability, non-Newtonian bank radius, and skin factor for the well test data in reservoirs with non-Newtonian power-law fluids. Martinez et al. [14] studied the transient pressure behaviors for a Bingham type fluid and the influence of the minimum pressure gradient. Escobar et al. [15, 16] studied transient pressure analysis for non-Newtonian fluids in naturally fractured formations modelled as double-porosity model. They [17] extended TDS technique to injection and fall-off tests of non-Newtonian pseudoplastic fluids. van den Hoek et al. [18] presented a simple and practical methodology to infer the in situ polymer rheology from PFO (Pressure Fall-Off) tests. To the problem of multilayered, Yu et al. [19] established a well testing model for polymer flooding and presented a numerical well testing interpretation technique to evaluate formation in crossflow double-layer reservoirs.

This paper presents a well-test model and the analytical solution for non-Newtonian polymer-flooding unsteady flow in multizone with crossflow layers and laid the foundation theory of field test data interpretation.

2. The Model Description

The reservoir model for the -layered system is shown in Figure 1. Each layer is assumed to be homogeneous and isotropic, with injected polymer non-Newtonian power-law fluids.

A symmetrically located well penetrates all the layers, and each layer has a skin of arbitrary value , wellbore storage coefficient is assumed to be constant, and crossflow may occur in the reservoir between any two adjacent layers.

Assuming that the fluid is slightly compressible, the compression coefficient is constant, the permeability, porosity, and thickness of each layer can be different, respectively, to distinguish between layer pairs with formation crossflow with and noncommunicating layers, and the reservoir is divided into (≤) zones. Between any two adjacent zones, there is no formation crossflow. Gravity and capillary forces can be neglected. Assuming weak formation crossflow, the flow is about interlayer pressure difference and has nothing to do with the shear rate, crossflow coefficient , on behalf of interlayer communicating ability. Formation crossflow is modeled as in the semipermeable-wall model of Deans and Gao [7], which assumes that all resistance to vertical flow is concentrated in the wall (layer top, bottom). Hence, the pressure difference between adjacent layers depends on only radial position and time, and flow within the layers is strictly horizontal, and assuming that each layer has the same initial pressure.

The flow in each layer () is governed by the following equation: where is given by where , and are thickness and vertical permeability of a nonperforated zone between layers and , and , where is the vertical permeability for layer that is the resistance to flow per unit length at the th layer interface. If there is no nonperforated zone between layers and , then the flow resistance on behalf of a layer interface unit length. If on the layer and the layer but perforated belt, then is zero. If there is no formation crossflow between layers and , then is zero.

The sand surface flow of each layer as a function of time:Assume that polymer solution viscosity and shear rate of power-law relations [4] are as follows: where is a constant and and , respectively, represent the shear rate and power-law index; when tends to one, fluid showed a Newtonian fluid properties.

Because of the flow in each layer changing over time, the crossflow rate is much smaller than stable flow rate, so the flow rate can be replaced by steady flow rate to calculate the shear rate:where is a constant and layer stable flow rate can be obtained by the late-time in unsteady flow.

With the viscosity of polymer solution into (1) for the shear rate representation, for radial flow, under cylindrical coordinate, dimensionless forms are obtained by finishing after

The boundary condition at the well is given by the following equations, which account for both skin and wellbore storage:For infinite-outer-boundary condition, For no-flow outer-boundary condition,For initial condition,For dimensionless layer flow rates,where the dimensionless variables are defined by the following:

3. The Solution of the Model

The above equation is given by Laplace transform on time.

Set The basic control equation solution is as follows:where the subscript on indexes the layer and the superscript indexes , , and are, respectively, the first and the two class of order modified Bessel function. is eigenvalue of real symmetric three diagonal positive definite matrices [], whereAccording to the boundary conditions, a function relationship between and is as follows:

Then, values for can be computed directly from the above recursion formula. Let zone () contain layer, the zone with a specific layer , dimensionless pressure:where each layer index, , in the sum refers to the same zone as layer .

Finally, the coefficients for each zone can be expressed as multiples of the coefficients, , , of the uppermost layer in zone with (12):External boundary condition implies the relationship between and is as follows:For infinite-outer-boundary condition,For no-flow outer-boundary conditions,According to the boundary conditions, the layer and layer which in the same zone have the following relations:The layer of zone and the layer of zone have the following relations:The wellbore pressure is as follows:Equation (31) is a general solution, and it is by no wellbore storage pressure solution conversion into the wellbore storage pressure solutions. Therefore, we will solve the cases of equations.

Equations (29)-(30) are linear equations with coefficient which can be solved by numerical method. The wellbore pressure without wellbore storage is given asIn addition, layer flow rate is given as follows:Distribution of radial pressure on each layer becomesFor the case of double-layer reservoirs with formation crossflow, assume that choosing parameters is as follows.

