## Abstract

Numerous studies have shown that careful particle size selection is the main parameter for reducing fluid loss when drilling permeable or fractured formations. The methods are generally built around either the D50 or D90 values of the particles in the fluid as a relative size to the pore openings of the formation to minimize fluid loss. A series of studies were conducted with the aim of assessing if analysis of fluid loss could be used to separate the formation of internal and external filter-cakes, thereby enabling a more accurate estimate of the permeabilities of the internal and external filter-cakes. It was concluded that conventional particle size methods were found to be adequate for designing a fluid for wellbore stabilization purposes. This led to higher solids invasion and a more impermeable internal filter-cake. However, for optimization of reservoir drilling fluids, a different particle size selection method was found to be more useful to prevent reservoir formation damage. This method involves selecting particles that are resistant towards shear-degradation and with a D90 particle size ⪞3/2 the pore size of the formation. By analyzing fluid loss regression data and correlating these with indicators of formation damage, such as disc mass and permeability change, it was found that a ratio defined as the relative plugging factor could provide insight into the extent of solids invasion into the formation and potential formation damage.

## 1 Introduction

Static fluid loss tests are conducted daily during drilling operations to assess the drilling fluid’s ability to seal off permeable formations. When conducting tests on ceramic discs, it is possible to measure the mass change of the discs and hence also obtain an indication of particle invasion into the near-wellbore part of the formation [1]. An extension of this is to also measure permeability changes of ceramic discs to obtain further information about potential formation damage, as demonstrated by Klungtvedt and Saasen [2].

It is known that polymers may control fluid loss in formations with pore sizes up to 10 *µ*m but fail to seal formations with 20 *µ*m pore sizes and pressures exceeding 500 psi, as shown by Khan et al. [3]. A typical water-based reservoir drilling fluid using starch and sized marble to control fluid loss and bridging was therefore designed as a basis for testing formations with pore openings up to 120 *µ*m. To also test the functionality and general validity of the fluid loss and formation damage models, two differently sized cellulose-based fiber products were included in the tests.

Experiments by Alvi et al. [4] have shown that it is possible to reduce filtration loss measured on filterpaper by addition of 0.5 wt% iron oxide nanoparticles to an oil-based drilling fluid. In a series of experiments, such filtration loss was nearly halved. Contreras et al. [5] found an optimum effect by addition of 0.5 wt% graphite together with 0.5 wt% nanoparticles based on iron or calcium. These additions seem to provide a formation strengthening effect. Razzaq et al. [6] demonstrated for water-based drilling fluids that silicon manganese fume can have a potential to reduce fluid losses in porous formations either alone or together with calcium carbonate. This fume seems to have a relatively large fraction of sub-micron particles. Hence, these results fall along the same results as the previous mentioned nano-particle effects. Nano-particle addition can be used in many respects to control drilling fluid performance. In the following, such effects will not be discussed. Several nano-particle additions can be evaluated based on the findings in the present article. A study by Jiang et al. [7] showed the results of expanding existing methods by including internal and external factors for improving the borehole during drilling to enhance, e.g., the cementation and cohesive forces between the rock particles and to transform capillary suction forces into resistance.

For a fluid loss test, the emphasis is often placed on the 30-min reading, and less importance is placed on the various readings during the period of the test. By logging the fluid loss using a digital scale, it is possible to get very accurate measurements of the mass of the fluid loss and the fluid loss rate at any point in time during the fluid loss test. The information gathered on the development of the fluid may be used together with the disc mass measurement to gain valuable insight into the formation of an internal filter-cake, as a deposit of particles into the permeable disc, and the formation of an external filter-cake.

Two approaches were used to model the formation of internal and external filter-cakes. One approach was to model the fluid loss as a linear function of the square root of time using a fluid loss coefficient and a spurt loss constant. The other approach was to analyze the fluid loss as a flow through a series of flow resistances, equivalent to an electric circuit where the resistances would be separated into the flow resistance of the formation, the flow resistance of the internal filter-cake and the flow resistance of the external filter-cake. A series of experiments was therefore conducted where the objectives were to:

Determine if the formation of internal and external filter-cakes may be adequately modeled using numerical models.

