Mitigation and Stimulation

Most of the models require input from core tests.
Barkman and Davidson's model could possibly be used without core test input data - if the filter cake permeability is estimated.
WID provides formulas to estimate coreflood parameters from historical information for situations when core test data is not available.
None of the models takes flow of oily water into account.
Only Khatib's model considers the filter cake to be compressible, and therefore the filter cake as a function of injection pressure and rate.

Most particle deposition models are based on the mass conservation equation for particle deposition (Herzig, et al., 1970), assuming incompressible flow and negligible changes to diffusion and porosity.
Iwasaki's deposition function, defining a filtration coefficient is also used commonly in models which attempt to fundamentally represent the physics of particle deposition.. Iwasaki proposed fundamental equations to describe filtration through a sand bed (Iwasaki, 1937). These incorporated velocity, concentrations of particles (in the flow stream and deposited), a damage factor and a filtration coefficient.
The filtration coefficient is a key factor, and it depends on a number of parameters. In most cases, the filtration coefficient is assumed to be constant and either derived from filter paper or core experiments or from empirical relationships (Sharma, et al., 1997).
To relate the particle deposition to permeability or injectivity reduction, the most common starting point is the Carman-Kozeny equation. Be extremely careful when you try to use this relationship to represent relationships between permeability, porosity, and filtration - particularly in poorly consolidated media.

Shell has published a series of papers describing their efforts towards predicting decline in water injectivity due to plugging of solids in the water. The most well-known models are summarized. Recently, van den Hoek et al. published a paper describing flow through wormholes, forming through the external filter cakes on fractured injection wells (van den Hoek, 1999).

Barkman and Davidson (1972) - Membrane Filter Tests Barkman and Davidson (B&D) suggested that impairment from suspended solids occurred by one or more of the four following mechanisms:

Wellbore narrowing: The solids form a filter cake on the face of the wellbore, an external filter cake.
Invasion: The solids invade the formation, bridge and form an internal filter cake.
Ų Perforation Plugging: The solids become lodged in the perforations.
Ų Wellbore fillup: The solids settle to the bottom of the well by gravity and decrease the net zone height.

They used pressure drop and material balance to develop equations for the time it takes the injectivity (II) to reduce to a fraction of the original injectivity. They referred to the time when the injectivity has reduced by 50% (a=½), as the "injector half-life" (t_1/2)

The model for invasion (INTERNAL FILTER CAKE) is based on the theory of deep bed filtration in which the superficial fluid velocity is a function of the position in the reservoir, as described by Herzig et al. (Herzig, et al., 1970). Davidson claimed that the impairment occurred in two stages:

Particles are deposited because of gravity or surface forces, causing restriction of the pores, and,
Bridging occurs within the formation, followed by the buildup of an internal filter cake.

The modelling of the wellbore narrowing (EXTERNAL FILTER CAKE), or formation of an external filter cake in an openhole completion, was considered.

The model for perforation plugging treats the perforation as a cylinder that is gradually filled from its innermost extremity towards the well. The increase in pressure drop because of this infill is taken as the original pressure drop plus the pressure drop through the filter plugs in the perforations. The permeability of the filter plug is taken as the external cake permeability.

In the wellbore fillup model, it is assumed that all of the solids fall to the bottom of the wellbore and decrease the injection interval. Fluid flow through the fill is neglected. Wellbore fillup, caused by water hammer effects in water injection wells in a Norwegian field, has been reported (Santarelli, et al., 1998). The mathematical model assumes that the injectivity decline is proportional to the height of the interval.

The key to the Barkman and Davidson injectivity prediction is the so-called "water quality ratio" or the solids concentration over the filter cake permeability (w/k_c). All of the models are functions of this ratio. Barkman and Davidson showed how this ratio could be found by simple on-site membrane testing.

If the cumulative filtrate volume is plotted against the square root of time (filtration curve), it will approach a straight line with slope S, assuming that a filter cake is formed. The same slope is obtained whether invasion takes place or not. Barkman and Davidson claimed that the characteristics of the filtration curve gave an indication of the plugging mechanisms involved.

When the suspended particles are larger than the pores of the filter medium, no invasion takes place. The extrapolation of the straight-line section of the filtration curve has a negative intercept on the y-axis.
When the suspended solids are much smaller than the pores of the filter medium, invasion takes place at least during the early part of the test. B&D's experiments showed that the extrapolation of the straight-line section will have a positive intercept on the y-axis.
The third type of filtration curve, the so-called "s-shaped curve," can apparently occur under several circumstances. One of them is when invasion, resulting in the formation of internal filter cake, is followed by formation of an external filter cake.

The Barkman and Davidson model was the first significant experimental model, providing a convenient on-site method to predict the rate of impairment.

The injector half-life model enjoys wide use but it has some limitations, mainly associated with the development of the filter cake properties. They are developed from membrane filter tests, which usually lack correlation with actual core flow behavior.

tHE model was tested against performance in the Ula field in the Norwegian sector of the North Sea (van der Zwaag and Ųyno, 1996). However, the flow in this situation is believed to have been dominated by thermally-induced fractures and application of the B&D model may be questionable. In Ula, fair prediction was reported but accuracy for half-life was less than a factor of two.

