Enhanced oil recovery as a stepping stone to carbon capture and sequestration

Abstract

Environmental concerns about carbon emissions coupled with the oil industry’s need to secure additional CO₂ for enhanced oil recovery (CO₂-EOR) projects have sparked interest in the potential that CO₂-EOR may have in jumpstarting carbon capture and sequestration (CCS). However, existing studies on the viability of coupling CO₂-EOR with CCS have generally placed more focus on either the engineering or economic aspects of the problem. Most engineering studies focus on the technical aspects of the CO₂-EOR project to produce the maximum amount of oil, while simultaneously storing the most CO₂ during the production process with the economics as an afterthought, while most economic studies found have focused on a singular aspect of the issue such as impacts of exogenously varying injection rates. Furthermore, modelling efforts have stopped at the end of the productive life of the field. We build a unique two-stage dynamic optimization model, which simultaneously addresses engineering and economic policy aspects, to study the viability of coupling CO₂-EOR transitioning into CCS. Our model includes a carbon tax for emissions, which becomes a subsidy for full scale sequestration after oil production has ceased; this allows us to explore the transition from CO₂-EOR, our first stage, to sole CO₂ sequestration in our second stage for a single field. We maximize the operator’s profits across both stages, while tracking the responsiveness of oil production and total carbon movements to both price and policy changes. We pair our optimization model with a reservoir simulation model, allowing us to mimic actual field behavior, giving our work a more realistic representation of both production and sequestration profiles. Our results suggest that small increases in the level of carbon tax can have large and discontinuous impacts on net sequestration. This stems from the observed transition from limited natural sources of CO₂ to more expensive captured CO₂ resulting from the implemented policy. With appropriate taxes, total volumes of captured CO₂ sequestered across both stages are equivalent to 30 to 40% of the emissions from the use of the oil produced. With the credits oil producers receive from sequestering CO₂, which equate to the tax, relatively high carbon taxes incentivize additional sequestration without significantly impacting the supply of oil. This, alongside maintaining a steady stream of profits, is a win-win situation for energy security and the climate.

Appendix: Model Setup and Optimality Conditions

The ordinary control theory formulation does not consider dynamic problems with more than a single optimization phase. The challenge arises in our two-stage setup in the selection of the transition point from one stage to the next, in addition to the selection of the optimal paths for the control and state variables in each stage of the optimization. We now present the setup and optimality conditions of our two-stage model that considers a price-taking producer that seeks to maximize the present value of profits with respect to a policy that could potentially make sole sequestration profitable. Our first integral in the objective function represents profits from CO₂-EOR, while the second represents profits generated by CCS. We assume that the transition from one stage to the next does not require any additional capital cost outside of shutting in producing wells. Our objective function (Eq. 11 from the text), associated constraints, and optimality or first-order conditions are listed below:

Objective function:

$$ \max \pi ={\int}_0^{T_1}{e}^{- rt}\left(\left[{p}_o-\beta \tau \right]{q}_p^o(t)-{w}_{CAP}{q}_{CAP}^c(t)-\left({w}_{NR}+\tau \right){q}_{NR}^c(t)-{w}_R{q}_R^c(t)+\tau {q}_S^c(t)-{F}_1\right) dt+{\int}_{T_1}^{T_2}{e}^{- rt}\left(\tau {q}_S^c(t)-{w}_{INJ}{q}_i^c(t)-{F}_2\right) dt $$

Substituting in the relationships between oil production and CO₂ injection and sequestration yields:

$$ \mathit{\max}\pi ={\int}_0^{T_1}{e}^{- rt}\left(\left[{p}_o-\beta \tau \right]\delta \left({q}_i^c(t)\right)R(t)-{w}_{CAP}{q}_{CAP}^c(t)-\left({w}_{NR}+\tau \right)\left({q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t)-{q}_{CAP}^c(t)\right)-{w}_R\left({q}_i^c(t)-{q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t)\right)+\tau {q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t)-{F}_1\right) dt+{\int}_{T_1}^{T_2}{e}^{- rt}\left(\tau {q}_i^c(t)-{w}_{INJ}{q}_i^c(t)-{F}_2\right) dt $$

(12)

Subject to the following constraints that will differ across each stage:

$$ 0\le t\le {T}_1 $$

$$ \dot{R(t)}=-{q}_p^o(t)=-\delta \left({q}_i^c(t)\right)R(t) $$

(13)

$$ \dot{X(t)}=-{q}_{NR}^c(t)=-\left({q}_s^c(t)-{q}_{CAP}^c(t)\right)={q}_{CAP}^c(t)-{q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t) $$

(14)

$$ \dot{S(t)}={q}_s^c(t)={q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t) $$

(15)

$$ R(0)=1\ \mathrm{million}\ \mathrm{rb}\ \mathrm{of}\ \mathrm{oil}\&R\left({T}_1\right)\ge 0 $$

