1. Asphaltene Near-Wellbore Formation Damage Modeling By Kosta J
Asphaltene Near-Wellbore Formation Damage Modeling By Kosta J. Leontaritis, Ph.D. AsphWax, Inc. Web:

2. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

3. Formation Damage
Mathematically, formation damage is a reduction in the flowing phase mobility, l: 13

4. Formation Damage
For the oil phase, the hydrocarbon effective mobility, lo, is: 13

5. Asphaltene-Induced Formation Damage Mechanisms
Absolute permeability impairment (k)
Wettability Changes (kro)
Viscosity (mo) increase due to:
Emulsion formation
Asphaltene particle increase near the borehole

6. Asphaltene-Induced Formation Damage Mechanisms
From the previous three mechanisms of asphaltene-induced formation damage the first one appears to be the dominant mechanism, although occasionally the second and third mechanisms do seem to play a role under certain circumstances. If there is no water production, which is the most likely case, then no emulsion of water-in-oil is expected. Hence, any viscosity increase measured in the laboratory would have to be attributed to asphaltene particle concentration increase as the reservoir fluid approaches the wellbore. Past experiments have shown that asphaltene flocculation in-of-itself does not result in a significant viscosity increase. Also, from experience, reservoirs that have asphaltene problems seem to be mixed-wet to oil-wet even before production commences. It evident then that the major cause of asphaltene-induced formation damage in asphaltenic reservoirs is the first mechanism.

7. Physical Blockage of Pore Throats Caused by In-Situ Asphaltene Deposition13

8. Reservoir characterization
Reservoir characterization is an enormous subject that consumes a lot of manpower energy in the oil industry. Some of the parameters that characterize a reservoir are:
k, permeability
f, porosity
RQI, Reservoir Quality Index (mean hydraulic radius)
Fluid Saturations
Wettability
Electrical Properties (Formation Factor and Resistivity Index)

9. Formation Mean Hydraulic Radius
Pore throat radii of a formation depend on many reservoir parameters and they certainly vary from one formation to another. However, some general statements about the distribution of hydraulic radii of formations can be made:
Largest distribution is to 100 micron
Usual distribution is 0.01 to 10 micron
Occasional distribution is 0.1 to 1 micron

10. Flocculated Asphaltene Micelles Forming an Asphaltene Particle
12 Å
45 Å

11. Obvious question: How can so small flocculated asphaltene particles plug pore throats of much larger size?

12. Physical Blockage of 100 m Pore Throat Caused by In-Situ Flocculation of Much Smaller Asphaltene Particles
100 m

13. Physical Blockage Caused by In-Situ Asphaltene Deposition-Sandstone13

14. Asphaltenes Adsorbed on the Rock Cause Wettability Changes12

15. Wettability Change Caused by In-Situ Asphaltene Deposition-Carbonate13

16. Asphaltene-Induced Formation Damage Near Production Wells
High draw-down
Miscible-gas breakthrough
Contact of oil with incompatible fluids during drilling, completion, stimulation, fracturing, and gravel packing operations
Drop in reservoir pressure below onset of asphaltene pressure

17. Well Producing with Asphaltene-Induced Formation Damage
1st Mechanism
2nd Mechanism

18. Well Producing with Asphaltene-Induced Formation Damage

19. Well Producing with Asphaltene-Induced Formation Damage
The previous slide shows an aerial view of a producing well suffering from asphaltene-induced formation damage. The well is on flow control from the choke. As Pw drops, when asphaltene deposition starts, the choke opens so that the lower Pw can push all of the oil flow, q, through the tubing. Pe and PAF remain constant. However, when the choke is completely open both re and rAF begin to decrease due to the production rate decrease caused by the ever-increasing asphaltene-induced formation damage. When rAF becomes equal to rw no additional formation damage is incurred. This is referred to as "true steady state condition". The production rate at this state, however, may not be economical.

20. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

21. Asphaltene Deposition Envelope

22. Asphaltene psd at P=2975 psia & T=122 °F

23. Asphaltene Particle Size and Sedimentation

24. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

25. Well Producing with Asphaltene-Induced Formation Damage
1st Mechanism
2nd Mechanism

26. Hydraulic Radius – Single Channel
The hydraulic radius of a single flow channel is given by:
Where:
S = x-sectional area of flow channel
LP = wetted perimeter of flow channel
L = length of flow channel

27. Hydraulic Radius – Core Plug
The hydraulic radius of a core plug is given by:
Where:
f = core plug porosity (= void volume/total volume)
SP = surface area of one core plug grain or particle
VP = volume of one core plug grain or particle

28. Hydraulic Radius
By further mathematical manipulation, the hydraulic radius of a core plug is given by:
Where:
f = core plug porosity (= void volume/total volume)
dg = average grain diameter. There are proprietary correlations for sandstones and carbonates that allow one to calculate dg from k and f.