The main limitations of the Barkman and Davidson method are that the model is based on membrane filter tests, which are not believed to correlate with actual core flooding measurements. Furthermore, it does not account for entrainment and re-deposition. Van Velzen and Leerlooijer claimed that the model had some serious shortcomings and therefore, in spite of its wide use, no case histories with good correlation between theoretical prediction and practical observation were known (van Velzen and Leerlooijer, 1992).

Davidson (1979) - Injection Half-Life In 1979, Davidson attempted to enhance the previous Barkman and Davidson model with theoretical and experimental modelling of particle invasion of suspended solids. He proposed using sandstone cores, instead of membrane filters, to describe particle invasion and forming of internal filter cake through deposition and bridging of pore throats.

The main applications of Davidson's approach were with the original Barkman and Davidson method which was designed as a simple approach for on-site experiments to predict injectivity performance.

This method has the same limitations as the B&D methods - not accounting for entrainment and re-deposition or wettability factors.

Eylander Model (1988) Eylander modified Barkman and Davidson's half-life calculations. He claimed that, in spite of membrane filtration having advantages for describing filter cake phenomena, it lacked correlation to actual rock cores needed to be valid for describing internal damage mechanisms. Therefore, he introduced a new approach, based on coreflood experiments, to identify internal formation plugging, by defining a filter cake porosity parameter.

Based on experimental results, Eylander developed a model to represent injectivity decline due to particle deposition during internal filtration, intended to be suitable for on-site application and recommended for validation against field performance data.

No specific method to determine the type of impairment was indicated. Therefore, in spite of its limitations, the 1/3 - 1/7th rule of thumb is recommended, even though Eylander's experiments showed that it only worked in 2 out of the 5 cores tested. Furthermore, the inflow equation does not include time-related depth of impairment (Eylander, 1988). It can be very difficult to specify invasion depth and the permeability of the internal filter cake, but both are needed prior to use. The method has not been validated on-site with field data.

Van Velzen Model (1992) van Velzen and Leerlooijer introduced a semi-empirical radial model to describe wellbore impairment by internal cake formation, caused by suspended solids in injection water. The model alleged to predict the half-life of water injection wells.

They claimed that their experiments confirmed the "1/3 - 1/7 rule-of-thumb" developed by Abrams (Abrams, 1977) as a criterion for water quality.

The model has been field-tested with data from Prudhoe Bay (Hsi, et al., 1994) and data from the Ula field (van der Zwaag and Ųyno, 1996). In both cases the model predicted worse performance than was actually observed. However, most injectors in both fields were believed to have been thermally fractured and the injected fluids were oily produced water. Therefore, the model was not fully applicable.

Van Velzen et al. claimed that, for both linear and radial flow geometry, the rate of impairment strongly depends on the inflow velocity at the injection surface, i.e., the rate of impairment decreases with increasing inflow velocity and is very sensitive to velocity changes at low velocity levels. Furthermore, they claimed that the rate of impairment also depends on the relationship between particle and pore sizes.

Khatib (1994) Khatib reported on a new model for calculating the combined matrix and filter cake resistances in perforations and fractures. This is an empirical correlation between permeability and porosity to predict the injectivity decline rate under reservoir conditions, for four common well completions:

Openhole,
Cased and perforated,
Perforated and gravel packed, and,
(Static hydraulic) fractured well.

It was also determined that the type of suspended particles and the compressibility of the filter cakes are important.

Experimental results for injecting oil-coated clay and silt solids, suspended in the injection water stream, suggested that additional permeability reduction [due to the oil] can be up to 50%. This was also confirmed with other types of solids.

Khatib also reported on a numerical model to predict the injectivity decline rates for four types of well completions; openhole, perforated, gravel packed and fractured. It assumes steady-state flow and constant injection and reservoir pressures. However, a "transient flow factor" is incorporated to take into account the reservoir pressure buildup, which is important, especially for tight reservoirs.

McCune's Model

Bayona, 1993 discussed McCune's technique to relate well injectivity to water quality. Bayona modified McCune's equation to a) relate total suspended solids (TSS) in the water and the well injectivity rate of decline and b) account for the portion of the TSS that does not cause reduction of well injectivity. He assumed that particles smaller than 1/10th of the formation pore throat diameter (TSSnp) can travel inside the pore space without bridging and causing reduction of permeability.
Bayona applied this method to a number of wells in the Ghawar field, a naturally fractured carbonate reservoir in Saudi Arabia, and discovered that plotting the decline rate against TSS resulted in a straight line. Unfortunately, Bayona did not indicate how well the slope matches the water quality ratio and the filter cake properties observed in core tests.
This is a very simple but effective relationship and only requires estimates of the Injectivity Index and the solids content. However, in most PWRI cases, solids plugging is not believed to be the only source of injectivity decline, as in the Ghawar field.

Sharma and Yortsos used an effective medium approximation to represent the flow field in a network of interconnected pores (Sharma, et al., 1991).
Pang and Sharma suggested that only two of Barkman and Davidson's mechanisms, external and internal filter cakes, realistically operate in the field (Pang and Sharma, 1994). They assumed that both wellbore fillup and perforation plugging were an extension of external filter cake growth, as described by Jiao and Sharma (Jiao and Sharma, 1993). To determine the nature of impairment, they collected information from eight previous coreflood experiments and plotted the RII against cumulative volume injected. Their experiments resulted in four different type curves for different type of impairment:

Straight line: Either incompressible external filter cake or pure internal filtration, as described by van Oort et al.
Curving with decreasing slope: compressible external filter cakes or internal filter cakes.
Curving with decreasing slope: No external filtration and deep particle invasion.
"S" shaped curves: several internal filtration mechanisms operating simultaneously.

