Derivation and application of mathematical model for well test analysis with variable skin factor in hydrocarbon reservoirs

Skin factor is often regarded as a constant in most of the mathematical model for well test analysis in oilfields, but this is only a kind of simplified treatment with the actual skin factor changeable. This paper defined the average permeability of a damaged area as a function of time by using the definition of skin factor. Therefore a relationship between a variable skin factor and time was established. The variable skin factor derived was introduced into existing traditional models rather than using a constant skin factor, then, this newly derived mathematical model for well test analysis considering variable skin factor was solved by Laplace transform. The dimensionless wellbore pressure and its derivative changed with dimensionless time were plotted with double logarithm and these plots can be used for type curve fitting. The effects of all the parameters in the expression of variable skin factor were analyzed based on the dimensionless wellbore pressure and its derivative. Finally, actual well testing data were used to fit the type curves developed which validates the applicability of the mathematical model from Sheng-2 Block, Shengli Oilfield, China.


I. INTRODUCTION
Skin factor is a parameter which is used to evaluate the degree of influence of formation permeability around the wellbore by external factors. 1,2 The conception of "thin-skin mathematical model" was initially proposed to adjust the error of bottom hole pressure between theoretical calculation and measured data by van Everdingen. 3 However, this conception does not have reasonable explanation in percolation mechanics theory when it was used to explain an increase in permeability around wellbore. Hawkins introduced an improved model called "thick-skin mathematical model" which was accepted by researchers and engineers and developed further into what is now called "skin factor". [2][3][4] There are many factors that causes skin effect which includes invasion of mud during drilling process, incomplete perforation, fines migration, hydraulic fracturing, acidification and so on. [5][6][7][8] Skin factor is an important parameter that is obtained in well test analysis. In most of the well test mathematical models, the skin factor is regarded as a constant. [9][10][11][12] In fact, due to turbulence effect of fluids and fines migration caused by the interaction between fluids and sands; hence the value of skin factor is variable. Some researchers believed that skin factor is related to turbulence effect of fluids due to the fact that high speed non-Darcy flow will produce additional pressure drop at the bottom of the well. [13][14][15][16][17] Moreover, the authors also proposed their own mathematical model of skin factor that changes with production rate. However, variable skin factor caused by sand migration lacked systematic study. Larsen and Kviljo studied cleanup effects which could change skin factor. 18 The variable skin factor was represented by a simple hyperbolic function of time which was absent in theoretical evidence. In addition, some researchers built filtration models which describe the particles capture and detachment. [19][20][21] As a result, these mathematical models are difficult to apply in well test analysis in oilfields.
The paper proposed mathematical model of the variable skin factor which was introduced into existing traditional well test models instead of using a constant skin factor to establish a new well test mathematical model. Also, the type curves generated from the new mathematical model came about as a result of using Laplace transform to solve this model. It can be used universally to conduct well test in single oil phase flow, especially in super-low and -high permeability oil reservoirs, their permeabilities around well area change significantly in short time for these two kinds of reservoirs. In effect, these results supplement the content of modern well test analysis.

A. Derivation of variable skin factor
In light of the definition of the skin factor in steady radial fluid flow of vertical wells, the mathematical equation is given as 4 Where S is skin factor, dimensionless; k is average permeability, µm 2 ; k s is average permeability of the damaged area, µm 2 ; r s is the radius of the damaged area, m; r w is the radius of wellbore, m.
Assuming that the radius and average permeability of a damaged area are constant, that is to say, r s and k s are changeless, so the skin factor is a constant which is the usual treatment of skin factor in most literatures and production. In fact, k s is a variable which changes with production time, and its change regulation is as follows.
In the early flow period, the rate of k s changing is fast, because the fines and uncemented particles are easily washed into the wellbore (the radius of particles are smaller than the radius of pores and throats. Generally, the average permeability of reservoir in this case is high) or accumulated around the wellbore (the radius of particles are larger than the radius of pores and throats. Generally, average permeability of reservoir in this case is low). Furthermore, with the elapse of time, these fines and uncemented particles continue to lessen, and the rate of k s changing become more and more slow until k s tends to a constant. This variation can be approximated as Where, k si is the average permeability of damaged area at initial time, µm 2 ; v i is initial rate of permeability changing, per day; t is time, day; β is a constant that is smaller than a unit, dimensionless.
The effects of β and v i on the average permeability of damaged area are shown in Figs. 1 and 2. According to Fig. 1, the limit of k si /k s is controlled by β, that is and if β is positive, it represents k s increasing gradually; conversely, if β is negative, it represents k s decreasing gradually. According to Fig. 2, the speed of tending to the limit of k si /k s is controlled by v i , the larger v i is, the shorter the time of k si /k s tending to 1-β becomes, the steeper is the curve, and the faster is the change speed of k s . If we define the initial skin factor as S i , Eq (1) can be rewritten as S i is a constant, S, however, is a variable which changes with time (because k s changes with time). Therefore, the relationship of S and S i can be established by combining Eqs (1), (2), and (4), it yields Therefore, the expression of variable skin factor was derived through the above analysis.