(16)

$$ X(0)=1\ \mathrm{million}\ \mathrm{rb}\ \mathrm{of}\ {\mathrm{CO}}_2\&X\left({T}_1\right)\ge 0 $$

(17)

$$ S(0)=0\ \mathrm{million}\ \mathrm{rb}\ \mathrm{of}\ {\mathrm{CO}}_2\kern0.5em \&S\left({T}_1\right)\ge 0 $$

(18)

$$ 0\le {q}_i^c(t)\le 1\ \mathrm{million}\ \mathrm{reservoir}\ \mathrm{barrels}/\mathrm{year} $$

(19)

Transversality conditions for our first stage:

$$ {H}_1\left({T}_1\right)=0 $$

(20)

$$ {\lambda}^R\left({T}_1\right)\ge 0\kern0.5em R\left({T}_1\right)\ge 0\kern0.5em {\lambda}^R\left({T}_1\right)\times R\left({T}_1\right)=0 $$

(21)

$$ {\lambda}^X\left({T}_1\right)\ge 0\kern0.5em X\left({T}_1\right)\ge 0\kern0.5em {\lambda}^X\left({T}_1\right)\times X\left({T}_1\right)=0 $$

(22)

$$ {\lambda}^S\left({T}_1\right)\ge 0\kern0.5em S\left({T}_1\right)\ge 0\kern0.5em {\lambda}^S\left({T}_1\right)\times S\left({T}_1\right)=0 $$

(23)

Moving to our second stage:

$$ {T}_1\le t\le {T}_2 $$

$$ \dot{S(t)}={q}_s^c\left({q}_i^c(t)\right)={q}_i^c(t) $$

(24)

$$ S\left({T}_1\right)=S\left({T}_1\right)\ \mathrm{from}\ \mathrm{stage}\ 1\kern0.5em \&S\left({T}_2\right)\le S\left({T}_1\right)+1.2\times {\int}_0^{T_1}\left({q}_p^o(t)\right) dt $$

(25)

$$ 0\le {q}_i^c(t)\le 1\ \mathrm{million}\ \mathrm{reservoir}\ \mathrm{barrels}/\mathrm{year} $$

(26)

The upper bound on sequestration for S(T₂) in Eq. 25 serves as our terminal conditions with free terminal time (T₂) and eliminates the need to identify transversality conditions for our second stage. This upper bound on sequestration also provides the link between the first and second stage of the model.

Modelling stage 1

$$ 0\le t\le {T}_1 $$

The current value Hamiltonian for period 1 is as follows:

$$ {H}_1(t)=\left[{p}_o-\beta \tau \right]\delta \left({q}_i^c(t)\right)R(t)-{w}_{CAP}{q}_{CAP}^c(t)-\left({w}_{NR}+\tau \right)\left({q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t)-{q}_{CAP}^c(t)\right)-{w}_R\left({q}_i^c(t)-{q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t)\right)+\tau {q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t)-{F}_1-{\lambda}^R\left(\mathrm{t}\right)\delta \left({q}_i^c(t)\right)R(t)+{\lambda}^X(t){q}_{CAP}^c(t)-{\lambda}^X(t){q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t)+{\lambda}^S(t){q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t) $$

(27)

Optimality conditions for stage 1

$$ \frac{\partial {H}_1(t)}{\partial {q}_i^c(t)}=\left[p-\beta \tau \right]{\delta}^{\prime}\left({q}_i^c(t)\right)R(t)-\left({w}_{NR}+\tau \right){q}_i^c(t){\delta}^{\prime}\left({q}_i^c(t)\right)R(t)-\left({w}_{NR}+\tau \right)\delta \left({q}_i^c(t)\right)R(t)-{w}_r+{w}_r{q}_i^c(t){\delta}^{\prime}\left({q}_i^c(t)\right)R(t)+{w}_r\delta \left({q}_i^c(t)\right)R(t)+\tau {q}_i^c(t){\delta}^{\prime}\left({q}_i^c(t)\right)R(t)+\tau \delta \left({q}_i^c(t)\right)R(t)-{\delta}^{\prime}\left({q}_i^c(t)\right)R(t)-{\lambda}^X(t){q}_i^c(t){\delta}^{\prime}\left({q}_i^c(t)\right)R(t)-{\lambda}^X(t)\delta \left({q}_i^c(t)\right)R(t)+{\lambda}^S(t){q}_i^c(t){\delta}^{\prime}\left({q}_i^c(t)\right)R(t)+{\lambda}^S(t)\delta \left({q}_i^c(t)\right)R(t)=0 $$

(28)

with

$$ {\delta}^{\hbox{'}}\left({q}_i^c(t)\right)=\frac{\partial \left({\delta}_w+{\delta}_1{q}_i^c(t)-{\delta}_2{q}_i^c{(t)}^2\right)}{\partial {q}_i^c}={\delta}_1(t)-2{\delta}_2(t){q}_i^c $$