They proposed the concept of transition time in order to model the deposition of both internal and external filter cakes simultaneously. Then, they developed three, complex semi-empirical models to explain injectivity in openhole, as well as perforated and fractured wells, although again, parameter estimation for the damage factor and a filtration coefficient were required a priori from coreflood experiments. These models were based on Stokesian dynamics theory and deep bed filtration.
Later they extended their model to be able to predict injectivity for fractured wells and developed the first version of the WID simulator (Water Injection Decline) (Pang and Sharma, 1995).
Wennberg carried forward the previous work of Pang and Sharma and developed version 3.1 of WID (Wennberg, 1998). He presented an injectivity decline equation in terms of resistances for various completion scenarios. Sharma et al. summarized the main features and limitations of the model behind WID 3.1 (Sharma, et al., 1997):

The model equations outlined have been implemented in a PC-based simulator.
The simulator can be used to predict injectivity decline in vertical and horizontal wells with openhole, perforated and gravel packed completions. For vertical wells, fractures and layering can also be handled.
The parameters used in the simulation can be obtained from core flow tests or from empirical correlations, built into the model.
The simulator can be run for either constant injection rate or for constant pressure.

Wennberg (1998) and Sharma et al. (1997) tested the simulator against actual field data from three fields. One of these is a highly unconsolidated Gulf of Mexico field, reported in paper SPE 38181. In this case, seawater (treated to high water quality) was injected below fracture pressure. The injectivity declined sharply in all wells. The model predicted the injectivity decline well in two of the six wells and for the first period in the third.

Roque et al. proposed a microscopic concept for particle deposition (Roque, et al., 1995). After an initial period of deposition on grains, especially at stagnation points, a phase of pore throat bridging is alleged to take over. This bridging also occurs when the particles are much smaller than the pore throats. Finally, internal and external accumulation (filter cake buildup) take place.
They performed an experimental series, injecting latex particles into sandstone - and they were able to observe these phases in some of the experiments. However, the last three phases often showed a considerable degree of overlap.
Rochon et al. derived equations along the same lines as Van Velzen et al. and Pang and Sharma (Rochon, et al., 1996).
Rochon et al also provided an equation for the cumulative volume injected (V*) when the damage switches from internal damage to growth of external cake. This corresponds to the number of particles injected when the near-wellbore porous medium is so damaged that solids can no longer enter the formation - they start to accumulate on the wellbore face.
They also defined an equation for permeability decline after external filter cake starts to form, including restriction due to both internal and external filter cakes and they further derived an equation for skin due to damage in conventional matrix injection:
The model was used to predict injectivity decline in two fields, one in the North Sea and another in the Guinea Gulf. The model predicted that the damage increased 5 to 6 times more rapidly for the Guinea Gulf case than for the North Sea conditions. If no action were taken, external filter cake would start to build up after 14 months in the North Sea but after only 3 months in the Guinea Gulf.
An important limitation of the IFP model is that it assumes constant flow rate. The authors claimed that they had found a model accounting for constant pressure gradient (Rochon, et al., 1996) to be "too complex for practical purposes".

This work has mainly attempted to describe the fundamental mechanisms causing near-wellbore damage with less emphasis on practical models to predict injectivity decline over field lifetimes.
Kumar and Todd (1988) described various factors in core testing to characterize water injection, including the impact of flow velocity on permeability. Kumar and Todd (1988) reported on experiments involving injection of solids in water into sandstone cores (Kumar, 1988, Kumar and Todd, 1988). Their experimental work resulted in two "phenomenological porosity models," based on the law of mass conservation of solid particles injected into a porous medium and the core flood experiments. The models were designed to account for spatial variation of volume flux and particle density. The main applications of the models were to predict the damaged permeability in a similar material, at any new flow velocity, provided that the flow is laminar region and no fracture is initiated during the injection. The main limitations of the first model are that it is very simple and it only applies to a uniform formation where the initial permeability is well known for a certain superficial flow velocity - it is therefore not very practical. The second method has not been validated against actual field data.
In 1994 Zhang, Todd and Somerville reported on injecting synthetic produced water into fractured and non-fractured cores (Zhang, 1994, Zhang, et al., 1993). In 1998, Al-Homadhi described injection of synthetic produced water through fractured core samples (Al-Homadhi, 1998).
Ehmad (1998) continued the work of Zhang and Todd, describing the effects of injection of oily produced water into fractured core.

Civan and others have published on formation damage modelling. Their models are very generic and include both chemical (scaling, clay swelling and fines release and migration) and mechanical processes (plugging). Ohen and Civan claimed that, in spite of all the research efforts towards increasing the quality of injection water, damage mechanisms other than plugging have more effect on injectivity (Ohen and Civan, 1991). Therefore, they concentrated on describing the chemical effects of salinity and pH on injectivity impairment. Among the mechanisms they claimed were dominant are:

Mobilization and migration of clay and/or siliceous fine particles by colloidal and hydrodynamical processes,
Swelling of clay minerals, reducing pore entry radius and impairing permeability, and,
Inorganic scaling.