Assess if factors used in the numerical modeling also can be correlated with other tests conducted to assess formation damage.

Assess the lost circulation material (LCM) particle size impact on formation damage.

Employ an experimental method for measuring the content of the fluid filtrate to assess if it is possible to trace polymers in the fluid filtrate.

## 2 Materials and Methods

Three fluid compositions were used for the tests. The basis for the study was a water-based reservoir drilling fluid composed with xanthan gum, starch, and sized ground marble (CaCO_{3}) as a bridging agent. The particle sizes were chosen to replicate a drilling fluid recipe used in a field operation. Two cellulose-based fluid loss materials were partly replacing ground marble for two of the fluid samples. One cellulose-based product is referred to as non-invasive fluid additive ultra-fine (NIF UF) with a D90 value of 75 *µ*m and another referred to as non-invasive fluid additive fine (NIF F) with a D90 value of 125 *µ*m, both having a density of 0.98 g/cm^{3}. The fluid compositions were as presented in Table 1.

The three fluids were hot-rolled for a period of 16 h at a temperature of 112 °C in a hot-rolling cell where a threaded steel rod was included to simulate mechanical wear. A wet-sieving study was conducted of the base fluid showing that >99% of the ground marble particles were finer than 53 *µ*m after hot-rolling. The hot-rolling temperature was selected to replicate a certain reservoir temperature. Fluid loss tests were thereafter conducted on 2.5 *µ*m filterpaper at 500 psi differential pressure and on ceramic discs with specified median pore sizes of 10, 20, 50, and 120 *µ*m, respectively.

The high temperature and high pressure (HTHP) fluid loss tests were conducted in accordance with ANSI/API 13B-1 [8] at 500 psi differential pressure, although the tests on ceramic discs were conducted with 1000 psi differential pressure, in order to identify if higher applied pressure could impact the sealing abilities and formation damage. The fluid loss tests were conducted at 90 °C, unless stated otherwise. By selecting a temperature below the boiling point, it was more practical to measure fluid loss accurately during the test.

## 3 Results and Discussion

### 3.1 Fluid Loss Tests and Regression Model.

The fluid loss for the test on the 120 *µ*m ceramic disc is plotted in Fig. 1. For the base fluid, the test was stopped after a short period of time as the fluid loss was high and uncontrolled. For the two other fluids, the fluid loss fell to a very low level shortly after an initial spurt loss. When plotting the fluid loss against the square root of time, the graph appears to be linear after the spurt loss is experienced, whereas models typically only describe it as a function of the square root of time, without considering the spurt loss separately [9].

^{0.5}). To separate the spurt loss from the linear loss phase, the first data point in the regression was the fluid loss value recorded after 15 s. Thereafter, a trendline was calculated using a linear regression model as presented in Eq. (1). The fluid loss model calculated the fluid loss (ml), FL

*, as a fluid loss coefficient*

_{T}*C*

_{FL}multiplied by the square root of time,

*T*

^{0.5}, plus a spurt loss constant, SL (ml). The fluid loss graphs for fluids NIF UF and NIF F on the 120

*µ*m discs are presented together with the linear regression models in Fig. 2. For both tests, the goodness of fit value,

*R*, was in excess of 0.998, thereby indicating that the linear regression describes the underlying data in a very good way.

^{2}Furthermore, similar regressions were made of the other fluid loss tests. The regressions of all three fluids on the 50 *µ*m ceramic disc are shown in Fig. 3, and similarly the regressions of the fluid loss on the 2.5 *µ*m filterpaper in Fig. 4. In contrast to a ceramic disc, it may be argued that a test on 2.5 *µ*m filterpaper presents less of formation plugging and thereby a clearer study of the external filter-cake itself.

