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When the horizontal well was acidized, in order to predict the productivity, through coordinate transformation method, based on the formation seepage model of horizontal well after acidification, we deduced the formula of horizontal well local skin factor , established the coupling model of reservoir and horizontal wellbore flow, and deduced the semi-analytical productivity model of horizontal well considering the change of local skin factor. Based on the example, the effect of horizontal well length, reservoir anisotropy coefficient, mud damage depth and damage degree on the production-increasing ratio after acidification were studied. The results showed reservoir with greater anisotropy, serious wellbore pollution and deeper mud invasion near wellbore, and that the stimulation effect of horizontal well after acidification was better .

While the horizontal well was produced, its productivity cannot reach the intended purpose, the reason partly is the formation near the horizontal well was damaged in the process of drilling and completion [

In order to accurately calculation the horizontal well skin factor after acidification, it is necessary to understand the seepage model of acidification formation. The study in document [

When acidification radius is greater than or equal to pollution zone radius ( r a ≥ r d ), pollution zone is improved by acidification, the acidification zone become a new “pollution zone”. The radius of acidification zone becomes new radius of “pollution zone”, the permeability of acidification zone becomes a new permeability of “pollution zone”.

Therefore, according to the hypothesis in reference [

S a ( x ) = ( K / K a ) ln [ 1 β + 1 ( r ah ( x ) r w + r ah 2 ( x ) r w 2 + β 2 + 1 ) ] (1)

S a ( x ) is local skin factor, f; K is reservoir permeability, mD; K a is acidification zone permeability, mD; r ah ( x ) is the distance of horizontal well heel in acidification zone, m; β is reservoir anisotropy coefficient, f; r w is wellbore radius, m.

When acidification radius is less than pollution zone radius ( r a < r d ), acids does not penetrate the whole pollution zone. In the original polluted zone, formed a composite zone consisting of acidification zone and polluted zone.

On the yz plane, the horizontal well can be regarded as the vertical well with radius r_{w} in a reservoir with thickness L, considering reservoir anisotropy, the steady seepage equation on yz plane is as follows:

K z ∂ 2 P ∂ z 2 + K y ∂ 2 P ∂ y 2 = 0 (2)

The internal boundary condition is:

z 2 + y 2 = r w ; P 2 = P w f 2 (3)

On the yz plane, the isobaric lines around the wellbore were concentric ellipses, uv coordinate system was introduced, the isobars around the wellbore become confocal concentric ellipses, as shown in

Introduced uv coordinate system, and conversed coordinate, there are:

u = ( K y / K z ) 1 / 4 z ; v = ( K z / K y ) 1 / 4 y (4)

Formula (2) can be changed to:

∂ 2 P ∂ u 2 + ∂ 2 P ∂ v 2 = 0 (5)

Formula (3) can be changed to:

( K z / K y ) 1 / 4 u 2 + ( K y / K z ) v 2 = r w 2 ; P 2 = P w f 2 (6)

In order to solve the above Laplace equation with elliptic inner boundary conditions, conformal transformation is introduced to transform the elliptic coordinate system into the ρθ coordinate system, there are:

u = b cosh ρ cos θ ; v = b sinh ρ sin θ (7)

When ρ = ρ w at the wellbore, derived from Equation (7):

u 2 b 2 cosh 2 ρ w + v 2 b 2 sinh 2 ρ w = 1 (8)

Comparison formula (5) and formula (7), there are:

tanh 2 ρ w = K z / K y ; b 2 = r w 2 ( K y − K z ) / ( K z K y ) 1 / 2 (9)

The average radius of the elliptical isobars can be defined as the average of the long and the short axes, there are:

r ¯ = a cosh ρ + a sinh ρ 2 = b 2 e ρ (10)

After coordinate transformation, as long as the average radius of the wellbore and the average radius of the reservoir damage zone are calculated separately, it can be solved by using the formula of skin factor calculation.

Comparison formula (9) and formula (10), the average radius of wellbore is:

r w ¯ = b 2 e ρ w = r w ( K y / K z + 1 ) / 2 ( K y / K z ) 1 / 4 (11)

The anisotropic problem is transformed into isotropic problem by coordinate transformation. Where the anisotropic coefficient β = K y / K z , then formula (11) becomes:

r w ¯ = r w ( β + 1 ) / 2 β (12)

Formula (4) can be changed to:

u = β z ; v = y / β (13)

The long and short axes of isobaric lines in equivalent isotropic formations are obtained from Equation (7) respectively:

u o = b cosh ρ ; v o = b sinh ρ (14)

For the elliptical polluted zone, combination formula (13) and formula (14), when ρ = ρ d , as follows:

ρ d = sinh − 1 ( v d / b ) = sinh − 1 ( r dh / b β ) (15)