29. Mean Hydraulic Pore-Throat Radius
A simple, quick and dirty way to estimate the Mean Hydraulic Pore-Throat Radius is via the following equation:

30. Retained Asphaltene Particle Diameter, dAP
The simplest rule-of-thump from filtration theory is that a filter retains particles with diameters 1/3 of the nominal rating of the filter. In this case, the filtration rule-of-thump means that:
Where:
dAP is the diameter of the average size asphaltene particle retained by the formation

31. Physical Blockage of 100 m Pore Throat Caused by In-Situ Flocculation of Much Smaller Asphaltene Particles
100 mm

32. Retained Asphaltene Particle Diameter, dAP
In the more general case, however, a more appropriate definition for dAP is:
Where:
a is a constant that accounts for the variation of the size of the asphaltene particle filtered by the formation. a varies from 0 to 1.
dH is the average hydraulic diameter of the producing horizon

33. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

34. Darcy's equation for steady state radial flow is:
Where:
m is viscosity, centipoise
q is reservoir barrels per day
k is permeability, Darcy
P is pressure, psia
r is distance from center of wellbore, feet

35. The Darcy equation applies to each radial segment Dr at location r

36. Initial Area Available to Flow Ainitial(r)
At time equal to zero, i.e., before any asphaltene plugging, the total area available to flow at a distance r from the center of the wellbore is:
Where:
h is the net thickness of the formation or the net pay zone
f is the initial average effective porosity of the formation

37. Net Area Available to Flow Anet(r,t)
The net area available to flow at location r, after asphaltene plugging for time t, is obtained as follows:
Where:
AAP(r,t) is the total area plugged by asphaltene particles at location r at time t. The calculation of AAP(r,t) is described next.

38. Total Area Plugged AAP(r,t)
The total area plugged by asphaltene particles at location r at time t is AAP(r,t):
Where:
DAAP(r,j) , is the incremental area plugged by asphaltene particles at location r within time interval j
N is the number of time intervals

39. Calculation of Incremental Area Plugged DAAP(r,j)
Very Effective Pore-Throat Plugging by Least Number of Asphaltene Particles
Pore-Throat
Asphaltene Particle

40. Calculation of Incremental Area Plugged DAAP(r,j)
The incremental area plugged by asphaltene particles at location r within time interval j, DAAP(r,j), is:
Where:
IMAT, is the number of incremental moles of asphaltene particles being trapped at location r within time interval j
MVA, is the molar volume of asphaltene particles at location r at time interval j
CSAAP, is the cross-sectional area of the average size asphaltene particle retained by the formation at location r at time interval j
VAP, is the volume of the average size asphaltene particle retained by the formation at location r at time interval j

41. Calculation of Incremental Area Plugged DAAP(r,j)
The equation giving the incremental area plugged by asphaltene particles at location r within time interval j, DAAP(r,j), is:
Where:
DAPtrap(r,t), is the number of incremental moles of asphaltene particles being trapped at location r within time interval j
uA(r,j) is the molar volume of asphaltene particles at location r at time interval j
dAP, the diameter of the average size asphaltene particle retained by the formation

42. Calculation of Incremental Area Plugged DAAP(r,j)
After rearrangement and substitution, the equation giving the incremental area plugged by asphaltene particles at location r within time interval j, DAAP(r,j), is:
Where:
DAPtrap(r,t), is the number of incremental moles of asphaltene particles being trapped at location r within time interval j
uA(r,j) is the molar volume of asphaltene particles at location r at time interval j
dAP, the diameter of the average size asphaltene particle retained by the formation

43. Total Area Plugged AAP(r,t)
Substitute into the equation giving the total area plugged by asphaltene particles at location r and time t, AAP(r,t), to get:

44. Total Area Plugged AAP(r,t)
Hence, to calculate the total area plugged by asphaltene particles at location r at time t, AAP(r,t), we need the following:
dAP, the diameter of the average size asphaltene particle retained by the formation
uA(r,j), the molar volume of average size asphaltene particles at location r at time increment j
DAPtrap(r,j), is the number of incremental moles of asphaltene particles being trapped at location r within time interval j

45. Average Diameter of Asphaltene Particles Retained
Remember that dAP is the diameter of the average size asphaltene particle retained by the formation and is given by:

46. Molar Volume of Asphaltene Particles Retained
uA(r,j), the molar volume of asphaltene particles at location r at time interval j, is calculated by the phase behavior model. In this case, it is calculated by the TC Model, AsphWax’s asphaltene phase behavior simulator.