The fluid loss coefficient data for the whole test-series are presented in Fig. 5. For the base fluid, the coefficient is more than double its value from the test on the filterpaper to the tests on the ceramic discs. In contrast, the coefficient only increases marginally for the fluid with NIF UF and NIF F. For the latter two, the coefficient gradually increases with higher disc pore size and reaches a maximum with the 50 *µ*m discs, before it falls marginally when using the 120 *µ*m discs.

The large change for the base fluid may be related to the increase in sealing pressure, from 500 psi on the filterpaper to 1000 psi on the ceramic discs. This may be natural as the ground marble particles are not believed to elastically compress, and hence the fluid loss may increase in a near linear fashion with pressure in the pressure range that has been investigated. In contrast, the increased pressure does not lead to a similar increase in fluid loss for NIF UF and NIF F, likely due to the compressibility of the cellulose fibers making them crate a compact filter-cake. For the test with similar pressures, it may be assumed that the permeability of the external filter-cake is consistent irrespective of the formation permeability and the internal filter-cakes, as long as the disc is sufficiently sealed during the spurt loss phase to facilitate the formation of an external filter-cake. For the base fluid, the *C*_{FL} is consistently higher than the other fluids in all the tests, indicating that the filter-cake of the base fluid is roughly twice as permeable as the filter-cakes of NIF UF and NIF F at 1000 psi.

For the regressions, the corresponding spurt loss constant data are presented in Fig. 6. The value for the base fluid on the 120 *µ*m disc of >70 ml was taken from the fluid loss measurement as no regression could be made as the test was aborted. The calculated SL values are as expected very low for the tests on filterpaper and highest on the tests with 120 *µ*m ceramic disc. The latter signifies that a larger volume of fluid was required to establish an effective internal filter-cake, against which an external filter-cake could be built. For the tests with 10, 20, and 50 *µ*m discs, the three SL values were in a relatively narrow range, for each fluid. The highest values in the tests were for the base fluid and the lowest for the NIF F fluid.

A high RPF would indicate that the fluid loss may be highly impacted by the spurt loss and thereby formation plugging. A high value may therefore be ideal for wellbore stabilization purposes, as a disturbance of the external filter-cake during drilling may have lower consequences. It is assumed that the formation of an internal filter-cake, as measured by the mass increase of discs or the estimated permeability of the internal filter-cake, will make the formation less exposed to disturbances in the wellbore. Circulation of drilling fluid or swabbing effects are less likely to lead to increased losses or differential sticking if the internal filter-cake limits the pressure communication between the wellbore and the formation. In contrast, a lower degree of formation plugging may be desired for reservoir drilling purposes, where formation plugging may lead to permanent permeability reduction and hence formation damage. Given that the ratio does not say anything about the absolute level of fluid loss, fluids should not be evaluated using the metric alone.

The RPF values for the tests are presented in Fig. 7. Given that the *C*_{FL} remained reasonably consistent for each fluid at a given pressure, the difference in SL also translates into the RPF readings. As expected, the RPF values for the tests on filterpaper are very low, indicating that the external filter-cake is the primary barrier towards fluid loss. This is in clear contrast with the values calculated for the 120 *µ*m ceramic disc tests. The high ratios indicate high particle migration and significant plugging of these discs during the HTHP tests. Another observation is that the RPF values for NIF F are considerably lower than for NIF UF in all tests. For practical purposes, these fluids are identical except for the different particle size distribution (PSD) of the cellulose fibers. This may indicate that the finer fibers of NIF UF need a slightly higher initial fluid loss to form an internal filter-cake, whereas the larger NIF F particles build an initial bridging faster.