Regardless of whether this is true or not, these other mechanisms must be considered.
They proposed a comprehensive numerical model describing formation damage due to invasion of drilling fluids, oil production and injection of saline water (Ohen and Civan, 1990).
The model was tested against injection into saline sandstone cores from the Ceuta field in Lake Maracaibo, Western Venezuela (Epry and Civan, 1992). The experiments focused both on chemical damage, due to the incompatibility between the constituents of mineral rock and the flowing fluids, and mechanical damage, caused either by external particle invasion or internal particle movement. They claimed that variation of the pH of the flowing fluid was an indicator of the extent of chemical activity occurring in the formation and that high pH promotes formation damage by particle deposition within the porous media and consequently particle bridging at the pore throats. They identified that the differential pressure continuously increased with the volume injected, suggesting plugging due to particle retention effects. Pulsing of the differential pressure was an indication of the occasional breaking of the particle bridges. The results from the core tests and model predictions were not confirmed with actual field performance.

Understand the potential damaging mechanisms. Recognize that there can be degradation due to phenomena other than TSS and OIW. Mitigate accordingly.
Use offset and analog information. The PWRI Toolbox has techniques for analysing injector performance, using existing data. Use these tools.
Use core information carefully. Frequently, core damage tests will indicate qualitative beahvior. More often than not, extrapolations of performance using coring information have been unsatisfactory.
Consider combining core information with field performance data. McCune's technique is a good example of how this can be successfully done.
Take it one step farther and determine if it is more appropriate, or at least feasible to inject under fracturing conditions and evaluate the comparative economics.
Future developments for simple matrix damage models could include modifying McCune's procedure to account for a two parameter damage model such as used in PWRAD (a tool in the PWRI Toolbox).