B. Assumption of the mathematical model
The assumptions of the mathematical model are as follows.
There is an infinite-acting reservoir in plane; its pay thickness is h (m), the reservoir is homogeneous, and there is a production well with complete perforation in the reservoir, whose production rate is q (m 3 /d). The initial pressure of the reservoir is p i (MPa); the bottom hole pressure is p wf (MPa); its porosity and permeability are φ (fraction) and k(µm 2 ) respectively; the radius of wellbore is r w (m); the viscosity of the fluid is µ(mPa·s); the volume factor of the fluid is B (m 3 /m 3 ); the total compression coefficient is c t (1/MPa); the skin factor is S (dimensionless); the wellbore storage coefficient is C (m 3 /MPa). Assuming that radial flow occurs in the formation and the flow follows Darcy's Law.

C. Establishment of the mathematical model
In the early flow period, the effect of wellbore storage is obvious, all or most of the fluid produced is supplied by the wellbore. The flow is not completely spread to the reservoir at this time, and the skin factor can be viewed as a constant in this period. However, when the effect of wellbore storage weakened gradually, the flow spreads completely to the reservoir, and the skin factor should be taken as variable. In light of Rammy et al., 23 the end time of the wellbore storage effect is Where, t α D is the time of wellbore storage effect terminates. The well testing model was established based upon above assumption and analysis. The dimensionless parameters are defined as follows.
The governing equation is: The initial condition is: The outer boundary condition is: The inner boundary condition is divided into two kinds of situation depends on time. 1) When t D < t α D , the effect of wellbore storage plays a major role and the skin factor is viewed as constant.
2) When t D ≥ t α D , the effect of wellbore storage can be neglected and the skin factor is viewed as variable.
Where, K 0 and K 1 are the zero and unit order modified Bessel functions respectively; s is the Laplace operator;p w D (s) is the dimensionless wellbore pressure in Laplace domain.

D. Solution of the mathematical model
Where, p w D (t D ) is the dimensionless wellbore pressure in real domain.

III. RESULTS AND DISCUSSION
The double logarithmic diagrams of p w D (t D ) and its derivatives (dp w D (t D ) /dln (t D )) versus dimensionless time (t D ) were obtained from the solution of the model. These diagrams were shown from Figs. 3-5. Four different initial skin factor (S i ) values were selected in each of the three Figures, which were 1, 5, 10, and 20, respectively. As shown in Figs. 3-5, the greater the value of S i is, the greater the value of the dimensionless wellbore pressure curve is, and the greater the peak value of derivative of dimensionless pressure is. The effect of each variable on p w D (t D ) and dp w D (t D ) /dln (t D ) was analyzed in next sections. Fig. 3 shows the effect of β on p w D (t D ) and dp w D (t D ) /dln (t D ), in which the value of r s D =40, v i D =0.00001, and C D =10, respectively. Three different values of β (-0.5, 0, and 0.5) were chosen to analyze the effect of β. The value of β reflects the ultimate changing degree of the average permeability of damaged area. The greater the β, the smaller the k si /k s , and the smaller the final skin factor becomes. This regular pattern was also reflected in Fig. 3.

A. Effect of β
As shown in Fig. 3(a), when β equals 0.5, p w D (t D ) decreases dramatically after wellbore storage effect finishes, then p w D (t D ) increases slowly with production. There will be negative among the values of dp w D (t D ) /dln (t D ), therefore, it is not continuous in the double logarithmic graph, and the larger the S i , the more the negative values among dp w D (t D ) /dln (t D ), so the wider the "gap" of dp w D (t D ) /dln (t D ) curve in Fig. 3(b). When β= -0.5, the trends of p w D (t D ) and dp w D (t D ) /dln (t D ) were opposite as compared with β = 0.5. p w D (t D ) will increase significantly after wellbore storage effect is complete, then the increasing speed becomes slow. The second peak value will appear in the curve of dp w D (t D ) /dln (t D ) in Fig. 3(b), and the larger the S i is, the greater the peak value is. When β=0, it represents that the skin factor is a constant. The p w D (t D ) will increase slowly after wellbore storage effect is complete, and the values of dp w D (t D ) /dln (t D ) equal 0.5 gradually after achieving the first peak.  As shown in Figs. 4(a) and 4(b), the larger the v i D , the faster the change rates of p w D (t D ), the earlier the second peak value ( β is negative) or "gap" ( β is positive) appears. However, when S i and β are invariant, all the curves of p w D (t D ) will eventually converge into one curve and all the curves of dp w D (t D ) /dln (t D ) will converge to 0.5 finally. (t D ) increases, the greater the ultimate value of p w D (t D ), and the greater the second peak value ( β is negative) or "gap" ( β is positive). However, as shown in Fig. 5(a), the effect of r s D on p w D (t D ) is small, especially on the case of the larger skin factor. According to the study of Mohamed et al. (2014), the radius of damaged area is smaller than 6 meters generally, and its range of variation is small as well. Therefore, the effect of r s D can be neglected.