### 3.2 Permeability and Disc Mass Measurements.

For the tests conducted on 10, 20, and 50 *µ*m ceramic discs, permeability tests were conducted using air both before the HTHP tests and after reverse flow through the discs to remove the external filter-cake and potentially parts of the internal filter-cake (see Figs. 8 and 9). For all the ceramic discs, the changes in mass were measured using a moisture analyzer after reverse flow with water to lift off the external filter-cake and remove loose deposits within the disc. Overall, the best permeability results were recorded on the 20 *µ*m ceramic discs. Considering the high concentration of CaCO_{3} in all three fluids, this may indicate that the size of the CaCO_{3} after degradation works most effectively in this range of pore sizes. Considering both disc mass increases and permeability measurements, NIF F appears to provide the best formation protection for any formation above 20 *µ*m, whereas the NIF UF may be more effective in the 10–20 *µ*m range. It should be noted how the permeability is clearly reduced with the base fluid on the 10 *µ*m disc. The result of 54% from the 30-min test was replicated with multiple tests with small variance.

In summary, the average retained permeability for the tests with the base fluid was 73%, with a standard deviation of 21%, whereas the tests with the cellulose-based fibers yielded an average retained permeability of 88%, with a standard deviation of 10%.

For the two test-series for fluids NIF UF and NIF F, it was possible to calculate both the RPF and measure the disc mass increase for all four disc grades. The correlations between the two variables were 0.998 for NIF UF and 0.99 for NIF F. When plotting the RPF against the ratio of the NIF particle D90 value to the median disc pore size, it is clear that the disc mass retains a relative stable level until the particle D90 to pore size ratio approaches the range of around 1.5–2.2. With lower ratios, the disc mass increase rises sharply, as presented in Figs. 10 and 11. This is a strong indicator that, with the applied concentration of the specific cellulose-based fibers and by selecting a D90 value ⪞ 3/2 times the pore opening, a low-permeability external filter-cake is created and the invasion of solid particles into the formation is limited.

When plotting the particle D90 to pore size ratio versus the RPF, a very similar pattern emerges, as presented in Figs. 12 and 13. When the particle D90 to pore size ratio falls below the 1.5–2.2 range, the RPF increases sharply, indicating that the particles of the fluid enter the formation, rather forming an external filter-cake. The RPF maintains a level in the range of 20–30 as long as the size ratio does not fall below the 1.5–2.2 range. This may indicate that 30 ⪞ RPF represents a limit where solids invasion is limited.

### 3.3 Polymer Content in the Fluid Filtrate.

An experimental analysis was conducted to measure the contents of the fluid filtrate relative to the drilling fluid before application. By using a series of test including turbidity, salinity, conductivity, and refractive index (BRIX), each of the components of the fluids were mapped and the values of the fluids and the fluid filtrates measured. Each component thus made a unique “fingerprint” in terms of relative readings on the different parameters measured. Using this method, it was possible to estimate the relative polymer concentrations in the fluid filtrates. This was calculated by measuring the BRIX value and subtracting the BRIX value resulting from other constituents in the filtrate such as salts. Furthermore, by combining the volume of the filtrate and the polymer concentration in the filtrate, it was possible to plot the polymer content in the filtrate for certain fluid loss tests.

Figure 14 presents the estimated polymer content in the filtrate from the 500 psi tests on filterpaper for the three respective fluids. Here the concentration of polymers in the test with the base fluid was around twice that of the NIF UF and the NIF F fluids. The area of each indicator reflects the multiple of the BRIX value and the fluid loss value to reflect the absolute volume of polymers. The equipment used is listed in the Appendix. For the first test-series, it was decided to use portable equipment that easily could be used in a small field laboratory.

The fluid loss test with 20 *µ*m ceramic discs had been conducted at a higher pressure than the test on filterpaper. The plot of the fluid loss and BRIX is presented in Fig. 15. For the tests with fluids NIF UF and NIF F, the estimated absolute volume of polymers was slightly lower than that found in the test with filterpaper. This may be due to some deposit of the polymers within the disc itself. In contrast, the volume of polymers calculated for the test with the base fluid nearly doubled relative to the test on filterpaper, likely indicating that more polymers escape the base fluid filter-cake as the pressure is doubled.

The analysis results of a third test conducted on 10 mm ceramic discs over a period of 24 h and with 500 psi applied pressure is presented in Fig. 16. In this test, the measured polymer content in the filtrate from the base fluid was noticeably smaller than presented in Figs. 14 and 15.