Abrams, A.: Mud Design to Minimize Rock Impairment due to Particle Invasion. Journal of Petroleum Technology. 1977.
Al-Homadhi, E. S.: A Study of the Formation Damage Associated with the Injection of Oil and Solids into Fractured and Non-Fractured Rock. Dept. of Petroleum Engineering, Heriot-Watt University. Edinburgh. 1998.
Barkman, J. H. and Davidson, D. H.: Measuring Water Quality and Predicting Well Impairment. JPT. 1972.
Bayona, H. J.: A Review of Well Injectivity Performance in Saudi Arabia's Ghawar Field Seawater Injection Program. SPE 25531. The SPE Middle East Oil Technical Conference and Exhibition. Bahrain. 1993.
Chauveteau, C., Nabzar, L. and Coste, J.-P.: Physics and Modelling of Permeability Damage Induced by Particle Deposition. SPE 39463. SPE International Symposium on Formation Damage Control. Lafayette, Louisiana. 1998.
Civan, F.: Evaluation and Comparison of the Formation Damage Models. SPE 23787. SPE International Symposium on Formation Damage Control. Lafayette, Louisiana. 1992.
Davidson, D. H.: Invasion and Impairment of Formation by Particulates. SPE 8210. 54th annual technical conference and exhibition of SPE. SPE. Las Vegas. 1979.
Epry, E. M. A. and Civan, F.: Characterization of Formation Damage by Particulate Process. The Journal of Canadian Petroleum Technology. 1992.
Eylander, J. G. R.: Suspended Solids Specifications for Water Injection from Coreflood Tests. SPE Reservoir Engineering. 1988.
Gruesbeck, C. and Collins, R. E.: Entrainment and Deposition of Fine Particles in Porous Media. SPE Journal. 1982.
Herzig, J. P., Leclerc, D. M. and Goff, P. L.: Application to Deep Filtration. Industrial and Engineering Chemistry. 1970.
Hsi, C. D., S., D. D., Lane, R. H., Buettner, J. W. and Neira, R. D.: Formation Injectivity Damage due to Produced Water Re-Injection. SPE 27395. International Symposium on Formation Damage Control. Lafayette, Louisiana. 1994.
Iwasaki, T.: Some Notes on Sand Filtration. Journal of American Water Works Association. 1937.
Jiao, D. and Sharma, M.: Mechanism of Cake Buildup in Crossflow Filtration of Colloidal Suspensions. Journal of Colloid and Interface Science. 1993.
Khatib, Z. I.: Prediction of Formation Damage due to Suspended Solids: Modelling Approach of Filter Cake Buildup in Injectors. SPE 28488 1994.
Khatib, Z. I. and Vitthal, S.: The Use of the Effective-Medium Theory and a 3D Nework Model to Predict Matrix Damage in Sandstone Formations. SPE 19649. SPE Production Engineering. 1991.
Kumar, T.: Modelling of Formation Damage due to Particle Invasion in Relation to Water Injection Schemes. Dept. of Petroleum Engineering. Heriot-Watt University. Edinburgh. 1988.
Kumar, T. and Todd, A. C.: A New Approach for Mathematical Modelling of Formation Damage due to Invasion of Solid Suspensions. SPE 18203. The 63rd Annual Technical Conference and Exhibition of the Society of Petroleum Engineers. Houston, Texas. 1988.
Nabzar, L., Coste, J.-P. and Chauveteau, C.: Water Quality and Well Injectivity. No 044. 9th European Symposium on Improved Oil Recovery. The Hague, The Netherlands. 1997.
Ohen, A. and Civan, F.: Simulation of Formation Damage in Petroleum Reservoir. The SPE Formation Damage Control Symposium. Lafayette, Louisiana. 1990.
Ohen, H. A. and Civan, F.: Predicting Skin Effects Due to Formatin Damage by Fines Migration. Production Operations Symposium. Oklahoma City, Oklahoma. 1991.
Pang, S. and Sharma, M.: A Model for Prediction Injectivity Decline in Water Injection Wells. SPE 28489. 69th Annual Technical Conference and Exhibition. New Orleans LA USA. 1994.
Pang, S. and Sharma, M. M.: Evaluating the performance of open-hole, perforated and fractured water injection wells. SPE 30127. SPE. 1995.
Pautz, J. F., Crocker, M. E. and Walton, C. G.: Relating Water Quality and Formation Permeability to Loss of Injectivity. SPE 18888. SPE. 1989.
Rege, S. D. and Fogler, H. S.: Competition Among Flow, Dissolution and Precipitation in Porous Media. AIChE Journal. 1989.
Rochon, J., Creusot, M. R., Rivet, P., Rocque, C. and Renard, M.: Water Quality for Water Injection Wells. SPE 31122. SPE. 1996.
Roque, C., Chauveteau, C. and Renard, M. e. a.: Mechanism of Formation Damage by Retention of Particles Suspended in Injection Water. SPE 30110. European Formation Damage Conference. den Haague, The Netherlands. 1995.
Santarelli, F. J., Skomedal, E., Markestad, P., Berge, H. I. and Nasvig, N.: Sand Production on Water Injectors: Just How Bad Can It Get? SPE/ISRM 47329. The SPE/ISRM Eurock '98. Trondheim, Norway. 1998.
Sharma, M. M., Chamoun, H., Sarma, D. S. H. S. R. and Schechter, R. S.: Factors Controlling the Hydrodynamic Detachment of Particles from Surfaces. Journal of Colloid and Interface Science. 1991.
Sharma, M. M., Pang, S., Wennberg, K. E. and Morgenthaler, L.: Injectivity Decline in Water Injection Wells: An Offshore Gulf of Mexico Case Study. SPE 38180. The 1997 SPE European Formation Damage Conference. The Hague, The Netherlands. 1997.
van der Zwaag, C. and Ųyno, L.: Comparison of Injectivity Prediction Models to Estimate Ula Field Injector Performance for Produced Water Reinjection. Produced Water 2: Environmental Issues and Mitigation Technologies. Plenum Press. New York. 1996.
van Oort, E., van Velzen, J. F. G. and Leerlooijer, K.: Impairment by Suspended Solids Invasion: Testing and Prediction. SPE (Can't read). SPE Production and Facilities. 1993.
van Velzen, J. F. G. and Leerlooijer, K.: Impairment of a Water Injection Well by Suspended Solids: Testing and Prediction. SPE 23822. Symposium on Formation Damage Control. Lafayette, Louisiana. 1992.
Wennberg, K. E.: Particle Retention in Porous Media. Department of Petroleum Engineering and Applied Goephysics. The Norwegian University of Science and Technology. Trondheim. 1998.
Wojtanowicz, A. K., Krilowv, Z. and Langlinais, J. P.: Study on the Effect of Pore Blocking Mechanisms on Formation Damage. SPE 16233. The SPE Production Operations Symposium. Oklahoma City, Oklahoma. 1987.
Zhang, N.: Study of Permeability Alterations Associated with the Invasion of Oil Droplets and Solid Particles in Produced Oily Water into Porous Media. Dept. of Petroleum Engineering. 1994.
Zhang, N. S., Somerville, J. M. and Todd, A. C.: An Experimental Investigation of the Formation Damage caused by Produced Oily Water Injection. SPE 26702. 1993.

Input:	Many of the models used for simulating degradation of injectivity under matrix conditions require a substantial number of parameters for their prediction. For many of the models, the required input parameters are not clear or cannot be easily determined. What are some of the considerations.
Most of the models require input from core tests. Barkman and Davidson's model could possibly be used without core test input data - if the filter cake permeability is estimated. WID provides formulas to estimate coreflood parameters from historical information for situations when core test data is not available. None of the models takes flow of oily water into account. Only Khatib's model considers the filter cake to be compressible, and therefore the filter cake as a function of injection pressure and rate.
Parameters:	Show me an example of some of the many parameters that are required for modeling. I should be able to see the complexity!