(29)

$$ \frac{\partial {H}_1(t)}{\partial {q}_{CAP}^c(t)}=-{w}_{CAP}+\left({w}_{NR}+\tau \right)+{\lambda}^X(t)=0 $$

(30)

$$ \dot{R(t)}=\frac{\partial {H}_1(t)}{\partial {\lambda}^R(t)}=-\delta \left({q}_i^c(t)\right)R(t) $$

(31)

$$ \dot{X(t)}=\frac{\partial {H}_1(t)}{\partial {\lambda}^X(t)}={q}_{CAP}^c(t)-{q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t) $$

(32)

$$ \dot{S(t)}=\frac{\partial {H}_1(t)}{\partial {\lambda}^S(t)}={q}_i^c(t)\delta \left({q}_i^c(t)\right)R(t) $$

(33)

$$ \dot{\lambda^R(t)}=-\frac{\partial {H}_1(t)}{\partial R(t)}+r{\lambda}^R(t) $$

(34)

Substituting in for $\frac{\partial {H}_1(t)}{\partial R(t)}$, we get the following:

$$ \dot{\lambda^R(t)}=r{\lambda}^R(t)-\left[\left(p-\beta \tau \right)-\left({w}_{NR}+\tau \right){q}_i^c(t)+{w}_r{q}_i^c(t)-{\lambda}^R(t)-{\lambda}^X(t){q}_i^c(t)+{\lambda}^S(t){q}_i^c(t)\right]\delta \left({q}_i^c(t)\right) $$

(35)

The above expression suggests that the user cost for oil is a complicated expression depending on the interactions of price, tax, CO₂ injection, and the shadow values of the other constraint. Our simulations show that the user cost falls.

For our natural CO₂, its shadow value follows:

$$ \dot{\lambda^X(t)}=-\frac{\partial {H}_1(t)}{\partial X(t)}+r{\lambda}^X(t)=r{\lambda}^X(t)\overset{\mathrm{yields}}{\to }\ {\lambda}^X(t)={\lambda}^X(0){e}^{rt} $$

(36)

We will see a growth in the shadow value of our natural CO₂ stock until it reaches the switch point to captured sources. The switch point to captured sources occurs at the difference in price between both sources. As such, we will see a growth in λ^X(t) at the rate of r until it reaches a value equal to the difference in price of both captured and natural sources of CO₂ (λ^X(0)e^rT = w _CAP  − (w _NR  + τ)). After this point, λ^X(t) will remain constant as we no longer make use of natural CO₂ and make sole use of CO₂ from captured sources. Captured sources can be viewed as a backstop technology. Since our prices and costs are held constant, we will observe our shadow price rise to the value that induces a switch from one source to the next.

Similarly, the shadow price for sequestration is as follows:

$$ \dot{\lambda^S(t)}=-\frac{\partial {H}_1}{\partial S(t)}+r{\lambda}^S(t)=r{\lambda}^S(t)\overset{\mathrm{yields}}{\to}\kern0.5em {\lambda}^S(t)={\lambda}^S(0){e}^{rt} $$

(37)

We will see a growth in the shadow value of our sequestration capacity until it equals the marginal cost of sequestration in our second stage.

Modelling stage 2

$$ {T}_1\le t\le {T}_2 $$

The current value Hamiltonian for period 2 is as follows:

$$ {H}_2(t)=\tau {q}_i^c(t)-{w}_{INJ}{q}_i^c(t)-{F}_2+{\lambda}^{S2}(t){q}_i^c(t) $$

(38)

Optimality conditions for stage 2

$$ \frac{\partial {H}_2(t)}{\partial {q}_i^c(t)}=\tau -{w}_{INJ}+{\lambda}^{S2}(t)=0 $$

(39)

$$ \dot{S(t)}=\frac{\partial {H}_2(t)}{\partial {\lambda}^{S2}(t)}={q}_i^c(t) $$

(40)

$$ \dot{\lambda^{S2}(t)}=-\frac{\partial H}{\partial S(t)}+r{\lambda}^{S2}(t)=r{\lambda}^{S2}(t)\overset{\mathrm{yields}}{\to}\kern0.5em {\lambda}^{S2}(t)={\lambda}^{S2}(0){e}^{rt} $$

(41)

The marginal cost of sequestration at the beginning of our second stage is ${\lambda}^{S_2}(0){e}^{rt}={w}_{INJ}-\tau$. This value grows as pore space increases until we hit the capacity constraint of the reservoir.

The complementarity model includes Eqs. 28 and 30 and discretized versions of Eqs. 31–33 and 35–37 for stage 1 and Eq. 39 and discretized versions of Eqs. 40 and 41 for stage 2.