The results from the test on filterpaper may indicate that the fluids containing cellulose fiber bind the polymers better in the filter-cake and hence release less polymers into the formation. Given that the cellulose particles and the polymers both exhibit polar properties, it may be that the polar interaction causes increased inter-particle adhesive and frictional forces. This hypothesis is also supported with the results from the test on 20 *µ*m ceramic discs, where the higher applied pressure nearly doubles the calculated polymer volume in the filtrate for the base fluid, whereas it remains relatively unchanged for the two fluids containing fibers.

The 24-h test on 10 *µ*m ceramic discs yielded lower calculated values of polymers in the filtrate for the base fluid. The cause of this may be that due to the finer pore openings of the disc, more polymers are deposited within the disc, and hence less are transferred as part of the fluid filtrate. The permeability measurements of the 24-h tests on 10 *µ*m discs were in line with those for the 10 *µ*m discs in the 30-min test presented in Fig. 8. A possible conclusion is therefore that the relative reduction in permeability for the base fluid on the 10 *µ*m discs is polymer invasion and partial plugging of the discs. In contrast, the two fluids with fibers hold the polymers firmer within the filter-cake and hence lead to a smaller formation damage.

### 3.4 Extension of Model for Filter-Cake Formation.

The three fluid samples all have a volume concentration of polymers of 2.25%, with a specific gravity of 0.95, whereas the volume concentration of solids (ground marble and fibers) was 4.23% for the base fluid, with a specific gravity of 2.7, and 4.63% for the fluids with NIF UF and NIF F, with an average specific gravity of 2.16. By combining these values with the calculated spurt loss constants, it is possible to analyze the creation of an internal filter-cake.

*U*= voltage,

*I*= current, and

*R*= resistance, as represented in Eq. (3). Converting this to a flow of fluid through a formation, the applied differential pressure, Δ

*P*, would be equivalent to the voltage,

*U*, and a volume flow of a specific Newtonian fluid,

*Q*, equivalent to the current,

*I*. Furthermore, the resistance to flow may be divided up into the flow resistance of the formation

*R*, the flow resistance of the internal filter-cake,

_{F}*R*

_{IF}, and the flow resistance of the external filter-cake

*R*

_{EF}, as per Eq. (4) modeling the elements as serial resistance to flow, and thereafter re-arranged into Eq. (5).

*K*is the permeability,

*Q*is the flowrate,

*η*is the assumed constant viscosity of the fluid, and

*A*is the area of flow. For simplicity, the area

*A*and the fluid viscosity

*η*can be set equal for the formation and the internal and external filter-cakes. By substitution, Eqs. (5) and (7) may be combined to form Eq. (8). Here, the length and permeabilities of formation, internal filter-cake, and external filter-cake are separated and named with subscript

*F*, IF, and EF.

When applying this to a study using ceramic discs, the area *A* is defined by the actual flow area through the discs, and Δ*L _{F}* as the thickness of the disc. The permeability of each disc,

*K*, can be assumed to be unchanged during the test. Following the logic of the linear regression model, Eq. (1), the flow resistance of the internal filter-cake is modeled as a function of the spurt loss and is assumed to be constant thereafter. With such an approach, the flow resistance of the external filter-cake is the only flow resistance factor changing after the initial spurt loss. The buildup of the flow resistance of the formation,

_{F}*R*, the internal filter-cake,

_{F}*R*

_{IF}, and the external filter-cake,

*R*

_{EF}, may hence be schematically presented as per Fig. 17. Due to the significant differences in value, the flow resistance is illustrated using a logarithmic scale.

During a fluid loss test, the content and behavior of the fluid filtrate normally change character over time. The higher the spurt loss, the more the initial filtrate will resemble the drilling fluid in look, content, and viscosity. After internal and external filter-cakes have been established, the fluid filtrate gradually changes to become similar to the base fluid. For a water-based fluid, this implies that the filtrate will gradually move towards showing Newtonian behavior.