General Models:	Tell me something about some of the original model developments for representing particulate damage to porous media.
Most particle deposition models are based on the mass conservation equation for particle deposition (Herzig, et al., 1970), assuming incompressible flow and negligible changes to diffusion and porosity. Iwasaki's deposition function, defining a filtration coefficient is also used commonly in models which attempt to fundamentally represent the physics of particle deposition.. Iwasaki proposed fundamental equations to describe filtration through a sand bed (Iwasaki, 1937). These incorporated velocity, concentrations of particles (in the flow stream and deposited), a damage factor and a filtration coefficient. The filtration coefficient is a key factor, and it depends on a number of parameters. In most cases, the filtration coefficient is assumed to be constant and either derived from filter paper or core experiments or from empirical relationships (Sharma, et al., 1997). To relate the particle deposition to permeability or injectivity reduction, the most common starting point is the Carman-Kozeny equation. Be extremely careful when you try to use this relationship to represent relationships between permeability, porosity, and filtration - particularly in poorly consolidated media.
Shell:	What are some of the features of models developed by Shell?
Shell has published a series of papers describing their efforts towards predicting decline in water injectivity due to plugging of solids in the water. The most well-known models are summarized. Recently, van den Hoek et al. published a paper describing flow through wormholes, forming through the external filter cakes on fractured injection wells (van den Hoek, 1999). Barkman and Davidson (1972) - Membrane Filter Tests Barkman and Davidson (B&D) suggested that impairment from suspended solids occurred by one or more of the four following mechanisms: Wellbore narrowing: The solids form a filter cake on the face of the wellbore, an external filter cake. Invasion: The solids invade the formation, bridge and form an internal filter cake. Ų Perforation Plugging: The solids become lodged in the perforations. Ų Wellbore fillup: The solids settle to the bottom of the well by gravity and decrease the net zone height. They used pressure drop and material balance to develop equations for the time it takes the injectivity (II) to reduce to a fraction of the original injectivity. They referred to the time when the injectivity has reduced by 50% (a=½), as the "injector half-life" (t_1/2) The model for invasion (INTERNAL FILTER CAKE) is based on the theory of deep bed filtration in which the superficial fluid velocity is a function of the position in the reservoir, as described by Herzig et al. (Herzig, et al., 1970). Davidson claimed that the impairment occurred in two stages: Particles are deposited because of gravity or surface forces, causing restriction of the pores, and, Bridging occurs within the formation, followed by the buildup of an internal filter cake. The modelling of the wellbore narrowing (EXTERNAL FILTER CAKE), or formation of an external filter cake in an openhole completion, was considered. The model for perforation plugging treats the perforation as a cylinder that is gradually filled from its innermost extremity towards the well. The increase in pressure drop because of this infill is taken as the original pressure drop plus the pressure drop through the filter plugs in the perforations. The permeability of the filter plug is taken as the external cake permeability. In the wellbore fillup model, it is assumed that all of the solids fall to the bottom of the wellbore and decrease the injection interval. Fluid flow through the fill is neglected. Wellbore fillup, caused by water hammer effects in water injection wells in a Norwegian field, has been reported (Santarelli, et al., 1998). The mathematical model assumes that the injectivity decline is proportional to the height of the interval. The key to the Barkman and Davidson injectivity prediction is the so-called "water quality ratio" or the solids concentration over the filter cake permeability (w/k_c). All of the models are functions of this ratio. Barkman and Davidson showed how this ratio could be found by simple on-site membrane testing. If the cumulative filtrate volume is plotted against the square root of time (filtration curve), it will approach a straight line with slope S, assuming that a filter cake is formed. The same slope is obtained whether invasion takes place or not. Barkman and Davidson claimed that the characteristics of the filtration curve gave an indication of the plugging mechanisms involved. When the suspended particles are larger than the pores of the filter medium, no invasion takes place. The extrapolation of the straight-line section of the filtration curve has a negative intercept on the y-axis. When the suspended solids are much smaller than the pores of the filter medium, invasion takes place at least during the early part of the test. B&D's experiments showed that the extrapolation of the straight-line section will have a positive intercept on the y-axis. The third type of filtration curve, the so-called "s-shaped curve," can apparently occur under several circumstances. One of them is when invasion, resulting in the formation of internal filter cake, is followed by formation of an external filter cake. The Barkman and Davidson model was the first significant experimental model, providing a convenient on-site method to predict the rate of impairment. The injector half-life model enjoys wide use but it has some limitations, mainly associated with the development of the filter cake properties. They are developed from membrane filter tests, which usually lack correlation with actual core flow behavior. tHE model was tested against performance in the Ula field in the Norwegian sector of the North Sea (van der Zwaag and Ųyno, 1996). However, the flow in this situation is believed to have been dominated by thermally-induced fractures and application of the B&D model may be questionable. In Ula, fair prediction was reported but accuracy for half-life was less than a factor of two. The main limitations of the Barkman and Davidson method are that the model is based on membrane filter tests, which are not believed to correlate with actual core flooding measurements. Furthermore, it does not account for entrainment and re-deposition. Van Velzen and Leerlooijer claimed that the model had some serious shortcomings and therefore, in spite of its wide use, no case histories with good correlation between theoretical prediction and practical observation were known (van Velzen and Leerlooijer, 1992). Davidson (1979) - Injection Half-Life In 1979, Davidson attempted to enhance the previous Barkman and Davidson model with theoretical and experimental modelling of particle invasion of suspended solids. He proposed using sandstone cores, instead of membrane filters, to describe particle invasion and forming of internal filter cake through deposition and bridging of pore throats. The main applications of Davidson's approach were with