47. Incremental Moles of Asphaltene Particles Retained
DAPtrap(r,j), the incremental moles of asphaltene particles retained, is obtained as follows:
At the pressure and temperature prevailing at location r at time t, the asphaltene phase behavior model (TCModel) calculates the moles of asphaltene particles per mole of reservoir fluid, s, and their psd, f(x), where x is the asphaltene particle diameter.
f(x) is then integrated from x = dAP to x = ¥ to obtain the moles of asphaltene particles being trapped, ftrap, per mole of reservoir fluid at location r within time interval Dt.
The total number of moles of reservoir fluid, MRF, flowing at location r is obtained by flowing the well at flowrate q for some specified production time interval Dt. 8

48. Incremental Moles of Asphaltene Particles Retained
Hence, from a material balance, the number of incremental moles of asphaltene particles being trapped at location r within time increment j, DAPtrap(r,j), is:

49. Total Area Plugged AAP(r,t)
Substitute into the previous equation to get the total area plugged by asphaltene particles at location r at time t, AAP(r,t), as:
Where:
g is a constant whose value is greater or equal to 1. it is related to a by the relation g=1/a. Recall that a varies from 0 to 1. g (or a) indicates the "efficiency" of plugging of the asphaltene particles. It may be used as a tuning parameter, if well history-matching data are available.

50. "Degree of Damage", DOD
It is convenient to introduce the "Degree of Damage", DOD, at each location r at time t. DOD may be defined as:

51. Permeability Impairment
Using the definition of DOD and applying Darcy’s law to both the damaged and initial situation one gets:

52. "Degree of Damage", DOD
After further substitution and simplification one gets:

53. Permeability Impairment
The previous equations yield the definition of damaged permeability:
The damaged permeability, kdam(r,t), at location r and time t is used in the Darcy equation to calculate the pressure drop in the formation. Although this gives the appearance that permeability is treated as a point property, in reality it is not. k does not really refer to a point at distance r but rather to an increment Dr at distance r from the center of the wellbore.

54. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

55. Porosity Loss
Porosity loss is modeled in a similar way to permeability. Using the definition of porosity we have:

56. Dividing both sides of the above equations, one gets:
Porosity Loss
Dividing both sides of the above equations, one gets:
If the radial interval Dr is taken very small we can assume that the area loss at this interval due to asphaltene flocculation is uniform across its thickness Dr. This implies that DOD(r,t), as previously defined, is an average value for the whole increment thickness Dr at location r.

57. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

58. Skin Factor
The familiar van Everdingen-Hurst skin factor is defined as:

59. Skin Factor
Since Pw(t) is calculated as a function of time, as described previously in the permeability section, DPs(t) (and consequently s(t)) can be calculated from the following equation:

60. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

61. Asphaltene Deposit Erosion
As plugging is occurring inside the asphaltene-inflicted formation, the interstitial velocity increases continuously at constant production rate (constant rate accomplished by opening the choke). If the system continues to produce undisturbed, the interstitial velocity eventually becomes equal to and surpasses the critical velocity at which point previously deposited asphaltene particles begin to move with the flow. At further velocity increases the rate of erosion becomes equal to the rate of deposition, hence, no additional asphaltene damage occurs at that location. Such a “pseudo-steady state condition” has been observed in the field in wells undergoing asphaltene deposition in the near-wellbore formation. The production rate at this steady state condition, however, may not be (and generally it is not) economically acceptable or viable. In general, when the steady state condition is reached or even before then, the well requires stimulation to restore production at economical rates. It should be noted that some of the smaller pore throats may plug up completely. Whereas the bigger pore throats may reach the steady state condition caused by the deposit erosion rate being equal to the deposition rate.

62. Well Producing with Asphaltene-Induced Formation Damage
1st Mechanism
2nd Mechanism

63. Asphaltene Deposit Erosion
At the previously described pseudo-steady state condition, the net area available to flow Anet(r,t) corresponds to some fraction of the initial area available to flow Ainitial(r). The following equation is recommended for achieving steady state condition in the simulation:

64. Asphaltene Deposit Erosion
There are certain salient features of the previous equation that make it suitable for representing asphaltene particle erosion at the steady state condition. First, it is evident that as the degree of damage, DOD, increases the value of DA(r,t) approaches B*Ainitial(r). This is equivalent to saying that:
Hence, the value of constant B is limited to 0 < B < 1. B places a limit on the maximum damage that can occur at location r. For those Dr segments or locations that reach the maximum deposition at some time t, DOD will remain constant after time t. This will show as a decline in the
damage rate increase, such as a decline in the skin factor increase, when a segment reaches its maximum deposition level specified by B. Factor B can be used for history-matching data, along with constant g.