A simplified model was used as an estimate of the viscosity of the fluid filtrate where the initial flowrate, during the spurt loss phase, was high and the viscosity near that of the drilling fluid, and as the fluid loss reduced, the viscosity moved asymptotically towards the viscosity of the base fluid.

*F*, at time,

_{T}*T*, is separated into a portion of solids, polymers, and some of the fluid base to form the filter-cake, denoted FC

*and a clear fluid portion which escapes the filter-cake and becomes the measured fluid loss, CL*

_{T}*after the initial spurt loss SL.*

_{T}*v*(%), the volumetric concentration of solids

_{P}*v*(%), and the average specific gravities of the polymers and solids

_{S}*ρ*and

_{P}*ρ*, we get Eq. (10).

_{S}*k*.

_{F}For the base fluid, this computes to a factor where *k _{F}* = 20% and substituting into Eq. (9), FC

*= 0.25 * CL*

_{T}*. For the NIF UF and NIF F fluids, the factors are almost similar as*

_{T}*k*= 19% and substituting into Eq. (9), FC

_{F}*= 0.234 * CL*

_{T}*.*

_{T}*can be represented as per Eq. (13). The thickness of the filter-cake is calculated as per Eq. (14) and the permeability,*

_{T}*K*, as per Eq. (15).

A numerical analysis was setup based on the assumptions described and Eqs. (1), (8), (14), and (15) together with the logging data from a fluid loss test to simulate the development of the internal and external filter-cakes and their permeabilities. Figure 18 presents such a data plot. For the test, a ceramic disc with median pore size of 50 *µ*m was used. The permeability to water was measured to be 22.4*D*. In contrast, the internal filter-cake permeability was calculated to be 69 mD and he external filter-cake <0.1 mD. In the modeling, the Δ*L*_{IF} was set to 4.5 times the median pore size of the disc. This was done after fracturing discs to make a visual inspection of particle invasion. By applying the regression data into Eq. (15), the permeability was calculated to be 0.054 mD, which was marginally lower than that of the numerical analysis. Equations (14) and (15) will be less useful if there is a substantial plugging into the disc during the spurt loss phase, as the internal filter-cake may then be the critical factor reducing the fluid loss. For the 120 *µ*m discs, the invasion was considerably larger, and for the test with the base fluid, deposits of CaCO_{3} were seen almost through the thickness of the disc even though the HTHP test was aborted within 2 min.

### 3.5 Discussion.

The fluid loss tests were conducted under static conditions. The linear regression model applied is consistent with common theory, where fluid loss is calculated as a constant multiplied by the square root of time [2]. In the current study, the model is slightly amended to separate the spurt loss phase from the steadier loss rate. During a static filtration test, the filter-cake is allowed to build steadily as there is no mechanical disturbance of the filter-cake surface. In contrast, a dynamic fluid loss test would experience a continuous disturbance to the wellbore side of the filter-cake due to circulation of fluid. In such a condition, an equilibrium condition is likely to be met where the rate of erosion of the filter-cake equals the rate of buildup due to fluid loss. Hence, in a dynamic condition, the fluid loss will remain higher, and the filter-cake will reach a maximum thickness depending on the rate of erosion. With the assumption that the filter-cake is tight enough to prevent particle migration, the difference between a static and a dynamic fluid loss test will not impact the formation of the internal filter-cake to any significant degree.