Combination formula (10) and formula (15), average radius of elliptical polluted zone is:

r d , eq ¯ = b 2 e ρ d = r w 2 β − 1 / β [ ( r dh r w β 2 − 1 ) 2 + 1 ] (16)

Formation permeability K and composite zone permeability

Combination formula (12) formula (16) and formula (17), skin factor of horizontal well is:

In document [

Formula (19) can be changed to:

Formula (18) can be changed to:

In reference [

assumed that the acidification zone and polluted zone radius d is tribute parabolically along the wellbore. Because the contact time between the toe of horizontal well and acid fluid is relatively short, it can be assumed that the formation is not polluted. When the radius of acidification zone reaches the maximum, the distribution functions of the radius of pollution zone and acidification zone along the wellbore direction are as follows:

Combination formula (20) formula (22) formula (23) and formula (21), the distribution pattern of local epidermal factors after acidification of horizontal well can be obtained.

From the toe to the heel of horizontal wellbore, the mass flow rate of fluid increased gradually. There is a coupling relationship between seepage flow in reservoir and in horizontal wellbore [

Assumed that the upper and lower boundaries of the reservoir were closed, z_{w} is the distance between the horizontal well and the lower boundaries of the reservoir, the horizontal section is divided into N micro-segments along the length direction, and the micro-lines on the micro-segments converge radially and uniformly, while the radial flow

According to the mirror reflection principle, the micro-element of the production section in section i(

Among,

Among:

The pressure of boundary is

Among, ^{3}/s; ^{3};

Assumed the flow in horizontal wellbore was single-phase incompressible fluid without gravity loss. The pressure loss of the i section in the production section were

The friction loss

Among, ^{3}/s; ^{3}/s; ^{3};

wellbore flow is laminar flow, when

Among,

turbulent,

Among,

The acceleration loss

Selected the segment

Among, ^{3}/d; K is the formation permeability, mD.

Combination formula (26) formula (27) formula (29), the pressure loss analysis model of the micro-element section considering the change of skin factor can be obtained:

For production section of horizontal well, there are the following relationships between flow rate and flow pressure:

Among,

Formula (25) formula (30) and formula (31) constitute a closed system of equations containing 2N equations and 2N unknown variables (

In the above productivity model, both

The basic parameters of a horizontal well were as follows: horizontal length

Used the formula (21) of the skin factor obtained in this paper, combined with the actual data above, the fig of the distribution of the skin factor can be drawn (

It can be seen from

Yield multiplier ratio was usually used to evaluate the acidification effect. The yield increase ratio is the ratio of the yield after acidification to that before acidification. According to the calculation results of the previous productivity model, analyzed the factors affecting the acidification effect of horizontal well.

1) Effect of horizontal well length on yield multiplier ratio of horizontal well

Considered the horizontal well length L = 100 m, 200 m, 300 m, 400 m, 500 m, other parameters in the above examples. The productivity model proposed in this paper is used to calculate yield multiplier ratio of horizontal well.

2) Effect of reservoir anisotropy coefficient on yield multiplier ratio of horizontal well

Considered the reservoir anisotropy coefficient β = 1, 3, 5, 7, 9 and other parameters in the above examples, used the productivity model proposed in this paper to yield increase ratio of horizontal well.

horizontal well. As can be seen from

3) Effect of pollution depth on yield multiplier ratio of horizontal well after acidification

Considered pollution depth (maximum pollution depth of mud) R_{m,max} = 1 m, 1.6 m, 2 m, 2.4 m, 2.8 m, and other parameters as the parameters of the above examples. The effect of maximum pollution depth of mud on the yield increase ratio after acidification is shown in

increases with the increase of pollution depth. This shows that the deeper the mud invasion near the wellbore, the better the effect of acidification on increasing production.

4) Effect of pollution degree on yield multiplier ratio of horizontal well after acidification

Considered the damage degree (the formation permeability/polluted area permeability) as 5, 10, 15, 20, 25, and other parameters as the parameters of the above examples,

1) Used coordinate transformation, the formula of local skin factor after acidification of horizontal well is obtained when the acidification radius is less than the radius of pollution zone. Combined with the coupled flow model of production section, a semi-analytical productivity model of horizontal well considering the change of local skin factor of acidification effect is derived.

2) Examples show that horizontal well length, formation anisotropy coefficient, mud pollution depth and pollution degree are the main factors affecting the yield increase ratio after acidification.

3) The more anisotropic, serious wellbore pollution and mud invasion near wellbore, the better acidification effect will be.

The authors declare no conflicts of interest regarding the publication of this paper.

Sun, E.H., Yang, W., Peng, Q., Meng, P. and Mu, S.R. (2020) The Productivity Model of Horizontal Well Considering Acidification Effect in Anisotropic Reservoirs. World Journal of Engineering and Technology, 8, 19-32. https://doi.org/10.4236/wjet.2020.81003