the original Barkman and Davidson method which was designed as a simple approach for on-site experiments to predict injectivity performance. This method has the same limitations as the B&D methods - not accounting for entrainment and re-deposition or wettability factors. Eylander Model (1988) Eylander modified Barkman and Davidson's half-life calculations. He claimed that, in spite of membrane filtration having advantages for describing filter cake phenomena, it lacked correlation to actual rock cores needed to be valid for describing internal damage mechanisms. Therefore, he introduced a new approach, based on coreflood experiments, to identify internal formation plugging, by defining a filter cake porosity parameter. Based on experimental results, Eylander developed a model to represent injectivity decline due to particle deposition during internal filtration, intended to be suitable for on-site application and recommended for validation against field performance data. No specific method to determine the type of impairment was indicated. Therefore, in spite of its limitations, the 1/3 - 1/7th rule of thumb is recommended, even though Eylander's experiments showed that it only worked in 2 out of the 5 cores tested. Furthermore, the inflow equation does not include time-related depth of impairment (Eylander, 1988). It can be very difficult to specify invasion depth and the permeability of the internal filter cake, but both are needed prior to use. The method has not been validated on-site with field data. Van Velzen Model (1992) van Velzen and Leerlooijer introduced a semi-empirical radial model to describe wellbore impairment by internal cake formation, caused by suspended solids in injection water. The model alleged to predict the half-life of water injection wells. They claimed that their experiments confirmed the "1/3 - 1/7 rule-of-thumb" developed by Abrams (Abrams, 1977) as a criterion for water quality. The model has been field-tested with data from Prudhoe Bay (Hsi, et al., 1994) and data from the Ula field (van der Zwaag and Ųyno, 1996). In both cases the model predicted worse performance than was actually observed. However, most injectors in both fields were believed to have been thermally fractured and the injected fluids were oily produced water. Therefore, the model was not fully applicable. Van Velzen et al. claimed that, for both linear and radial flow geometry, the rate of impairment strongly depends on the inflow velocity at the injection surface, i.e., the rate of impairment decreases with increasing inflow velocity and is very sensitive to velocity changes at low velocity levels. Furthermore, they claimed that the rate of impairment also depends on the relationship between particle and pore sizes. Khatib (1994) Khatib reported on a new model for calculating the combined matrix and filter cake resistances in perforations and fractures. This is an empirical correlation between permeability and porosity to predict the injectivity decline rate under reservoir conditions, for four common well completions: Openhole, Cased and perforated, Perforated and gravel packed, and, (Static hydraulic) fractured well. It was also determined that the type of suspended particles and the compressibility of the filter cakes are important. Experimental results for injecting oil-coated clay and silt solids, suspended in the injection water stream, suggested that additional permeability reduction [due to the oil] can be up to 50%. This was also confirmed with other types of solids. Khatib also reported on a numerical model to predict the injectivity decline rates for four types of well completions; openhole, perforated, gravel packed and fractured. It assumes steady-state flow and constant injection and reservoir pressures. However, a "transient flow factor" is incorporated to take into account the reservoir pressure buildup, which is important, especially for tight reservoirs.
Chevron:	What are some of the features of models developed by Chevron?
McCune's Model Bayona, 1993 discussed McCune's technique to relate well injectivity to water quality. Bayona modified McCune's equation to a) relate total suspended solids (TSS) in the water and the well injectivity rate of decline and b) account for the portion of the TSS that does not cause reduction of well injectivity. He assumed that particles smaller than 1/10th of the formation pore throat diameter (TSSnp) can travel inside the pore space without bridging and causing reduction of permeability. Bayona applied this method to a number of wells in the Ghawar field, a naturally fractured carbonate reservoir in Saudi Arabia, and discovered that plotting the decline rate against TSS resulted in a straight line. Unfortunately, Bayona did not indicate how well the slope matches the water quality ratio and the filter cake properties observed in core tests. This is a very simple but effective relationship and only requires estimates of the Injectivity Index and the solids content. However, in most PWRI cases, solids plugging is not believed to be the only source of injectivity decline, as in the Ghawar field.
University of Texas:	What are some of the features of models developed by the University of Texas?
Sharma and Yortsos used an effective medium approximation to represent the flow field in a network of interconnected pores (Sharma, et al., 1991). Pang and Sharma suggested that only two of Barkman and Davidson's mechanisms, external and internal filter cakes, realistically operate in the field (Pang and Sharma, 1994). They assumed that both wellbore fillup and perforation plugging were an extension of external filter cake growth, as described by Jiao and Sharma (Jiao and Sharma, 1993). To determine the nature of impairment, they collected information from eight previous coreflood experiments and plotted the RII against cumulative volume injected. Their experiments resulted in four different type curves for different type of impairment: Straight line: Either incompressible external filter cake or pure internal filtration, as described by van Oort et al. Curving with decreasing slope: compressible external filter cakes or internal filter cakes. Curving with decreasing slope: No external filtration and deep particle invasion. "S" shaped curves: several internal filtration mechanisms operating simultaneously. They proposed the concept of transition time in order to model the deposition of both internal and external filter cakes simultaneously. Then, they developed three, complex semi-empirical models to explain injectivity in openhole, as well as perforated and fractured wells, although again, parameter estimation for the damage factor and a filtration coefficient were required a priori from coreflood experiments. These models were based on Stokesian dynamics theory and deep bed filtration. Later they extended their model to be able to predict injectivity for fractured wells and developed the first version of the WID simulator (Water Injection Decline) (Pang and Sharma, 1995). Wennberg carried forward the previous work of Pang and Sharma and developed version 3.1 of WID (Wennberg, 1998). He presented an injectivity decline equation in