The hot-rolling procedure included a threaded steel rod to simulate mechanical degradation. Studies have shown that CaCO_{3} particles degrade during circulation and exposure to mechanical shear [10]. In the same study, a cellulose-based LCM was found to show very low levels of particle size degradation. Applying these findings to the three fluids used in these studies, it may be assumed that the CaCO_{3} particles, with an initial D90 value in the region of 125 *µ*m and D50 value of 50 *µ*m, may have been ground down in size and that the largest particles in fluids NIF UF and NIF F may be the cellulose-based fibers. The test with the base fluid did not effectively seal the 120 *µ*m disc, thus suggesting that the new D50 value of the CaCO_{3} is less than 120/3 or 40 *µ*m, following the Abrams rule [11]. Similarly, the base fluid sealed the 50 *µ*m disc, suggesting that the D50 is likely larger than 50/3 *µ*m or the D90 value is ⪞45 *µ*m after the hot-rolling process, applying the findings of Alsaba et al. [12]. The tests thus showed that if a circulating fluid is exposed to mechanical wear like that of the applied hot-rolling process, the PSD of CaCO_{3} before circulation cannot be applied using known particle size selection methods. For the 50 *µ*m discs, the tests with NIF F provided the lowest fluid loss, disc mass increase, and permeability reduction. For this test, the NIF UF fluid would likely contain particles around 3/2 times the pore opening, whereas NIF F would likely contain particles ⪞3/2 times the pore opening, indicating that the latter would be preferrable to limit formation damage.

The tests presented in Sec. 3.1 showed that with particles present in the fluid that were equal to or larger than the pore size of the discs, the mass increase of each disc was between 48 and 121 mg. For the 120 *µ*m discs, where the particle size was equal to or smaller than the pore size, the mass increase was in the region of 615–1495 mg. Also, comparing the tests with the fluids NIF UF and NIF F, it is clear that the larger particles in NIF F better protects against solids invasion and fluid loss with pore openings in the range from 50 to 120 *µ*m, whereas the differences between NIF UF and NIF F are small in the range from 20 *µ*m and smaller pore openings.

The experimental method applied to identify and measure polymer concentration in fluid filtrate showed consistent differences between the fluids with fibers and the base fluid. It is an interesting observation which may be explained by polar interaction and thereby increased adhesive and frictional forces between the cellulose-based fibers and the polymers used for fluid loss and viscosity. Such interaction may also be consistent with increased tensile strength or cohesive strength of the filter-cake. Further analysis should be conducted to better verify the experimental method and its applicability. Further studies should also be conducted to understand potential interaction between cellulose-based fibers and polymers and its impact on filter-cake strength and also the impact on dynamic fluid loss tests. With higher filter-cake cohesion, less erosion should be expected in a dynamic condition, and hence the fluid loss could be further reduced.

The extension of the model for analysis of filter-cake formation increases the complexity of the modeling. It was possible to calculate estimates of the permeability of internal and external filter-cakes, but with very significant increases in computation relative to the regression model. The extension of the model indicates that, once established, the external filter-cake was the dominant factor in controlling the fluid loss for the tests conducted in this experiment. This observation was also expected given that the fluid compositions used were designed for reservoir drilling purposes. If similar modeling had been conducted with a fluid designed for wellbore strengthening purposes, the results of the modeling might be different. As such, with such a numerical model being established it may provide useful information for further understanding of how the internal and external filter-cakes are being built.

The linear regression model obtained very high goodness of fit values which may be useful to predict fluid loss under static conditions. Applying this together with Eq. (14), one can also predict how the thickness of the external filter-cake will evolve over time. As an example, under the test conditions for the test on 10 *µ*m discs, a 72-h test could be forecasted to yield 180 ml of fluid loss for the base fluid and 84 ml of fluid loss for each of NIF UF and NIF F. The corresponding filter-cake thicknesses would be 11.1 mm for the base fluid and 4.9 mm for NIF UF and NIF F.

The relative plugging factor appeared to give meaningful information regarding when a fluid changed from primarily producing an external filter-cake to when the discs were plugged through solids migration. This inflection point was observed when the RPF was around 30. The tested fluids had relatively similar concentrations of solids. It could be expected that a higher solids concentration would lead to a lower spurt loss. A higher concentration of the same solids may, however, not necessarily cause a change in the filter-cake permeability, and hence in the coefficient of fluid loss, as the same distribution of particles would be present to build the filter-cake. Therefore, it would be natural that a fluid with higher volumetric concentration of solids would produce an inflection point for the RPF at values less than 30.