IV. APPLICATIONS IN THE OILFIELDS
There is a well in Sheng-2 Block, Shengli Oilfield, China which was used to validate the variable skin factor well test model derived. A layer of this well has been tested in 1994. The thickness of the layer is 5.6 meters; the radius of the wellbore is 0.1 meters. Daily oil production of this well is 450 m 3 /d during the test; the oil viscosity is 4.2 mPa•s, the volume coefficient of the oil is 1.10 m 3 /m 3 , and its compression coefficient is 8.9×10 −4 MPa −1 . The bottom-hole pressure is always higher than the bubble point pressure of the oil during the test, so it is single phase of oil flow during production period.
The following procedures depict how type curve matching is employed to calculate the essential parameters of reservoirs in oilfields. Firstly, prepare two pieces of log-log graph paper which are exactly the same, one of which is plotted by pressure difference (∆p) and its derivative value (d∆p) versus test time of well test data, the other one is plotted by a family of type curve. Secondly, put the graph contained test data to the type curve graph paper, and then move the paper to obtain the best match of data with one of the type curves. Finally, read from a random match point. Fig. 6 shows the result of fitting the type curves of the proposed model in this paper with the oilfield well test data. The parameters of the type curve fitted well to the well test data were as follows: v i D = 0.001, β = 0.7, C D =100, S i = 9.0 and r s D =40. Choosing one point from the plot of the type curve overlapped the well test data arbitrarily. The coordinate of this point in the type curve is (10 6 , 5.12) and that of the well test data is (18, 4.66). Therefore, unknown parameters can be calculated as follows.
The calculated results show that the initial skin factor is 9.0, and the initial permeability of damaged area is 0.75 µm 2 . However, according to the theory of variable skin factor in this paper, the ultimate permeability around the wellbore should be 0.75/(1-β) = 2.5 µm 2 . This result agrees closely with the 2.8 µm 2 which was verified by production performance in the later stage, and it indicated that the model used in this case is suitable.
As it was shown in Fig. 6 and the parameters of the reservoir, fluid flow rate was fast during well test, and the permeability of the layer was high. The cleanup effects made the particles around the wellbore washed into it resulting in skin factor decreasing gradually. 18,21 The variation of skin factor with time is shown in Fig. 7.
As shown in Fig. 7 and above analysis, the permeability around wellbore increased from 0.75 µm 2 to about 2.5 µm 2 and the skin factor decreased from 9 to a small value with an increase in production time. If there is no account for the variation of the skin around the well, it may cause too small oil production allocation or unnecessary operations (plugging removal, fracturing, and so on) resulting in economic losses.
Because different pressure differences occur, it is important to note that the type curve of dp w D (t D ) /dln (t D ) is not continuously smooth and the values sometimes are negative. The different type curve results from the piecewise function. In the former part of the piecewise function, the skin factor is considered as a constant whereas that is variable in the latter part of the piecewise function.
Strictly speaking, skin factor should be viewed as variable parameter in the former part as well, but if we do so, the equation is too difficult to solve. In addition, the time that represented by the  former part of the function is too short, only occurs in the beginning of the production process. Finally, the data used in type curve matching procedure are usually in the middle and late periods. So the early period data is not as important as the data of middle and late periods. Therefore, this model has some drawbacks, but they do not affect the applicability of the type curves.

V. CONCLUSIONS
1) This paper established a well test analysis model of considering variable skin factor with the model expressed as a relationship between skin factor and time by defining two parameters that are β and v i D . It was proven that the definitions of these two parameters were reasonable by case analysis.
2) Laplace transform was used to solve the model and the type curves were plotted based upon the results. These plots were variation of p w D (t D ) and dp w D (t D ) /dln (t D ) with t D . The influence of the parameters in the expression of the variable skin factor ( β, v i D , and r s D ) on p w D (t D ) and dp w D (t D ) /dln (t D ) were discussed.
3) The flow of the reservoir is not fully extended to the formation in early flow period in the proposed mode. The effect of wellbore storage played a major role and the skin factor in this period was considered as a constant. When the effect of wellbore storage is neglected, the skin factor can be viewed as a variable. 4) A case analysis of well test data of a well was conducted to illustrate and validate the applicability of the model of Sheng-2 Block in Shengli Oilfield, China.
5) The solution in Laplace domain is a piecewise function, and there exist the cases of dimensionless pressure drop. Therefore, the curve of dp w D (t D ) /dln (t D ) will not be smooth and sometimes the values are negative, but these shortcomings do not affect the applicability of the type curves.