terms of resistances for various completion scenarios. Sharma et al. summarized the main features and limitations of the model behind WID 3.1 (Sharma, et al., 1997): The model equations outlined have been implemented in a PC-based simulator. The simulator can be used to predict injectivity decline in vertical and horizontal wells with openhole, perforated and gravel packed completions. For vertical wells, fractures and layering can also be handled. The parameters used in the simulation can be obtained from core flow tests or from empirical correlations, built into the model. The simulator can be run for either constant injection rate or for constant pressure. Wennberg (1998) and Sharma et al. (1997) tested the simulator against actual field data from three fields. One of these is a highly unconsolidated Gulf of Mexico field, reported in paper SPE 38181. In this case, seawater (treated to high water quality) was injected below fracture pressure. The injectivity declined sharply in all wells. The model predicted the injectivity decline well in two of the six wells and for the first period in the third.
Institut Francais de Petrole:	What are some of the features of models developed by IFP?
Roque et al. proposed a microscopic concept for particle deposition (Roque, et al., 1995). After an initial period of deposition on grains, especially at stagnation points, a phase of pore throat bridging is alleged to take over. This bridging also occurs when the particles are much smaller than the pore throats. Finally, internal and external accumulation (filter cake buildup) take place. They performed an experimental series, injecting latex particles into sandstone - and they were able to observe these phases in some of the experiments. However, the last three phases often showed a considerable degree of overlap. Rochon et al. derived equations along the same lines as Van Velzen et al. and Pang and Sharma (Rochon, et al., 1996). Rochon et al also provided an equation for the cumulative volume injected (V*) when the damage switches from internal damage to growth of external cake. This corresponds to the number of particles injected when the near-wellbore porous medium is so damaged that solids can no longer enter the formation - they start to accumulate on the wellbore face. They also defined an equation for permeability decline after external filter cake starts to form, including restriction due to both internal and external filter cakes and they further derived an equation for skin due to damage in conventional matrix injection: The model was used to predict injectivity decline in two fields, one in the North Sea and another in the Guinea Gulf. The model predicted that the damage increased 5 to 6 times more rapidly for the Guinea Gulf case than for the North Sea conditions. If no action were taken, external filter cake would start to build up after 14 months in the North Sea but after only 3 months in the Guinea Gulf. An important limitation of the IFP model is that it assumes constant flow rate. The authors claimed that they had found a model accounting for constant pressure gradient (Rochon, et al., 1996) to be "too complex for practical purposes".
Heriot-Watt University:	What are some of the features of models developed by Heriot-Watt University?
This work has mainly attempted to describe the fundamental mechanisms causing near-wellbore damage with less emphasis on practical models to predict injectivity decline over field lifetimes. Kumar and Todd (1988) described various factors in core testing to characterize water injection, including the impact of flow velocity on permeability. Kumar and Todd (1988) reported on experiments involving injection of solids in water into sandstone cores (Kumar, 1988, Kumar and Todd, 1988). Their experimental work resulted in two "phenomenological porosity models," based on the law of mass conservation of solid particles injected into a porous medium and the core flood experiments. The models were designed to account for spatial variation of volume flux and particle density. The main applications of the models were to predict the damaged permeability in a similar material, at any new flow velocity, provided that the flow is laminar region and no fracture is initiated during the injection. The main limitations of the first model are that it is very simple and it only applies to a uniform formation where the initial permeability is well known for a certain superficial flow velocity - it is therefore not very practical. The second method has not been validated against actual field data. In 1994 Zhang, Todd and Somerville reported on injecting synthetic produced water into fractured and non-fractured cores (Zhang, 1994, Zhang, et al., 1993). In 1998, Al-Homadhi described injection of synthetic produced water through fractured core samples (Al-Homadhi, 1998). Ehmad (1998) continued the work of Zhang and Todd, describing the effects of injection of oily produced water into fractured core.
University of Oklahoma:	What are some of the features of models developed by the University of Oklahoma?
Civan and others have published on formation damage modelling. Their models are very generic and include both chemical (scaling, clay swelling and fines release and migration) and mechanical processes (plugging). Ohen and Civan claimed that, in spite of all the research efforts towards increasing the quality of injection water, damage mechanisms other than plugging have more effect on injectivity (Ohen and Civan, 1991). Therefore, they concentrated on describing the chemical effects of salinity and pH on injectivity impairment. Among the mechanisms they claimed were dominant are: Mobilization and migration of clay and/or siliceous fine particles by colloidal and hydrodynamical processes, Swelling of clay minerals, reducing pore entry radius and impairing permeability, and, Inorganic scaling. Regardless of whether this is true or not, these other mechanisms must be considered. They proposed a comprehensive numerical model describing formation damage due to invasion of drilling fluids, oil production and injection of saline water (Ohen and Civan, 1990). The model was tested against injection into saline sandstone cores from the Ceuta field in Lake Maracaibo, Western Venezuela (Epry and Civan, 1992). The experiments focused both on chemical damage, due to the incompatibility between the constituents of mineral rock and the flowing fluids, and mechanical damage, caused either by external particle invasion or internal particle movement. They claimed that variation of the pH of the flowing fluid was an indicator of the extent of chemical activity occurring in the formation and that high pH promotes formation damage by particle deposition within the porous media and consequently particle bridging at the pore throats. They identified that the differential pressure continuously increased with the volume injected, suggesting plugging due to particle retention effects. Pulsing of the differential pressure was an indication of the occasional breaking of the particle bridges. The results from the core tests and model predictions were not confirmed with actual field performance.