The two approaches used for modeling of the filter-cake permeabilities clearly show that already within the first seconds of the tests, the original formation permeability becomes insignificant in controlling the fluid loss, relative to the much lower permeabilities of the internal and external filter-cakes. As such, the microflow of polymers through the permeable discs were not studied in detail. Other studies such as Ref. [13] have been conducted to understand the viscoelastic flow of polymer fluids in permeable formations, and in particular the microflow mechanisms of polymer displacements. Such findings may bring further insight if applied to the study of fluid filtrate when using water-based drilling fluids with polymers.

## 4 Conclusion

Numerical modeling of the formation of internal and external filter-cakes provided to be a useful approach. New information was discovered, and further studies should be conducted to assess if additional insight into filter-cake formation might be gained.

It was verified that a modified linear regression model could describe a static fluid loss test with very high goodness of fit by separating the factors into a spurt loss constant and a coefficient of fluid loss and plotting against the square root of time.

The modified regression model enables a separation of the calculation of the internal and external filter-cake permeabilities.

The calculated relative plugging factor provided consistent results with the measurements of disc mass increases. This indicates that 30 ⪞ relative plugging factor represents situations where fluid loss, for the tested fluids, is primarily controlled by an external filter-cake and that formation damage is limited.

Average retained permeability for the tests with the base fluid was 73%, whereas the tests with cellulose-based additives showed an average retained permeability of 88%.

It was found that cellulose particles with size ⪞3/2 the pore size limited polymer and solids invasion into the formation, whereas invasion of particles were higher when the largest particles were equal to or smaller than the pore openings. Using the experimental analysis of fluid filtrate, it was found that the presence of cellulose particles in the filter-cake led to reduced polymer invasion into the formation relative to a fluid with only CaCO

_{3}used as bridging material.

The experimental analysis of fluid filtrate should be further applied and analyzed to determine its consistency and accuracy.

## Acknowledgment

The authors would like to thank Jan Kristian Vasshus, Bjørn Berglind, Swapan Mandal, and Nicola Santarelli for advice and the Research Council of Norway for financially supporting the project through RCN# 320646.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## Nomenclature

*A*=cross-sectional flow area (m

^{2})*I*=current (A)

*K*=permeability (m

^{2}), where 1 Darcy = 1*µ*m^{2}*Q*=flowrate (m

^{3}/s)*R*=electrical resistance (ohm)

*R*=flow resistance (Pa s/m

^{3})*U*=voltage (V)

*C*_{FL}=coefficient of fluid loss (ml/s

^{0.5})*R*=_{X}flow resistance (Pa s/m

^{3}), where the subscript refers to the medium*R*^{2}=goodness of fit for the regression model

- BRIX =
degrees of refraction (°Bx)

- RPF =
relative plugging factor (s

^{0.5})- SL =
spurt loss constant (ml)

- FL
=_{T} fluid loss at time

*T*(ml)- Δ
*L*= length (m)

- Δ
*P*= applied differential pressure (Pa)

*η*=viscosity (Pa s)

### Appendix

The equipment setup was as follows:

Equipment used for testing:

Hamilton Beach Mixer

Ohaus Pioneer Precision PX3202

Ofite Filter Press HTHP 175 ml, double capped

Ofite Viscometer model 900

Ofite roller-oven #172-00-1-C

Apera pH90, pH meter

Ohaus MB120 Moisture Analyser

Custom built transparent acrylic cell for enabling of reverse flow of fluid through the ceramic discs

Festo Pressure Regulator LRP-1/4-2.5 and LRP-1/4-0.25

Festo Pressure Sensor SPAN-P025R and SPAN-P10R

Festo Flowmeter SFAH-10U

Nitrogen source and manifold for pressure up to 1350 psi, Ofite #171-24

Vacuum machine, DVP EC.20-1

Thermo Scientific, Eutech Expert CTS

Hanna, HI96801 Refractometer