, , , , , and , through the numerical inversion of the Laplace transform of Stehfest [20]. Pressure and flow in the real space are obtained. The wellbore pressure curve is shown in Figure 1. The wellbore pressure curve is shown in Figure 2. It can be seen that the pressure derivative curve is unit slope straight line in the early-time; namely, linear unit slope represents pure effect of wellbore storage; in the medium term, the curve is concave interporosity flow transition characteristics; in the late stage, the pressure derivative curve is approximately straight line up, and the slope is related to the power law index. The slope is . And it can be seen that the pressure derivative rises and increases with the reduction of power-law index steepened.

In Figures 3 and 4 flow curve can be seen; for the first layer with high quasi capacity coefficient, the dimensionless flow rate increases with the time increasing and finally tends to be stable flow rate ; for the second layer with low quasi capacity coefficient, the dimensionless flow rate increases with the time increasing. In a certain period, due to high permeability layer crossflow, pressure tends to be balanced, and the crossflow that is more and more weak, finally, tends to be stable flow rate .

4. The Late Stable Flow Model

For both cases stable layer flow-rates were discussed, including (1) without formation crossflow and (2) with formation crossflow. We only discussed the behavior in late time, so ignore the effect of wellbore storage effect.(1)Without Formation Crossflow.for the infinite-out-boundary, the stable flow rate, for layer , is as follows:For the no-flow-boundary, the stable flow rate, for layer , is as follows:(2)With Formation Crossflow.Regarding the eigenvalues of matrix , there is a value to meet , and the rest is independent of and and depends only on and . Assume that . Stable layer flow rate through the determinant value can be expressed aswhere det is determinant of matrix .The matrix element is as follows:The matrix is used in matrix in column replacement for , where superscript is the vector transpose.

5. Conclusions

Establishing the multilayer reservoir well-test model for polymer flooding gives the formation and wellbore transient pressure and the layer flow-rate finally. The expressions of late time stable flow rate in each layer are given.

The unsteady well-test model provides theoretical method for polymer flooding well test analysis of multilayer reservoir; the experimental data and the theoretical curve fitting can be used to determine the permeability, skin factor and effective interlayer vertical permeability, and other important parameters.

Nomenclature

:Matrix elements defined in (22)
:Coefficient for th layer, th root defined in (23)
,:Coefficient for th layer, th root, in zone , defined in (25)
:Coefficient for outer boundary condition defined in (27) or (28)
:Wellbore-storage coefficient (cm3/kPa)
:Total compressibility in layer (kPa−1)
:Matrix elements defined in (38)
:Constant defined in (5)
:Formation thickness in Layer (cm)
Constant defined in (4)
:Modified order Bessel function of the first and second kind
:Thickness of a nonperforated zone between layers and (cm)
:Horizontal permeability in Layer , μm2
:Vertical permeability of a tight zone between layers and (μm2)
:Vertical permeability in layer (μm2)
:Constant defined in (20)
:Number of layers in Zone
:Power-law index
:Number of layers in reservoir system
:Number of Zones in reservoir system
:Reservoir pressure in layer (kPa)
:Reservoir pressure in layer in Zone (kPa)
:Dimensionless reservoir pressure in layer defined in (12)
:Initial reservoir pressure (kPa)
:Dimensionless bottomhole pressure in Laplace space
:Dimensionless bottomhole pressure without wellbore storage in Laplace space
:Surface production rate (cm3/s)
:Flow rate for layer (cm3/s)
:Stable flow rate for layer (cm3/s)
:Radial distance (cm)
:Dimensionless radius, defined in (18)
:Reservoir outer radius (cm)
:Dimensionless outer reservoir radius, defined in (19)
:Wellbore radius (cm)
:Wellbore skin factor for layer
:Time (s)
:Elapsed time after rate change (s)
:Dimensionless time referenced to producing layer
:Shear rate in layer (s−1)
:Crossflow coefficient between layers and defined in (2)
:Resistance to flow per unit length at the th layer interface
:Laplace space variable
:Coefficient for layer , Root , defined in (23)
:Coefficient for layer , Root , defined in Zone , defined in (25)
:Coefficient for layer , defined in (14)
:Dimensionless semipermeability between layers and , defined in (16)
:Dynamic viscosity (mpa·s)
:Porosity fraction for layer
:Coefficient for layer , defined in (15)
:Shear rate for layer defined in (5)
:Determinant of matrix .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project was supported by the National Key Science and Technology Project (2011ZX05009-006), the Major State Basic Research Development Program of China (973 Program) (no. 2011CB707305), and the China Scholarship Council, CAS Strategic Priority Research Program (XDB10030402).