65. Asphaltene Deposit Erosion
It is evident that at time zero the degree of damage is equal to 1. That is DODt=0=1. Also, at time zero Anet(r,t=0) = Ainitial(r), because AAP(r,t) = 0. Hence, from the previous equations one can derive that:
A + B = 1
A consequence of the above equation is that the constants A and B are not independent. If one is specified the other is obtained from the above equation. Also, since 0 < B < 1, it follows that 0 < A < 1.

66. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

67. Solution Algorithm Where:
The key to solving the set of equations in this formation damage model lies in solving the equation shown below numerically. As already mentioned, the near well formation is broken into small equal segments Dr and the production time in small intervals Dt, i.e., this is a finite difference approach.
Where:
i, refers to radial segment Dr at position i in the formation
j, refers to time intervals j and varies from 1 to N.
N,is the total number of time intervals. Hence, total production time is equal to N*Dt.

68. Solution Algorithm
The near-wellbore region is divided into radial segments Dr. The formation damage calculation is performed for all segments at each time interval Dt.

69. Solution Algorithm
Pe and PAF remain constant while PW is declining. Flow, q, is constant. During time interval j, PW remains constant. Say for time interval 6, PW is P6, for time interval 4, PW is P4, etc.

70. Solution Algorithm
The previous figure shows the decline in the bottom hole pressure Pw with time caused by asphaltene-induced formation damage. An analogous decline is taking place in the entire region affected by asphaltene deposition (r£rAF). One can assume that during a time interval Dt the pressure is equal to an average value Pavg. This allows one to calculate the thermodynamic properties of the reservoir fluid and utilize the previous equation as if during time interval Dt the system was at steady state (although during time interval Dt some incremental formation damage is taking place). The accuracy of this approach obviously improves with smaller time interval Dt and radius segment Dr, because the smaller the time interval and radius segment the closer the system is to steady state. This is true with any other numerical finite difference approach.

71. Solution Algorithm The calculation proceeds as follows:
Using the well productivity index, PI, at time t=0 calculate a starting Pw that corresponds to the production rate q. Then, the formation hydraulic diameter dH and average retained asphaltene particle diameter dAP are calculated. Also, the mole rate of reservoir fluid, MRF, corresponding to q is calculated.
Assuming no damage present, the pressure profile is calculated from the wellbore to the well drainage radius, re, using the Darcy equation (i.e., DPs=0). This gives us the undamaged pressure profile at time t=0.
At the first time interval Dt, i=1, the FDModel starts the calculation at r=rAF, because if q remains constant rAF remains constant as well. Hence, the pressure profile at locations with r³rAF remains constant during the pseudo-steady state period (i.e., there is no damage at r³rAF). But the pressure profile changes continuously at r£rAF.

72. Solution Algorithm
For every radial segment inside the damaged area (r£rAF), the asphaltene model is used to calculate the psd f(x), the mole fraction S, and molar volume uA of the flocculated asphaltene particles.
The asphaltene psd, f(x), is then integrated for x=dAP to infinity to obtain the mole fraction of asphaltene particles trapped, ftrap.
The derived equation is used to calculate the area plugged by the asphaltene particles at location i and time interval j, AAP(r,t).
DOD is calculated next.
Then, kdam, fdam, DPdam, DPs, and s are calculated.
The damaged permeability kdam obtained in the previous step is used in the Darcy equation to calculate a new pressure profile for the region r£rAF.
The last 6 steps are repeated for the next time interval Dt, j+1.

73. Solution Algorithm
The calculation proceeds until one of the following happens:
The time t=j*Dt reaches tmax, the total production time specified at the beginning of the calculation.
the choke opens all the way and the flow rate cannot be kept constant. At this time, the pressure profile at r³rAF begins to change also due to the decreasing flowrate. This corresponds to some low Pw. This low Pw must be specified to the model as a model stopping criteria.

74. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

75. FDModel Application

76. FDModel Application

77. FDModel Application

78. FDModel Application

79. FDModel Application

80. FDModel Application

81. FDModel Application Damage limit specified by factor B
reached in Dr next to the wellbore.

82. Asphaltene Near-Wellbore Formation Damage Modeling
Asphaltene-Induced Formation Damage
Asphaltene Particle-Size Distribution (psd)
Hydraulic Radius
Permeability Impairment
Porosity Loss
Skin Factor
Asphaltene Deposit Erosion
Solution Algorithm
FDModel Application

83. Asphaltene Near-Wellbore Formation Damage Modeling By Kosta J
Asphaltene Near-Wellbore Formation Damage Modeling By Kosta J. Leontaritis, Ph.D. AsphWax, Inc. Web: